diff --git a/-NE0T4oBgHgl3EQfxAEX/content/tmp_files/2301.02639v1.pdf.txt b/-NE0T4oBgHgl3EQfxAEX/content/tmp_files/2301.02639v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..30f2e7eacdd8340ee6741403986bce5b14c14d66 --- /dev/null +++ b/-NE0T4oBgHgl3EQfxAEX/content/tmp_files/2301.02639v1.pdf.txt @@ -0,0 +1,948 @@ +arXiv:2301.02639v1 [math.RA] 6 Jan 2023 +Filtered skew derivations on simple artinian rings +Adam Jones, William Woods +January 9, 2023 +Abstract +Given a complete, positively filtered ring (R, f) and a compatible skew derivation (σ, δ), +we may construct its skew power series ring R[[x; σ, δ]]. Due to topological obstructions, +even if δ is an inner σ-derivation, in general we cannot “untwist” it, i.e. reparametrise to +find a filtered isomorphism R[[x; σ, δ]] ∼= R[[x′; σ]], as might be expected from the theory +of skew polynomial rings; similarly when σ is an inner automorphism. We find general +conditions under which it is possible to untwist the multiplication data, and use this to +analyse the structure of R[[x; σ, δ]] in the simplest case when R is a matrix ring over a +(noncommutative) noetherian discrete valuation ring. +Contents +1 +Introduction +2 +1.1 +Maximal orders in semisimple artinian rings . . . . . . . . . . . . . . . . . . . . +2 +1.2 +Untwisting skew derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +4 +1.3 +Ideal contraction and simplicity . . . . . . . . . . . . . . . . . . . . . . . . . . . +5 +1.4 +Uniform dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +5 +2 +Preliminaries +6 +2.1 +Filtered rings and discrete valuation rings +. . . . . . . . . . . . . . . . . . . . . +6 +2.2 +Skew derivations on semisimple artinian rings +. . . . . . . . . . . . . . . . . . . +6 +2.3 +Compatible filtrations and skew power series rings . . . . . . . . . . . . . . . . . +7 +2.4 +Discrete valuation rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +8 +3 +Reparametrising filtered skew derivations +9 +3.1 +Conditions for identical filtrations . . . . . . . . . . . . . . . . . . . . . . . . . . +9 +4 +Skew derivations on semisimple artinian rings +11 +4.1 +Reducing to orbits +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +11 +4.2 +Untwisting inner automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . +13 +4.3 +Untwisting inner σ-derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . +14 +5 +Applications +15 +5.1 +Polynomial elements +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +15 +5.2 +Uniform dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +16 +1 + +1 +Introduction +Let R be a ring and (σ, δ) a skew derivation on R: that is, σ is an automorphism of R, and δ +is a left σ-derivation on R, i.e. a linear map R → R satisfying δ(ab) = δ(a)b + σ(a)δ(b) for all +a, b ∈ R. Then we may define the skew polynomial ring R[x; σ, δ] as the unique ring which is +equal to R[x] as a left R-module and whose multiplication is given by xr = σ(r)x + δ(r) for all +r ∈ R. +In the theory of skew polynomial rings (see §1.2 below), it is well known that, if σ is an +inner automorphism of R, then there exists some x′ ∈ R[x; σ, δ] such that R[x; σ, δ] = R[x′; δ′] +for some derivation δ′; and likewise, if δ is an inner σ-derivation of R, then there exists some +x′ ∈ R[x; σ, δ] such that R[x; σ, δ] = R[x′; σ]. This is a crucial and frequently used simplification +in the theory: see e.g. [7, §2.3(iii), Theorem 14.9] or [6, Proposition 3.10]. +We are primarily interested in skew power series rings, where there are many extra topological +difficulties to deal with. +Firstly: given an arbitrary ring R and an arbitrary skew derivation (σ, δ) on R, it is in general +not true that there exists a well-defined multiplication of the above form on the left module +R[[x]] without imposing some kind of convergence condition on the multiplication data. Our +primary motivation comes from studying the completed group algebras of certain finite-rank +pro-p groups (these completed group algebras are also known as Iwasawa algebras), where the +following notions are appropriate. If (R, v) is a complete, N-filtered ring, and (σ, δ) is compatible +with v (in the sense of Definition 2.4 below), then we can define the skew power series ring +R[[x; σ, δ]] = +�� +n≥0 +rnxn : rn ∈ R +� +, +which is also a complete N-filtered ring. (If v has values in Z∪{∞} rather than N ∪{∞}, then +we can define an appropriate notion of bounded skew power series ring – see [10] for details – +but we do not deal with such rings in this paper.) +Secondly: suppose that there exists t ∈ R such that δ(r) = tr − σ(r)t, an inner σ-derivation. +Then, if we reparametrise the skew polynomial ring by changing our variable from x to x′ = x+t, +we find that R[x; σ, δ] = R[x′; σ]: in passing from x to x′, we will say that we have untwisted δ +from R[x; σ, δ]. This is beneficial as rings of the form R[x′; σ] (with zero derivation) are typically +much easier to understand. (There is a similar procedure by which inner automorphisms σ can +be untwisted.) However, under a reparametrisation like this, it is generally not true that we will +have R[[x]] = R[[x′]] even as left R-modules (see Example 3.1), meaning that this simplification +is not always available. +1.1 +Maximal orders in semisimple artinian rings +The main result of this paper uses this notion of untwisting to analyse the structure of skew +power series rings. To state this result, we need to impose further conditions on the base ring, +since general filtered rings are too pathological for us to be able to say much of consequence. +The rings of interest, which we denote by O, will typically be specific maximal orders in certain +complete, filtered semisimple artinian rings Q. These are often well-behaved enough that inner +parts of the multiplication data (σ, δ) can always be untwisted from O[[x; σ, δ]], making a study +of these skew power series rings tractable using our methods. +Rings of this form are highly abundant, since beginning with a sufficiently nice filtered ring R, +it is possible to produce such a ring O which is closely related to R due to [1, §3, Theorem C +2 + +and proof]. For instance, the authors of the present paper proved the results of [10] by relating +skew power series rings over R to skew power series rings over Q(O). +In short, we will take Q to be a semisimple artinian ring throughout, and we will assume that +it is complete with respect to a filtration vQ. Naturally, by the Artin-Wedderburn theorem, Q +is isomorphic to a finite direct product of full matrix rings over division rings F1, · · · , Fd, and +we will usually construct our maximal order O in Q by simply taking maximal orders in the Fi. +More specifically, we will assume that the data (Q, O, vQ) satisfies some or all of the following +hypotheses: +Hypotheses. +(H1) We can realise Q as a product Q = A1 × · · · × Ad, where the rings A1, . . . , Ad form the +minimal non-zero ideals of Q, O = O1 × · · · × Od for some maximal order Oi in Ai. and +for each i = 1, . . . , d, we are given +• a complete discrete valuation ring Di (defined as in §2.4 below), +• its Goldie ring of quotients Fi (with its induced filtration vFi: see §2.4), +• the full matrix rings Mn(Di) ⊆ Mn(Fi) (with the matrix filtration Mn(vFi): see +Definition 2.1.2), +• filtered ring isomorphisms ιi : Ai → Mn(Fi) such that Oi = ι−1 +i (Mn(Di)). +In this context, we will write D and F for the products of the Di and Fi respectively, and +they will be given their respective product filtrations (see Definition 2.1.1), which we will +sometimes denote vD and vF. +It will also sometimes be convenient to identify Mn(F) = Mn(F1) × · · · × Mn(Fd). We +will write ι : Q → Mn(F) for the induced filtered isomorphism, and we assume that +vQ = Mn(vF) ◦ ι. +(H2) The skew derivation (σ, δ) is compatible with vQ (defined as in §2.3 below). +(H3) The automorphism σ permutes the minimal nonzero ideals A1, . . . , Ad of Q transitively. +It will additionally be convenient to name the following hypothesis: +(S) In the context of (H1), d = 1: that is, Q is simple artinian, F is a division ring, D is a +complete discrete valuation ring, etc. +Remarks. +• Hypotheses (H1) + (S) are the context of [1, §3, particularly 3.14]: our Q is there called +Q(B), and it can be realised as a simple quotient of the artinian ring called �Q. +• It follows from (H1) that vF is the J(D)-adic filtration on F, and hence vQ is the J(O)-adic +filtration on Q. +• Hypothesis (H2) is a crucial hypothesis when working with filtered skew power series rings: +many natural and important examples of filtered skew power series rings satisfy some +kind of compatibility criterion (see e.g. [10,13,19], [20, §§2.4–2.5]), and this compatibility +criterion ensures that the ring multiplication is well defined (see §1.2). Note that, in +particular, together with (H1) it implies that σ preserves O. +• Hypothesis (H3) is a mild simplification. In fact, our results can also deal with the more +general case where Q ∼= �d +i=1 Mni(Fi) (note that the ni may be different!) and σ has +multiple orbits, by applying the techniques of §4.1 to reduce easily to a case satisfying +(H3). +Assuming these hypotheses, our main result allows us to realise the skew power series ring +O[[x; σ, δ]] in a form that allows us to reduce to the study of skew-power series rings over the +division rings Di, a much less daunting task: +3 + +Theorem A. If (Q, O, vQ) satisfy hypotheses (H1-3), with F, D, vF, ι defined as in the state- +ments of the hypotheses, then there exists a skew derivation (τ, θ) on F, compatible with vF, +and an isomorphism of filtered rings ϕ : O[[x; σ, δ]] → Mn(D[[y; τ, θ]]) extending ι|O, where y +is the image of ax − t for some a ∈ O×, t ∈ J(O). +Note that this statement makes sense, because if (τ, θ) is compatible with vF, which is the +J(D)-adic filtration, then it follows that τ and θ preserve D, and hence (τ, θ) restricts to a +compatible skew-derivation of D. +It also follows from Theorem A that the Krull dimension of O[[x; σ, δ]] is equal to the Krull +dimension of D[[y; τ, θ]], which is 2, by similar methods to those of [20, §3.1 and Theorem 3.3]. +1.2 +Untwisting skew derivations +In order to prove our main result, we first find general conditions under which inner parts of +the multiplication data (σ, δ) may be untwisted from R[[x; σ, δ]]: +Notation. Given a ring R and an invertible element a ∈ R×, we will write ca for the inner +automorphism of R defined by ca(r) = ara−1 for all r ∈ R. Also, given an element t ∈ R, we +will write dσ,t for the inner σ-derivation of R defined by dσ,t(r) = tr − σ(r)t for all r ∈ R. +Let R be a ring and (σ, δ) a skew derivation on R, and fix a ∈ R× and t ∈ R. In the study +of skew polynomial rings, it is often useful to reparametrise the ring R[x; σ, δ], i.e. replace the +variable x with a new, more convenient variable y ∈ R[x; σ, δ], usually taken to be y = ax or +y = x − t. It is easy to see that R[ax] = R[x − t] = R[x] as R-modules, and a calculation of +the multiplication data shows that +(ax)r = a(σ(r)x + δ(r)) = (aσ(r)a−1)(ax) + aδ(r) +and +(x − t)r = σ(r)x + δ(r) − tr = σ(r)(x − t) + δ(r) − (tr − σ(r)t), +from which we can conclude that +• R[x; σ, δ] = R[ax; caσ, aδ], +• R[x; σ, δ] = R[x − t; σ, δ − dσ,t] +as rings. This reparametrisation is a powerful tool in the study of skew polynomial rings, as it +effectively implies that inner automorphisms and σ-derivations can be “untwisted” to become +trivial. +In the case of filtered skew power series rings R[[x; σ, δ]], we can no longer reparametrise arbi- +trarily due to the topology: that is, given a prospective new variable y ∈ R[[x; σ, δ]], it is no +longer clear when R[[x]] = R[[y]] as modules. Our second main result gives clear and broadly +applicable sufficient conditions. +Theorem B. Let (R, v) be a complete filtered ring, and suppose that (σ, δ) is a compatible +skew derivation on R. Fix a ∈ R× and t ∈ R. +(i) If v(a) = v(a−1) = 0, then Rb[[x; σ, δ]] = Rb[[ax; caσ, caδ]] as filtered rings. +(ii) If v(t) ≥ 1, then Rb[[x; σ, δ]] = Rb[[x − t; σ, δ + dσ,t]] as filtered rings. +(By the phrase “as filtered rings” here, we mean that they are equal as rings, and that the +standard filtrations as defined in (2.1) are equal: i.e. in the notation of (2.1), we have fv,x = fv,ax +in part (i) and fv,x = fv,x−t in part (ii).) +4 + +This is proved at the end of §3. +1.3 +Ideal contraction and simplicity +In §5 we will prove some results that follow as consequences from our main theorems. The first +of these addresses the following question: when is R[[x; σ, δ]] a simple ring? +Let R ⊆ S be rings. Then we will say that an ideal I ✁ S is R-disjoint if I ∩ R = 0. +Let R be a simple ring and (σ, δ) a skew derivation on R. It is often useful to ask when R[x; σ, δ] +is also a simple ring: see e.g. [5, §3] or [18]. This is clearly equivalent to the statement that +R[x; σ, δ] has no nonzero R-disjoint ideals. However, the ideal generated by x is a nonzero +R-disjoint ideal in the case when δ = 0, and similarly – by untwisting – there exist nonzero +R-disjoint ideals more generally when δ is inner. This suggests that inner derivations have a +role to play in the simplicity of R[x; σ, δ]. +The correct generalisation of “inner” is as follows. The following are equivalent [12, Theorem +2.6, Corollary 2.7]: +(i) R[x; σ, δ] has nonzero R-disjoint ideals, +(ii) δ is a quasi-algebraic σ-derivation, i.e. there exists an endomorphism θ of R, an inner +θ-derivation D of R, and elements 0 ̸= an, an−1, . . . , a1, b ∈ R (for some n ≥ 1) such that +anδn(r) + an−1δn−1(r) + · · · + a1δ(r) = bD(r) +for all r ∈ R. (In fact, n and θ can be chosen so that θ = σn [12, §2].) +There are also equivalent conditions phrased in the language of invariant and semi-invariant +polynomials. Many further such results, and references to the historical literature on these +matters, are given in [11,12]. See [9, Theorem 3.4] or [4, §3] for examples of the usefulness of +conditions involving R-disjoint ideals. +Theorem C. If we assume that the data (Q, O, vQ) satisfies Hypotheses (H1–3) + (S), then +O[[x; σ, δ]] has no nonzero O[x; σ, δ]-disjoint ideals. It follows that if Q[x; σ, δ] is a simple ring, +then Q ⊗O O[[x; σ, δ]] is a simple ring. +1.4 +Uniform dimension +Recall that the uniform dimension (also called Goldie dimension or Goldie rank) of a right +R-module M is defined as follows. We set udim(MR) = n if and only if there are uniform +submodules U1, . . . , Un ≤ M, pairwise intersecting in zero, such that U1 ⊕ · · · ⊕ Un ≤ M is an +essential submodule [17, 2.2.9]. If R is a ring, we write r.udim(R) := udim(RR) for its (right) +uniform dimension. +Uniform dimension is preserved under skew polynomial extensions in many cases of interest. +For instance, Goodearl and Letzter showed that r.udim(R[x; σ, δ]) = r.udim(R) if R is a prime +noetherian ring [7, Lemma 1.2], and Matczuk [16] showed that this equality holds in an even +broader range of cases, including the case where R is semiprime right Goldie. +More recently, the study of uniform dimension under skew power series extensions was initi- +ated by Letzter and Wang in the paper [14]. Let S = R[[x; σ]] be a pure automorphic skew +power series extension (i.e. δ = 0): then, if R is semiprime right noetherian, we have that +r.udim(S) = r.udim(R) by [14, Theorem 2.8]. +5 + +Of course, if (R, v) is a complete positively filtered ring, (σ, δ) is a compatible skew deriva- +tion on R, and δ happens to be an inner σ-derivation of the form described in Theorem +A(ii), say δ = dσ,t for some t ∈ R satisfying v(t) ≥ 1, then it follows from Theorem A that +R[[x; σ, δ]] = R[[y; σ]] after setting y = x + t. This puts us immediately into the context of [14], +allowing us to conclude that r.udim(R[[x; σ, δ]]) = r.udim(R) in this context too. +Now assume Hypotheses (H1–3) + (S) and their notation: in particular, recall that O ∼= Mn(D). +We prove the following result only under these rather stringent restrictions, but this is (to our +knowledge) the first such result for skew power series extensions with nontrivial derivations, and +unlike the previous paragraph, it covers the case of some outer σ-derivations using methods +unlike those of [14]. We hope that, combined with the localisation process for filtered rings +outlined in [1, §3 and Theorem C], this will spark further research for more general filtered +skew power series rings. In §5.2, we prove: +Theorem D. If we assume that the data (Q, O, vQ) satisfies Hypotheses (H1-3) + (S), then +r.udim(O[[x; σ, δ]]) = r.udim(O). +2 +Preliminaries +2.1 +Filtered rings and discrete valuation rings +Our conventions for filtrations (which, in this paper, are always separated Z-filtrations) are as +follows. +A (ring) filtration on a ring R is a function f : R → Z∪{∞} satisfying the following properties +for all r, s ∈ R: +(i) f(1) = 0, +(ii) f(r + s) ≥ min{f(r), f(s)}, +(iii) f(rs) ≥ f(r) + f(s), +(iv) f(r) = ∞ if and only if r = 0. +We will say that (R, f) is a filtered ring for short. If f takes values in N ∪ {∞}, we will say +that (R, f) is N-filtered or positively filtered. +Definition 2.1. +1. If (A, fA) and (B, fB) are filtered rings, the product filtration f := fA × fB on R = A × B +is given by f(a, b) = min{fA(a), fB(b)}. +2. If (A, f) is a filtered ring and n ≥ 2 is an integer, the matrix filtration g := Mn(f) on +Mn(A) is given by g(� aijeij) = mini,j{f(aij)}, where {eij}1≤i,j≤n is the standard set of +matrix units of Mn(A). +2.2 +Skew derivations on semisimple artinian rings +Let R be a ring. The pair (σ, δ) is called a skew derivation on R if σ ∈ Aut(R) and δ is a (left) +σ-derivation of R, which means that δ is a linear map satisfying δ(rs) = δ(r)s + σ(r)δ(s) for +all r, s ∈ R. +Here are some basic properties: +6 + +Suppose that Q is a semisimple artinian ring (without topology), say Q = �d +i=1 Ai as a product +of two-sided ideals, where each Ai ∼= Mni(Fi) as rings for some positive integers ni and division +rings Fi. Suppose that (σ, δ) is a skew derivation on Q. We list some well-known facts. +Properties 2.2. +1. There exists a permutation ρ of the indices {1, . . . , d} such that σ(Ai) = Aρ(i) and +δ(Ai) ⊆ Ai + Aρ(i). Hence, if S is an orbit of ρ, then setting B := � +i∈S Ai and σ′ = σ|B, +δ′ = δ|B, we get that (σ′, δ′) is a skew derivation of B. [3, 1.1–1.3] +2. Suppose that ρ permutes {1, . . . , d} transitively. Then n1 = · · · = nd (= n, say), so that +there exists an isomorphism ι : Q → Mn(F1 × · · · × Fd). Writing F := F1 × · · · × Fd, +we can then write σ as η ◦ Mn(τ)ι, where η is an inner automorphism of Q and τ is an +automorphism of F. [3, 2.1–2.4] (Here, and elsewhere, Mn(τ)ι means ι−1Mn(τ)ι.) +3. Suppose further that η is trivial, so σ = Mn(τ)ι. Then δ = ε + Mn(θ)ι, where ε is an +inner σ-derivation of Q and θ is a τ-derivation of F. [3, 2.5] +Remark 2.3. In fact, in the context of Property 2.2.3, if d > 1 then something stronger holds: +θ can be taken to be the zero map, so that δ itself is an inner σ-derivation [3, 1.4]. However, in +the context of filtered rings, we will allow θ to be nonzero, as this extra flexibility is crucial for +ensuring that the decomposition δ = ε + Mn(θ)ι behaves well with respect to the filtration. +2.3 +Compatible filtrations and skew power series rings +Definition 2.4. Let (R, v) be a filtered ring and (σ, δ) a skew derivation on R. We will say +that (σ, δ) is (weakly) compatible with v if v(σ(r)) = v(r) and v(δ(r)) > v(r) for all 0 ̸= r ∈ R. +Remark 2.5. This is more general than the notion of “compatibility” used by the authors in [10], +which could be called strong compatibility. +Definition 2.6. Let (R, v) be a complete, positively filtered ring, i.e. R is a ring admitting a +separated discrete filtration v : R → N ∪ {∞} with respect to which R is complete. +The set R[[x]] := +� +n≥0 +Rxn, whose elements are formal sums r0 + r1x + r2x2 + . . . over arbitrary +ri ∈ R, is a left R-module. This is a complete filtered R-module with standard filtration +f := fv,x : R[[x]] → N ∪ {∞}, +given by +f +�� +i≥0 +rixi +� += inf +i≥0{v(ri) + i}, +(2.1) +which is separated and discrete. Note that R[x] is dense in R[[x]]. +A skew derivation (σ, δ) on R makes R[x] into a ring, with multiplication determined uniquely +by the rule xr = σ(r)x + δ(r). We write this ring as R[x; σ, δ]. +If (σ, δ) is compatible with v, then it induces a well-defined associative multiplication on R[[x]] +in the same way: see [10, Proposition 1.17] (cf. [19, Lemma 2.1] or [13, §3.4]), and we denote this +ring by R[[x; σ, δ]]. The function f on R[[x; σ, δ]] as defined above is a positive ring filtration, +and R[[x; σ, δ]] is complete with respect to f. +7 + +2.4 +Discrete valuation rings +Definition 2.7. Following Ardakov [1], we will say that a discrete valuation ring is a noetherian +domain D with the property that, for every nonzero x ∈ Q(D) (the division ring of quotients), +we have either x ∈ D or x−1 ∈ D. +We begin by showing that D has properties very similar to those of commutative discrete +valuation rings. +Lemma 2.8. Let D be a discrete valuation ring. +(i) D is a local ring. +(ii) All right (resp. left) ideals of D are principal. +(iii) The lattice of right (resp. left) ideals of D is totally ordered. +(iv) All right (resp. left) ideals of D are two-sided. +Proof. In the language of [15], D is a noetherian total subring of the skew field Q(D), and +statements (i–iii) follow from [15, Proposition 1.2.15]. +The proof of (iv) below is adapted from [2, Lemma 1]. We give the proof for right ideals; the +proof for left ideals is of course similar. +Suppose there exist right ideals of D that are not two-sided, and let J be the maximal such +right ideal. By (ii), J = aD for some a ∈ D: then, for some r ∈ D, we have ra =: b ̸∈ aD by +assumption. By (iii), this implies aD ⊊ bD, and so a = bs for some s ∈ D. Combining these +two equations, we can see that b = rbs. +Now, by the maximality of J, we have that Db ⊆ DbD = bD, so that rb = bt for some t ∈ D. +In particular, b = bts, and so b(1 − ts) = 0. But b cannot be zero, so as D is a domain, we +must have ts = 1, and hence (as noetherian rings are Dedekind-finite) st = 1. It follows that +at = b, contradicting the assumption that b ̸∈ aD. +Proposition 2.9. Let D be a discrete valuation ring. +(i) J(D) = πD for some normal element π. +(ii) Every nonzero ideal of D has the form πnD for some n ∈ N. +Proof. (i) is an immediate consequence of Lemma 2.8. +To show (ii): note that Lemma 2.8(iv) also implies that D is an FBN ring [17, 6.4.7], and so +�∞ +n=1 πnD = 0 by [8, Theorem 9.13]. We now argue exactly as in the commutative case: indeed, +a nonzero ideal aD must satisfy πn+1D ⊊ aD ⊆ πnD for some n by Lemma 2.8(iii), from which +it follows that a = πnu for some u ∈ D \ πD, which must be a unit by Lemma 2.8(i). +An element π as in the above proposition will be called a uniformiser of D. +In the following, D will continue to denote a complete discrete valuation ring, and we will also +set F = Q(D), O = Mn(D) and Q = Mn(F). Also write vF for the induced J(D)-adic filtration +on F, and suppose that (τ, θ) is a skew derivation on F compatible with vF; likewise write vQ +for the J(O)-adic filtration on Q, and suppose that (σ, δ) is a skew derivation on Q compatible +with vQ. This puts us essentially in the situation of Hypotheses (H1–3) + (S). The following is +now routine to check. +Corollary 2.10. +8 + +(i) Let S be the multiplicatively closed set in D generated by π. Then F = S−1D = DS−1. +Moreover, π is normal in D[[y; τ, θ]], and F⊗DD[[y; τ, θ]] = S−1D[[y; τ, θ]] = D[[y; τ, θ]]S−1. +(ii) Let S be the multiplicatively closed set in O generated by π (where we identify D with +its diagonal embedding in O). Then Q = S−1O = OS−1. Moreover, π is normal in +O[[x; σ, δ]], and Q ⊗O O[[x; σ, δ]] = S−1O[[x; σ, δ]] = O[[x; σ, δ]]S−1. +3 +Reparametrising filtered skew derivations +Throughout this section, let (R, v) be a complete, positively filtered ring. Suppose also that +R admits a skew derivation (σ, δ) which is compatible with v, and take y ∈ R[x] such that +R[x] = R[y] (an equality of left R-modules). +Given an element �m +i=0 riyi ∈ R[y], we may define the function +fv,y : +m +� +i=0 +riyi �→ inf +i≥0{v(ri) + i}. +Note that fv,x is the standard ring filtration defined in (2.1). In contrast, for arbitrary elements +y ∈ R[x], the function fv,y will not always be a ring filtration, and even when it is, it will +generally not be equivalent to fv,x. +Example 3.1. Take y := x + 1 ∈ Zp[x], and v the p-adic valuation on Zp. Then for all n, we +have fv,x((x+1)n) = 0 but fv,y((x+1)n) = n. In particular, Zp[x] = Zp[y] but Zp[[x]] ̸= Zp[[y]]. +In this section, we identify two families of elements y ∈ R[x] for which fv,x and fv,y are equal +as functions. +3.1 +Conditions for identical filtrations +With notation as above, we first show that fv,x and fv,y are equal when y = x − t for some +t ∈ R satisfying v(t) ≥ 1. +Write (x − t)n = xn + βn,1xn−1 + · · · + βn,n−1x + βn,n for all n. +Lemma 3.2. For all n, i, we have v(βn,i) ≥ i. +Proof. Firstly, note that (x − t)βn,ixn−i = σ(βn,i)xn+1−i + (δ(βn,i) − tβn,i)xn−i, so (writing +βn,0 := 1 for ease of notation) we may calculate (x − t)n+1 as +(x − t) +� n +� +i=0 +βn,ixn−i +� += +n +� +i=0 +σ(βn,i)xn+1−i + +n +� +j=0 +(δ(βn,j) − t(βn,j))xn−j += xn+1 + +n +� +i=1 +(σ(βn,i) + δ(βn,i−1) − tβn,i−1)xn+1−i + (δ(βn,n) − tβn,n), +by setting j = i + 1 in the second sum. That is, + + + + + +βn+1,0 = 1, +βn+1,i = σ(βn,i) + δ(βn,i−1) − tβn,i−1 +(1 ≤ i ≤ n), +βn+1,n+1 = δ(βn,n) − tβn,n, +from which the claim follows by induction on n. +9 + +Lemma 3.3. Let p(x) ∈ R[x] be a polynomial, and t ∈ R such that v(t) ≥ 1. +Then +fv,x(p(x − t)) ≥ fv,x(p(x)). +Proof. Write p(x) = r0 + r1x + · · · + rmxm. Then +p(x − t) = +m +� +n=0 +rn(x − t)n = +m +� +n=0 +rn +� n +� +i=0 +βn,ixn−i +� +setting βn,0 := 1 += +m +� +j=0 +� m +� +n=j +rnβn,n−j +� +xj +where j := n − i, +and so +fv,x(p(x − t)) = fv,x +� m +� +j=0 +� m +� +n=j +rnβn,n−j +� +xj +� += +inf +0≤j≤m +� +v +� m +� +n=j +rnβn,n−j +� ++ j +� +≥ +inf +0≤j≤m +inf +j≤n≤m {v(rn) + v(βn,n−j) + j} +as v is a filtration +≥ +inf +0≤j≤m +inf +j≤n≤m {v(rn) + n} +by Lemma 3.2 += +inf +0≤n≤m {v(rn) + n} += fv,x +� m +� +n=0 +rnxn +� += fv,x(p(x)). +Proposition 3.4. Let t ∈ R such that v(t) ≥ 1. Then fv,x = fv,x−t. +Proof. Take an arbitrary element p(x) ∈ R[x]. Then +fv,x(p(x)) ≤ fv,x(p(x + t)) +applying Lemma 3.3 to − t += fv,x−t(p(x)) +changing variables x �→ x − t throughout +≤ fv,x−t(p(x − t)) +applying Lemma 3.3 to t += fv,x(p(x)) +changing variables x �→ x + t throughout, +from which we can conclude that fv,x(p(x)) = fv,x−t(p(x)). +Next, we show that fv,x and fv,y are equal when y = ax where a ∈ R× and v(a) = v(a−1) = 0. +Write (ax)n = γn,0xn + γn,1xn−1 + · · · + γn,n−1x + γn,n. +Lemma 3.5. v(γn,i) ≥ i. +Proof. As in Lemma 3.2, calculating (ax)n+1 = ax(γn,0xn +γn,1xn−1 +· · ·+γn,n−1x+γn,n) gives + + + + + +γn+1,0 = aσ(γn,0), +γn+1,i = aσ(γn,i) + aδ(γn,i−1) +(1 ≤ i ≤ n), +γn+1,n+1 = aδ(γn,n), +and we may perform induction as in Lemma 3.2. +10 + +Lemma 3.6. fv,x(p(ax)) ≥ fv,x(p(x)). +Proof. +p(ax) = +m +� +n=0 +rn(ax)n = +m +� +n=0 +rn +� n +� +i=0 +γn,ixn−i +� += +m +� +j=0 +� m +� +n=j +rnγn,n−j +� +xj +where j := n − i, +and so +fv,x(p(ax)) = fv,x +� m +� +j=0 +� m +� +n=j +rnγn,n−j +� +xj +� +, +and the proof now proceeds exactly as in the proof of Lemma 3.3. +Proposition 3.7. fv,x = fv,ax. +Proof. Take an arbitrary element p(x) ∈ R[x]. Then +fv,x(p(x)) ≤ fv,x(p(a−1x)) +applying Lemma 3.6 to a−1 += fv,ax(p(x)) +changing variables x �→ ax throughout +≤ fv,ax(p(ax)) +applying Lemma 3.6 to a += fv,x(p(x)) +changing variables x �→ a−1x throughout, +from which we can conclude that fv,x(p(x)) = fv,ax(p(x)). +Finally, we fix any y ∈ R[x] such that R[x] = R[y] and fv,x = fv,y. Denote this common +filtration by f. It follows that: +Theorem 3.8. R[[x]] = R[[y]] as filtered left R-modules. +Proof of Theorem B. In case (i), set y = ax, so that f := fv,x = fv,y by Proposition 3.7; in +case (ii), set y = x − t, and use Proposition 3.4. In both cases, R[x] = R[y]. Now Theorem 3.8 +implies that R[[y]] and R[[x]] can be identified as filtered modules, and the multiplication data +has already been calculated in §1.2, so the conclusion follows. +4 +Skew derivations on semisimple artinian rings +4.1 +Reducing to orbits +For now, we do not assume any of the Hypotheses (H1–3), and we let (R, v) be an arbitrary +filtered ring such that R admits a decomposition R ∼= B × C (as unfiltered rings). +We will abuse notation and write R = B × C (as unfiltered rings). We will also write B (resp. +C) for the ideal B × 0 (resp. 0 × C) of R, so that there are inclusion maps jB : B → R and +jC : C → R and projection maps πB : R → B and πC : R → C. +Suppose further that the filtration on R is complete and positive, and that R admits a skew +derivation (σ, δ) which restricts to skew derivations on B and C. (That is, setting σB = πBσjB +and δB = πBδjB, we have that (σB, δB) is a skew derivation on B, and likewise for C.) +11 + +Write vB and vC for the restrictions of v to B and C respectively. In general, even if (σ, δ) is +compatible with v, it may not be true that (σB, δB) is compatible with vB, and so we must +restrict to the case in which the decomposition R ∼= B × C and the filtration v interact nicely. +The following lemma is an immediate consequence of the definition of the product filtration, +as in Definition 2.1. +Lemma 4.1. If v = vB × vC, then (σB, δB) is compatible with vB, and (σC, δC) is compatible +with vC. +Hence, under the assumption v = vB × vC, we may define the filtered rings +• B[[xB; σB, δB]], with filtration fB, satisfying fB|B = vB and fB(xB) = 1, +• C[[xC; σC, δC]], with filtration fC, satisfying fC|C = vC and fC(xC) = 1, +as in (2.1). +Proposition 4.2. Suppose that v = vB × vC. Then there is an isomorphism of filtered rings +ϕ : R[[x; σ, δ]] → B[[xB; σB, δB]] × C[[xC; σC, δC]]. +Proof. It is straightforward to check that the maps +ϕ : R[[x; σ, δ]] → B[[xB; σB, δB]] × C[[xC; σC, δC]] +� +i≥0 +rixi �→ +�� +i≥0 +πB(ri)xi +B, +� +i≥0 +πC(ri)xi +C +� +and +θ : B[[xB; σB, δB]] × C[[xC; σC, δC]] → R[[x; σ, δ]] +�� +i≥0 +bixi +B, +� +i≥0 +cixi +C +� +�→ +� +i≥0 +(jB(bi), jC(ci))xi +are the mutually inverse filtered isomorphisms as required. +Let Q′ be an arbitrary semisimple artinian filtered ring admitting a skew derivation (σ, δ), and +write the minimal nonzero ideals of Q′ as A1, . . . , Ae, so that Q′ = �e +i=1 Ai. In the case where +the Ai fall into several σ-orbits, write ρ for the permutation of the indexing set {1, . . . , e} +induced on the set {A1, . . . , Ae} by σ as in Property 2.2.1. Let S be a union of (some) orbits of +ρ and S′ = {1, . . . , e}\S, and suppose (to avoid trivial cases) that both S and S′ are nonempty. +Now set B′ = � +i∈S Ai and C′ = � +i∈S′ Ai: it follows from Property 2.2.1 that (σB′, δB′) restricts +to a skew derivation on B′, and likewise for C′. +Assume now that each Ai ∼= Mni(Fi), where ni ≥ 1 is some positive integer and Fi is the Goldie +ring of quotients of a complete discrete valuation ring Di. Set Oi to be the preimage in Ai of +Mni(Di), so that O′ = O1 × · · · × Oe is a maximal order in Q′. Moreover, if j = ρ(i) then +Aj = σ(Ai) so Mni(Fi) ∼= Mnj(Fj), which implies that ni = nj and Fi ∼= Fj. +Suppose that all of these rings are given their natural filtrations: that is, +• each Di retains its discrete valuation, and Fi inherits the J(Di)-adic valuation (see §2.4), +• Mni(Di) and Mni(Fi) are given the corresponding matrix filtrations (see Definition 2.1.2), +• Oi and Ai inherit their filtrations from Mni(Di) and Mni(Fi) under the above isomor- +phisms, and +12 + +• O′ and Q′ are given the product filtrations (see Definition 2.1.1) from the Oi and Ai +respectively. +Then O′, B′ and C′ as defined above will satisfy the hypotheses of Proposition 4.2. Moreover, if +S is taken to be a single orbit of ρ with |S| = d, then (after renumbering so that S = {1, . . . , d} +and writing n = n1 = · · · = nd) the ring O := O′ ∩ B′ as defined above, its Goldie ring of +quotients Q := B′, etc. will satisfy Hypotheses (H1–3). +4.2 +Untwisting inner automorphisms +In this subsection, we assume the full force of Hypotheses (H1–3) and adopt their notation. +Without loss of generality, reordering the Ai if necessary, we will set σ(Ai) = Ai+1 for 1 ≤ i ≤ d−1 +and σ(Ad) = A1. +We may now invoke Property 2.2.2. In particular, there is a decomposition σ = η ◦ Mn(τ)ι, +where η is an inner automorphism of Q, say η = ca for some a ∈ Q×, and τ is an automorphism +of F. +Lemma 4.3. Both η and Mn(τ)ι preserve O. +Proof. Since η is an inner automorphism of Q, it will preserve each Ai. But σ(Ai) = Ai+1 +(with indices interpreted modulo d), so Mn(τ)ι must send Ai to Ai+1, and hence τ sends +Fi to Fi+1. However, since σ and η are continuous, it follows that τ is continuous, and so +Di+1 = Oi+1 ∩ Fi+1 = τ(Di), hence Mn(τ)ι preserves O. Now it follows that η = σ ◦ Mn(τ −1)ι +preserves O. +Our aim in this subsection is to “untwist” η by making a change of variables x �→ x′, i.e. find +an element x′ ∈ O[[x; σ, δ]] such that +O[[x; σ, δ]] = O[[x′; Mn(τ)ι, δ′]]. +By Theorem B(i), it would suffice if v(a) = v(a−1) = 0: this would imply that a, a−1 ∈ O, and +we could then set x′ = a−1x, giving δ′ = a−1δ as in §1.2. +Of course, in general, a will not necessarily have this property: for instance, if O is a complete +discrete valuation ring with central uniformiser π, then ca = cπra for all r ∈ Z, and v(πra) will +usually not be zero. Surprisingly, this naive obstruction is the only kind of obstruction that +occurs. +Proposition 4.4. There exist an element b ∈ Q×, an inner automorphism η′ = cb of Q, and +an automorphism τ ′ of F such that σ = η′ ◦ Mn(τ ′)ι and v(b) = v(b−1) = 0. +Proof. Suppose a = (a1, . . . , ad) ∈ Q×: then, by Lemma 4.3, we have aiOia−1 +i += Oi for each +i. Let ki = v(ai), i.e. ai ∈ J(Oi)ki \ J(Oi)ki+1. So, if πj is a uniformiser of Dj, Ij is the +identity matrix of Mn(Dj), and ˜πj := ι−1(πjIj), then we have aj = bj˜π +kj +j for some bj ∈ Oj with +v(bj) = 0. Set b = (b1, . . . , bd), so that v(b) = 0. +By Corollary 2.10(ii), ˜πj is normal in Oj, so the right ideal bjOj is in fact a two-sided ideal. +Moreover, by Proposition 2.9(ii) and Morita equivalence, as the ideal bjOj contains the element +bj of value 0, it must be equal to Oj. Hence bj is a unit in Oj, and we have v(bj) = v(b−1 +j ) = 0. +Now set η′(r) = brb−1 for all r ∈ Q and τ ′(s) = Πτ(s)Π−1, where Π = (πk1 +1 I1, . . . , πkd +d Id). The +claim now follows from a short calculation. +13 + +4.3 +Untwisting inner σ-derivations +In this subsection, let (R, f) be an arbitrary Z-filtered ring admitting a compatible skew deriva- +tion (σ, δ), and write the f-level sets of R as FnR for n ∈ Z. We will also suppose that +• R = Mn(A) for some ring A, +• f = Mn(g) for some filtration g on A, +• σ = Mn(τ) for some automorphism τ of A, and +• δ = Mn(θ) + ε, where θ is a τ-derivation of A and ε is an inner σ-derivation of R, say +ε = dσ,u for some u ∈ R. +(Compare Property 2.2.3.) We will write the standard set of matrix units in R as {eij}1≤i,j≤n. +Our aim in this subsection is to “untwist” ε by making a change of variables x �→ x′, i.e. find +an element x′ ∈ R[[x; Mn(τ), δ]] such that +R≥0[[x; Mn(τ), δ]] = R≥0[[x′; Mn(τ), Mn(θ)]] +where of course, R≥0 is the positively filtered subring of R. As before, we would be done by +Theorem B(ii) if we had f(u) ≥ 1. This is also unreasonable to expect, albeit this time for +slightly less obvious reasons: for instance, if A is the division ring of fractions of a complete +discrete valuation ring with uniformiser π, then Mn(θ)+dMn(τ),u = Mn(θ+dτ,πr)+dMn(τ),u−πrI +for all r ∈ Z, and v(u − πrI) can be less than 1. However, again, under mild conditions this is +the only obstruction that occurs. +Proposition 4.5. In the above setup, there exist an element u′ ∈ R, a τ-derivation θ′ of A, +and an inner σ-derivation ε′ = inn(u′) of R such that δ = Mn(θ′) + ε′ and f(u′) ≥ 1. +Proof. Write u = � +i,j uijeij for some coefficients uij ∈ A. Let 1 ≤ p, q ≤ n be arbitrary, and +consider the matrix unit epq ∈ R. By assumption, δ(epq) ∈ F1R, and so +ε(epq) ≡ −Mn(θ)(epq) +mod F1R. +(4.1) +But we can calculate the left-hand side of this congruence (4.1) explicitly as +ε(epq) = +� +i,j +(uijeijepq − epquijeij) = +� +i +uipeiq − +� +j +uqjepj, +and the right-hand side of (4.1) is just −θ(1A)epq, which is zero. So we may rewrite (4.1) as +� +i uipeiq − � +j uqjepj ≡ 0 mod F1R, and equate corresponding entries, to get + + + + + +uip ∈ F1A +i ̸= p +uqj ∈ F1A +j ̸= q +upp − uqq ∈ F1A, +and so, as p and q were arbitrary, we get u ≡ u111R mod F1R. +Now setting u′ := u − u111R, and defining θ′(a) := θ(a) + u11a − τ(a)u11 and ε′ := dMn(τ),u′, we +are done. +Upshot: in this case, using Theorem B(i) we can pass to the case when δ = Mn(θ). +Proof of Theorem A. Firstly, Proposition 4.4 shows that there exist some τ ∈ Aut(F) and some +b ∈ Q× satisfying v(b) = v(b−1) = 0 such that σ = cb ◦ Mn(τ)ι, and both of these preserve O +14 + +by the same argument as in Lemma 4.3. So by Theorem B(i), we may set x′ = b−1x to get +O[[x; σ, δ]] = O[[x′; Mn(τ)ι, δ′]] for the Mn(τ)ι-derivation δ′ := b−1δ. +Secondly, Proposition 4.5 shows that there exist some τ-derivation θ of F and some u ∈ Q +satisfying v(u) ≥ 1 such that δ′ = Mn(θ)ι + dMn(τ)ι,u. +So by Theorem B(ii), we may set +x′′ = x′ − u to get O[[x′; Mn(τ)ι, δ′]] = O[[x′′; Mn(τ)ι, Mn(θ)ι]]. +Finally, the maps +O[[x′′; Mn(τ)ι, Mn(θ)ι]] → Mn(D)[[y; Mn(τ), Mn(θ)]] +� +i≥0 +qi(x′′)i �→ +� +i≥0 +ι(qi)yi +and +Mn(D)[[y; Mn(τ), Mn(θ)]] → Mn(D[[z; τ, θ]]) +� +i≥0 +� +n +� +j,k=1 +cijkejk +� +yi �→ +n +� +j,k=1 +�� +i≥0 +cijkzi +� +ejk +can now be checked to be filtered ring isomorphisms, and vO = Mn(vD) ◦ ι. +5 +Applications +Throughout this section, we assume our data (Q, O, vQ) satisfies Hypotheses (H1–3) + (S). +5.1 +Polynomial elements +Definition 5.1. A polynomial element of R[[x]] (resp. R[[x; σ, δ]]) is an element of R[x] (resp. +R[x; σ, δ]). +We first recall the following important generalisation of the Weierstrass preparation theorem +for skew power series rings, essentially due to Venjakob. +Theorem 5.2. Every nonzero right ideal of D[[y; τ, θ]] contains a nonzero polynomial element. +Proof. Let v be the J(D)-adic filtration and π a uniformiser for D. Then every nonzero ele- +ment r ∈ D[[y; τ, θ]] can be written as r = (s0 + s1y + s2y2 + . . . )πm, where all si ∈ D and +infi≥0{v(si)} = 0, by Corollary 2.10(i). Hence s = s0 + s1y + s2y2 + . . . satisfies the hypotheses +of [19, Theorem 3.1], and so a right-hand version of [19, Corollary 3.2] tells us that s can be +expressed uniquely as s = Pu, where u ∈ D[[y; τ, θ]]× and P ∈ D[y; τ, θ]. +In particular, if A is a nonzero right ideal of D[[y; τ, θ]], then given any nonzero r ∈ A, we can +write it as r = Puπm as above. Since π is normal in D[[y; τ, θ]] by Corollary 2.10(i), this is just +r = Pπmu′ for some unit u′ ∈ D[[y; τ, θ]], and hence the polynomial element Pπm is also an +element of A. +Remark. Corollary 2.10(i) also implies a similar result for F ⊗D D[[y; τ, θ]]. +Corollary 5.3. Every nonzero (two-sided) ideal of O[[x; σ, δ]] contains a nonzero polynomial +element. +15 + +Proof. Theorem A tells us that there exists an isomorphism ϕ : O[[x; σ, δ]] → Mn(D[[y; τ, θ]]) +extending ι, such that y = ϕ(ax − t). In particular, if P ∈ D[[y; τ, θ]] is polynomial in the +variable y, then ϕ−1(PI) (where I is the identity matrix) is polynomial in the variable x. The +result now follows from Theorem 5.2 and by Morita equivalence. +Proof of Theorem C. The first statement is simply restating Corollary 5.3. +For the second statement, take a nonzero ideal I of Q ⊗O O[[x; σ, δ]]. Then J := I ∩ O[[x; σ, δ]] +is clearly an ideal of O[[x; σ, δ]]. By Corollary 2.10(ii), multiplying any nonzero element of I by +an appropriate power of the regular element π will give an element of J, so J ̸= 0. +Hence, by the first statement, we know that J ∩ O[x; σ, δ] ̸= 0, and so I ∩ Q[x; σ, δ] ̸= 0. But +as we are assuming that Q[x; σ, δ] is simple, it must follow that I ∩ Q[x; σ, δ] = Q[x; σ, δ]. In +particular, 1 ∈ I, and so I = Q ⊗O O[[x; σ, δ]]. +5.2 +Uniform dimension +This subsection continues the work of [14] on uniform dimensions (Goldie ranks) of skew power +series rings. As we have already remarked in the Introduction, Theorem B sometimes allows +us to reduce directly to the results of [14] when the derivation is inner. However, in the special +case of Hypotheses (H1–3) + (S), we can now prove similar results about skew power series +rings over O for arbitrary derivations. +Proof of Theorem D. By Theorem A, we know that O[[x; σ, δ]] ∼= Mn(D[[y; τ, θ]]) for some ap- +propriate skew derivation (τ, θ), and hence that r.udim(O[[x; σ, δ]]) = n(r.udim(D[[y; τ, θ]])) [17, +Example 2.11(iii)]. In the same way, r.udim(O) = n(r.udim(D)), and of course r.udim(D) = 1 +as D is a noetherian integral domain [17, Example 2.11(i)]. +It remains to show that r.udim(D[[y; τ, θ]]) = 1. So let U be a uniform right ideal of D[[y; τ, θ]], +and let I be an arbitrary nonzero right ideal. +By Theorem 5.2, U ∩ D[y; τ, θ] ̸= 0 and +I ∩ D[y; τ, θ] ̸= 0, and so as D[y; τ, θ] is a prime ring [17, Theorem 1.2.9(iii)], their inter- +section (U ∩ I) ∩ D[y; τ, θ] is also nonzero. In particular, this shows that U is an essential right +ideal of D[[y; τ, θ]], and so r.udim(D[[y; τ, θ]]) = 1. +References +[1] K. Ardakov. +Prime ideals in nilpotent Iwasawa algebras. +Inventiones mathematicae, +190(2):439–503, 2012. +[2] H.-H. Brungs. Generalized discrete valuation rings. Canad. J. 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Skew power series rings over a prime base ring (preprint). +https://arxiv.org/abs/2112.10242. +[11] T. Y. Lam, K. H. Leung, A. Leroy, and J. Matczuk. Invariant and semi-invariant poly- +nomial rings. In L. Rowen, editor, Ring Theory, pages 247–261. Weizmann Science Press, +1989. +[12] Andr´e Leroy and Jerzy Matczuk. The extended centroid and X-inner automorphisms of +Ore extensions. Journal of Algebra, 145:143–177, 1992. +[13] Edward S. Letzter. Prime ideals of noetherian skew power series rings. Israel J. Math., +192:67–81, 2012. +[14] Edward S. Letzter and Linhong Wang. Goldie ranks of skew power series rings of auto- +morphic type. arXiv:0812.2010v3, 2011. +[15] Hidetoshi Marubayashi and Freddy van Oystaeyen. Prime Divisors and Noncommutative +Valuation Theory. Lecture Notes in Mathematics, 2059. Springer, 2012. +[16] Jerzy Matczuk. Goldie rank of Ore extensions. Comm. Alg., 23(4):1455–1471, 1995. +[17] J.C. McConnell and J.C. Robson. Noncommutative Noetherian Rings. American Mathe- +matical Society, 2001. +[18] Johan ¨Oinert, Johan Richter, and Sergei D. Silvestrov. Maximal commutative subrings +and simplicity of Ore extensions. J. Algebra Appl., 12(4), 2013. +[19] Otmar Venjakob. A noncommutative Weierstrass preparation theorem and applications +to Iwasawa theory. J. Reine Angew. Math., 559:153–191, 2003. +[20] William Woods. +Dimension theory in iterated local skew power series rings. +Algebr. +Represent. Theory. Published online: https://doi.org/10.1007/s10468-022-10144-3, +2022. +17 + diff --git a/-NE0T4oBgHgl3EQfxAEX/content/tmp_files/load_file.txt b/-NE0T4oBgHgl3EQfxAEX/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..aba8bc5938c4b1b92dc56135265dc77be08854fd --- /dev/null +++ b/-NE0T4oBgHgl3EQfxAEX/content/tmp_files/load_file.txt @@ -0,0 +1,1247 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf,len=1246 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='02639v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='RA] 6 Jan 2023 Filtered skew derivations on simple artinian rings Adam Jones, William Woods January 9, 2023 Abstract Given a complete, positively filtered ring (R, f) and a compatible skew derivation (σ, δ), we may construct its skew power series ring R[[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Due to topological obstructions, even if δ is an inner σ-derivation, in general we cannot “untwist” it, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' reparametrise to find a filtered isomorphism R[[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ]] ∼= R[[x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ]], as might be expected from the theory of skew polynomial rings;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' similarly when σ is an inner automorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' We find general conditions under which it is possible to untwist the multiplication data, and use this to analyse the structure of R[[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ]] in the simplest case when R is a matrix ring over a (noncommutative) noetherian discrete valuation ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Contents 1 Introduction 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='1 Maximal orders in semisimple artinian rings .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='2 Untwisting skew derivations .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' 14 5 Applications 15 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='1 Polynomial elements .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' 15 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='2 Uniform dimension .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' 16 1 1 Introduction Let R be a ring and (σ, δ) a skew derivation on R: that is, σ is an automorphism of R, and δ is a left σ-derivation on R, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' a linear map R → R satisfying δ(ab) = δ(a)b + σ(a)δ(b) for all a, b ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Then we may define the skew polynomial ring R[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ] as the unique ring which is equal to R[x] as a left R-module and whose multiplication is given by xr = σ(r)x + δ(r) for all r ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' In the theory of skew polynomial rings (see §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='2 below), it is well known that, if σ is an inner automorphism of R, then there exists some x′ ∈ R[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ] such that R[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ] = R[x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' δ′] for some derivation δ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' and likewise, if δ is an inner σ-derivation of R, then there exists some x′ ∈ R[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ] such that R[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ] = R[x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' This is a crucial and frequently used simplification in the theory: see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' [7, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='3(iii), Theorem 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='9] or [6, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' We are primarily interested in skew power series rings, where there are many extra topological difficulties to deal with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Firstly: given an arbitrary ring R and an arbitrary skew derivation (σ, δ) on R, it is in general not true that there exists a well-defined multiplication of the above form on the left module R[[x]] without imposing some kind of convergence condition on the multiplication data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Our primary motivation comes from studying the completed group algebras of certain finite-rank pro-p groups (these completed group algebras are also known as Iwasawa algebras), where the following notions are appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' If (R, v) is a complete, N-filtered ring, and (σ, δ) is compatible with v (in the sense of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='4 below), then we can define the skew power series ring R[[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ]] = �� n≥0 rnxn : rn ∈ R � , which is also a complete N-filtered ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' (If v has values in Z∪{∞} rather than N ∪{∞}, then we can define an appropriate notion of bounded skew power series ring – see [10] for details – but we do not deal with such rings in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=') Secondly: suppose that there exists t ∈ R such that δ(r) = tr − σ(r)t, an inner σ-derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Then, if we reparametrise the skew polynomial ring by changing our variable from x to x′ = x+t, we find that R[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ] = R[x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ]: in passing from x to x′, we will say that we have untwisted δ from R[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' This is beneficial as rings of the form R[x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ] (with zero derivation) are typically much easier to understand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' (There is a similar procedure by which inner automorphisms σ can be untwisted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=') However, under a reparametrisation like this, it is generally not true that we will have R[[x]] = R[[x′]] even as left R-modules (see Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='1), meaning that this simplification is not always available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='1 Maximal orders in semisimple artinian rings The main result of this paper uses this notion of untwisting to analyse the structure of skew power series rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' To state this result, we need to impose further conditions on the base ring, since general filtered rings are too pathological for us to be able to say much of consequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' The rings of interest, which we denote by O, will typically be specific maximal orders in certain complete, filtered semisimple artinian rings Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' These are often well-behaved enough that inner parts of the multiplication data (σ, δ) can always be untwisted from O[[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ]], making a study of these skew power series rings tractable using our methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Rings of this form are highly abundant, since beginning with a sufficiently nice filtered ring R, it is possible to produce such a ring O which is closely related to R due to [1, §3, Theorem C 2 and proof].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' For instance, the authors of the present paper proved the results of [10] by relating skew power series rings over R to skew power series rings over Q(O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' In short, we will take Q to be a semisimple artinian ring throughout, and we will assume that it is complete with respect to a filtration vQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Naturally, by the Artin-Wedderburn theorem, Q is isomorphic to a finite direct product of full matrix rings over division rings F1, · · · , Fd, and we will usually construct our maximal order O in Q by simply taking maximal orders in the Fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' More specifically, we will assume that the data (Q, O, vQ) satisfies some or all of the following hypotheses: Hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' (H1) We can realise Q as a product Q = A1 × · · · × Ad, where the rings A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' , Ad form the minimal non-zero ideals of Q, O = O1 × · · · × Od for some maximal order Oi in Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' and for each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' , d, we are given a complete discrete valuation ring Di (defined as in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='4 below), its Goldie ring of quotients Fi (with its induced filtration vFi: see §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='4), the full matrix rings Mn(Di) ⊆ Mn(Fi) (with the matrix filtration Mn(vFi): see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='2), filtered ring isomorphisms ιi : Ai → Mn(Fi) such that Oi = ι−1 i (Mn(Di)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' In this context, we will write D and F for the products of the Di and Fi respectively, and they will be given their respective product filtrations (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='1), which we will sometimes denote vD and vF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' It will also sometimes be convenient to identify Mn(F) = Mn(F1) × · · · × Mn(Fd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' We will write ι : Q → Mn(F) for the induced filtered isomorphism, and we assume that vQ = Mn(vF) ◦ ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' (H2) The skew derivation (σ, δ) is compatible with vQ (defined as in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='3 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' (H3) The automorphism σ permutes the minimal nonzero ideals A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' , Ad of Q transitively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' It will additionally be convenient to name the following hypothesis: (S) In the context of (H1), d = 1: that is, Q is simple artinian, F is a division ring, D is a complete discrete valuation ring, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Hypotheses (H1) + (S) are the context of [1, §3, particularly 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='14]: our Q is there called Q(B), and it can be realised as a simple quotient of the artinian ring called �Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' It follows from (H1) that vF is the J(D)-adic filtration on F, and hence vQ is the J(O)-adic filtration on Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Hypothesis (H2) is a crucial hypothesis when working with filtered skew power series rings: many natural and important examples of filtered skew power series rings satisfy some kind of compatibility criterion (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' [10,13,19], [20, §§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='4–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='5]), and this compatibility criterion ensures that the ring multiplication is well defined (see §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Note that, in particular, together with (H1) it implies that σ preserves O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Hypothesis (H3) is a mild simplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' In fact, our results can also deal with the more general case where Q ∼= �d i=1 Mni(Fi) (note that the ni may be different!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=') and σ has multiple orbits, by applying the techniques of §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='1 to reduce easily to a case satisfying (H3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Assuming these hypotheses, our main result allows us to realise the skew power series ring O[[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ]] in a form that allows us to reduce to the study of skew-power series rings over the division rings Di, a much less daunting task: 3 Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' If (Q, O, vQ) satisfy hypotheses (H1-3), with F, D, vF, ι defined as in the state- ments of the hypotheses, then there exists a skew derivation (τ, θ) on F, compatible with vF, and an isomorphism of filtered rings ϕ : O[[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ]] → Mn(D[[y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' τ, θ]]) extending ι|O, where y is the image of ax − t for some a ∈ O×, t ∈ J(O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Note that this statement makes sense, because if (τ, θ) is compatible with vF, which is the J(D)-adic filtration, then it follows that τ and θ preserve D, and hence (τ, θ) restricts to a compatible skew-derivation of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' It also follows from Theorem A that the Krull dimension of O[[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ]] is equal to the Krull dimension of D[[y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' τ, θ]], which is 2, by similar methods to those of [20, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='1 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='2 Untwisting skew derivations In order to prove our main result, we first find general conditions under which inner parts of the multiplication data (σ, δ) may be untwisted from R[[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ]]: Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Given a ring R and an invertible element a ∈ R×, we will write ca for the inner automorphism of R defined by ca(r) = ara−1 for all r ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Also, given an element t ∈ R, we will write dσ,t for the inner σ-derivation of R defined by dσ,t(r) = tr − σ(r)t for all r ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Let R be a ring and (σ, δ) a skew derivation on R, and fix a ∈ R× and t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' In the study of skew polynomial rings, it is often useful to reparametrise the ring R[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' replace the variable x with a new, more convenient variable y ∈ R[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ], usually taken to be y = ax or y = x − t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' It is easy to see that R[ax] = R[x − t] = R[x] as R-modules, and a calculation of the multiplication data shows that (ax)r = a(σ(r)x + δ(r)) = (aσ(r)a−1)(ax) + aδ(r) and (x − t)r = σ(r)x + δ(r) − tr = σ(r)(x − t) + δ(r) − (tr − σ(r)t), from which we can conclude that R[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ] = R[ax;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' caσ, aδ], R[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ] = R[x − t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ − dσ,t] as rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' This reparametrisation is a powerful tool in the study of skew polynomial rings, as it effectively implies that inner automorphisms and σ-derivations can be “untwisted” to become trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' In the case of filtered skew power series rings R[[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ]], we can no longer reparametrise arbi- trarily due to the topology: that is, given a prospective new variable y ∈ R[[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ]], it is no longer clear when R[[x]] = R[[y]] as modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Our second main result gives clear and broadly applicable sufficient conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Let (R, v) be a complete filtered ring, and suppose that (σ, δ) is a compatible skew derivation on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Fix a ∈ R× and t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' (i) If v(a) = v(a−1) = 0, then Rb[[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ]] = Rb[[ax;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' caσ, caδ]] as filtered rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' (ii) If v(t) ≥ 1, then Rb[[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ]] = Rb[[x − t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ + dσ,t]] as filtered rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' (By the phrase “as filtered rings” here, we mean that they are equal as rings, and that the standard filtrations as defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='1) are equal: i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' in the notation of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='1), we have fv,x = fv,ax in part (i) and fv,x = fv,x−t in part (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=') 4 This is proved at the end of §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='3 Ideal contraction and simplicity In §5 we will prove some results that follow as consequences from our main theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' The first of these addresses the following question: when is R[[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ]] a simple ring?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Let R ⊆ S be rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Then we will say that an ideal I ✁ S is R-disjoint if I ∩ R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Let R be a simple ring and (σ, δ) a skew derivation on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' It is often useful to ask when R[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ] is also a simple ring: see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' [5, §3] or [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' This is clearly equivalent to the statement that R[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ] has no nonzero R-disjoint ideals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' However, the ideal generated by x is a nonzero R-disjoint ideal in the case when δ = 0, and similarly – by untwisting – there exist nonzero R-disjoint ideals more generally when δ is inner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' This suggests that inner derivations have a role to play in the simplicity of R[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' The correct generalisation of “inner” is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' The following are equivalent [12, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='6, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='7]: (i) R[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ] has nonzero R-disjoint ideals, (ii) δ is a quasi-algebraic σ-derivation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' there exists an endomorphism θ of R, an inner θ-derivation D of R, and elements 0 ̸= an, an−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' , a1, b ∈ R (for some n ≥ 1) such that anδn(r) + an−1δn−1(r) + · · · + a1δ(r) = bD(r) for all r ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' (In fact, n and θ can be chosen so that θ = σn [12, §2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=') There are also equivalent conditions phrased in the language of invariant and semi-invariant polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Many further such results, and references to the historical literature on these matters, are given in [11,12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' See [9, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='4] or [4, §3] for examples of the usefulness of conditions involving R-disjoint ideals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' If we assume that the data (Q, O, vQ) satisfies Hypotheses (H1–3) + (S), then O[[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ]] has no nonzero O[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ]-disjoint ideals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' It follows that if Q[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ] is a simple ring, then Q ⊗O O[[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ]] is a simple ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='4 Uniform dimension Recall that the uniform dimension (also called Goldie dimension or Goldie rank) of a right R-module M is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' We set udim(MR) = n if and only if there are uniform submodules U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' , Un ≤ M, pairwise intersecting in zero, such that U1 ⊕ · · · ⊕ Un ≤ M is an essential submodule [17, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' If R is a ring, we write r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='udim(R) := udim(RR) for its (right) uniform dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Uniform dimension is preserved under skew polynomial extensions in many cases of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' For instance, Goodearl and Letzter showed that r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='udim(R[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ]) = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='udim(R) if R is a prime noetherian ring [7, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='2], and Matczuk [16] showed that this equality holds in an even broader range of cases, including the case where R is semiprime right Goldie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' More recently, the study of uniform dimension under skew power series extensions was initi- ated by Letzter and Wang in the paper [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Let S = R[[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ]] be a pure automorphic skew power series extension (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' δ = 0): then, if R is semiprime right noetherian, we have that r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='udim(S) = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='udim(R) by [14, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' 5 Of course, if (R, v) is a complete positively filtered ring, (σ, δ) is a compatible skew deriva- tion on R, and δ happens to be an inner σ-derivation of the form described in Theorem A(ii), say δ = dσ,t for some t ∈ R satisfying v(t) ≥ 1, then it follows from Theorem A that R[[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ]] = R[[y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ]] after setting y = x + t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' This puts us immediately into the context of [14], allowing us to conclude that r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='udim(R[[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ]]) = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='udim(R) in this context too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Now assume Hypotheses (H1–3) + (S) and their notation: in particular, recall that O ∼= Mn(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' We prove the following result only under these rather stringent restrictions, but this is (to our knowledge) the first such result for skew power series extensions with nontrivial derivations, and unlike the previous paragraph, it covers the case of some outer σ-derivations using methods unlike those of [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' We hope that, combined with the localisation process for filtered rings outlined in [1, §3 and Theorem C], this will spark further research for more general filtered skew power series rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' In §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='2, we prove: Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' If we assume that the data (Q, O, vQ) satisfies Hypotheses (H1-3) + (S), then r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='udim(O[[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ]]) = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='udim(O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' 2 Preliminaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='1 Filtered rings and discrete valuation rings Our conventions for filtrations (which, in this paper, are always separated Z-filtrations) are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' A (ring) filtration on a ring R is a function f : R → Z∪{∞} satisfying the following properties for all r, s ∈ R: (i) f(1) = 0, (ii) f(r + s) ≥ min{f(r), f(s)}, (iii) f(rs) ≥ f(r) + f(s), (iv) f(r) = ∞ if and only if r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' We will say that (R, f) is a filtered ring for short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' If f takes values in N ∪ {∞}, we will say that (R, f) is N-filtered or positively filtered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' If (A, fA) and (B, fB) are filtered rings, the product filtration f := fA × fB on R = A × B is given by f(a, b) = min{fA(a), fB(b)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' If (A, f) is a filtered ring and n ≥ 2 is an integer, the matrix filtration g := Mn(f) on Mn(A) is given by g(� aijeij) = mini,j{f(aij)}, where {eij}1≤i,j≤n is the standard set of matrix units of Mn(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='2 Skew derivations on semisimple artinian rings Let R be a ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' The pair (σ, δ) is called a skew derivation on R if σ ∈ Aut(R) and δ is a (left) σ-derivation of R, which means that δ is a linear map satisfying δ(rs) = δ(r)s + σ(r)δ(s) for all r, s ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Here are some basic properties: 6 Suppose that Q is a semisimple artinian ring (without topology), say Q = �d i=1 Ai as a product of two-sided ideals, where each Ai ∼= Mni(Fi) as rings for some positive integers ni and division rings Fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Suppose that (σ, δ) is a skew derivation on Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' We list some well-known facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Properties 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' There exists a permutation ρ of the indices {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' , d} such that σ(Ai) = Aρ(i) and δ(Ai) ⊆ Ai + Aρ(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Hence, if S is an orbit of ρ, then setting B := � i∈S Ai and σ′ = σ|B, δ′ = δ|B, we get that (σ′, δ′) is a skew derivation of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' [3, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='1–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='3] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Suppose that ρ permutes {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' , d} transitively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Then n1 = · · · = nd (= n, say), so that there exists an isomorphism ι : Q → Mn(F1 × · · · × Fd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Writing F := F1 × · · · × Fd, we can then write σ as η ◦ Mn(τ)ι, where η is an inner automorphism of Q and τ is an automorphism of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' [3, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='1–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='4] (Here, and elsewhere, Mn(τ)ι means ι−1Mn(τ)ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=') 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Suppose further that η is trivial, so σ = Mn(τ)ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Then δ = ε + Mn(θ)ι, where ε is an inner σ-derivation of Q and θ is a τ-derivation of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' [3, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='5] Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' In fact, in the context of Property 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='3, if d > 1 then something stronger holds: θ can be taken to be the zero map, so that δ itself is an inner σ-derivation [3, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' However, in the context of filtered rings, we will allow θ to be nonzero, as this extra flexibility is crucial for ensuring that the decomposition δ = ε + Mn(θ)ι behaves well with respect to the filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='3 Compatible filtrations and skew power series rings Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Let (R, v) be a filtered ring and (σ, δ) a skew derivation on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' We will say that (σ, δ) is (weakly) compatible with v if v(σ(r)) = v(r) and v(δ(r)) > v(r) for all 0 ̸= r ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' This is more general than the notion of “compatibility” used by the authors in [10], which could be called strong compatibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Let (R, v) be a complete, positively filtered ring, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' R is a ring admitting a separated discrete filtration v : R → N ∪ {∞} with respect to which R is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' The set R[[x]] := � n≥0 Rxn, whose elements are formal sums r0 + r1x + r2x2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' over arbitrary ri ∈ R, is a left R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' This is a complete filtered R-module with standard filtration f := fv,x : R[[x]] → N ∪ {∞}, given by f �� i≥0 rixi � = inf i≥0{v(ri) + i}, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='1) which is separated and discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Note that R[x] is dense in R[[x]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' A skew derivation (σ, δ) on R makes R[x] into a ring, with multiplication determined uniquely by the rule xr = σ(r)x + δ(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' We write this ring as R[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' If (σ, δ) is compatible with v, then it induces a well-defined associative multiplication on R[[x]] in the same way: see [10, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='17] (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' [19, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='1] or [13, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='4]), and we denote this ring by R[[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' The function f on R[[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ]] as defined above is a positive ring filtration, and R[[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ]] is complete with respect to f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='4 Discrete valuation rings Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Following Ardakov [1], we will say that a discrete valuation ring is a noetherian domain D with the property that, for every nonzero x ∈ Q(D) (the division ring of quotients), we have either x ∈ D or x−1 ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' We begin by showing that D has properties very similar to those of commutative discrete valuation rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Let D be a discrete valuation ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' (i) D is a local ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' (ii) All right (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' left) ideals of D are principal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' (iii) The lattice of right (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' left) ideals of D is totally ordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' (iv) All right (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' left) ideals of D are two-sided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' In the language of [15], D is a noetherian total subring of the skew field Q(D), and statements (i–iii) follow from [15, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' The proof of (iv) below is adapted from [2, Lemma 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' We give the proof for right ideals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' the proof for left ideals is of course similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Suppose there exist right ideals of D that are not two-sided, and let J be the maximal such right ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' By (ii), J = aD for some a ∈ D: then, for some r ∈ D, we have ra =: b ̸∈ aD by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' By (iii), this implies aD ⊊ bD, and so a = bs for some s ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Combining these two equations, we can see that b = rbs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Now, by the maximality of J, we have that Db ⊆ DbD = bD, so that rb = bt for some t ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' In particular, b = bts, and so b(1 − ts) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' But b cannot be zero, so as D is a domain, we must have ts = 1, and hence (as noetherian rings are Dedekind-finite) st = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' It follows that at = b, contradicting the assumption that b ̸∈ aD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Let D be a discrete valuation ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' (i) J(D) = πD for some normal element π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' (ii) Every nonzero ideal of D has the form πnD for some n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' (i) is an immediate consequence of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' To show (ii): note that Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='8(iv) also implies that D is an FBN ring [17, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='7], and so �∞ n=1 πnD = 0 by [8, Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' We now argue exactly as in the commutative case: indeed, a nonzero ideal aD must satisfy πn+1D ⊊ aD ⊆ πnD for some n by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='8(iii), from which it follows that a = πnu for some u ∈ D \\ πD, which must be a unit by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='8(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' An element π as in the above proposition will be called a uniformiser of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' In the following, D will continue to denote a complete discrete valuation ring, and we will also set F = Q(D), O = Mn(D) and Q = Mn(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Also write vF for the induced J(D)-adic filtration on F, and suppose that (τ, θ) is a skew derivation on F compatible with vF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' likewise write vQ for the J(O)-adic filtration on Q, and suppose that (σ, δ) is a skew derivation on Q compatible with vQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' This puts us essentially in the situation of Hypotheses (H1–3) + (S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' The following is now routine to check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' 8 (i) Let S be the multiplicatively closed set in D generated by π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Then F = S−1D = DS−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Moreover, π is normal in D[[y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' τ, θ]], and F⊗DD[[y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' τ, θ]] = S−1D[[y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' τ, θ]] = D[[y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' τ, θ]]S−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' (ii) Let S be the multiplicatively closed set in O generated by π (where we identify D with its diagonal embedding in O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Then Q = S−1O = OS−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Moreover, π is normal in O[[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ]], and Q ⊗O O[[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ]] = S−1O[[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ]] = O[[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ]]S−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' 3 Reparametrising filtered skew derivations Throughout this section, let (R, v) be a complete, positively filtered ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Suppose also that R admits a skew derivation (σ, δ) which is compatible with v, and take y ∈ R[x] such that R[x] = R[y] (an equality of left R-modules).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Given an element �m i=0 riyi ∈ R[y], we may define the function fv,y : m � i=0 riyi �→ inf i≥0{v(ri) + i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Note that fv,x is the standard ring filtration defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' In contrast, for arbitrary elements y ∈ R[x], the function fv,y will not always be a ring filtration, and even when it is, it will generally not be equivalent to fv,x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Take y := x + 1 ∈ Zp[x], and v the p-adic valuation on Zp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Then for all n, we have fv,x((x+1)n) = 0 but fv,y((x+1)n) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' In particular, Zp[x] = Zp[y] but Zp[[x]] ̸= Zp[[y]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' In this section, we identify two families of elements y ∈ R[x] for which fv,x and fv,y are equal as functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='1 Conditions for identical filtrations With notation as above, we first show that fv,x and fv,y are equal when y = x − t for some t ∈ R satisfying v(t) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Write (x − t)n = xn + βn,1xn−1 + · · · + βn,n−1x + βn,n for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' For all n, i, we have v(βn,i) ≥ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Firstly, note that (x − t)βn,ixn−i = σ(βn,i)xn+1−i + (δ(βn,i) − tβn,i)xn−i, so (writing βn,0 := 1 for ease of notation) we may calculate (x − t)n+1 as (x − t) � n � i=0 βn,ixn−i � = n � i=0 σ(βn,i)xn+1−i + n � j=0 (δ(βn,j) − t(βn,j))xn−j = xn+1 + n � i=1 (σ(βn,i) + δ(βn,i−1) − tβn,i−1)xn+1−i + (δ(βn,n) − tβn,n), by setting j = i + 1 in the second sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' That is, \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 βn+1,0 = 1, βn+1,i = σ(βn,i) + δ(βn,i−1) − tβn,i−1 (1 ≤ i ≤ n), βn+1,n+1 = δ(βn,n) − tβn,n, from which the claim follows by induction on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' 9 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Let p(x) ∈ R[x] be a polynomial, and t ∈ R such that v(t) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Then fv,x(p(x − t)) ≥ fv,x(p(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Write p(x) = r0 + r1x + · · · + rmxm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Then p(x − t) = m � n=0 rn(x − t)n = m � n=0 rn � n � i=0 βn,ixn−i � setting βn,0 := 1 = m � j=0 � m � n=j rnβn,n−j � xj where j := n − i, and so fv,x(p(x − t)) = fv,x � m � j=0 � m � n=j rnβn,n−j � xj � = inf 0≤j≤m � v � m � n=j rnβn,n−j � + j � ≥ inf 0≤j≤m inf j≤n≤m {v(rn) + v(βn,n−j) + j} as v is a filtration ≥ inf 0≤j≤m inf j≤n≤m {v(rn) + n} by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='2 = inf 0≤n≤m {v(rn) + n} = fv,x � m � n=0 rnxn � = fv,x(p(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Let t ∈ R such that v(t) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Then fv,x = fv,x−t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Take an arbitrary element p(x) ∈ R[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Then fv,x(p(x)) ≤ fv,x(p(x + t)) applying Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='3 to − t = fv,x−t(p(x)) changing variables x �→ x − t throughout ≤ fv,x−t(p(x − t)) applying Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='3 to t = fv,x(p(x)) changing variables x �→ x + t throughout, from which we can conclude that fv,x(p(x)) = fv,x−t(p(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Next, we show that fv,x and fv,y are equal when y = ax where a ∈ R× and v(a) = v(a−1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Write (ax)n = γn,0xn + γn,1xn−1 + · · · + γn,n−1x + γn,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' v(γn,i) ≥ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' As in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='2, calculating (ax)n+1 = ax(γn,0xn +γn,1xn−1 +· · ·+γn,n−1x+γn,n) gives \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 γn+1,0 = aσ(γn,0), γn+1,i = aσ(γn,i) + aδ(γn,i−1) (1 ≤ i ≤ n), γn+1,n+1 = aδ(γn,n), and we may perform induction as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' 10 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' fv,x(p(ax)) ≥ fv,x(p(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' p(ax) = m � n=0 rn(ax)n = m � n=0 rn � n � i=0 γn,ixn−i � = m � j=0 � m � n=j rnγn,n−j � xj where j := n − i, and so fv,x(p(ax)) = fv,x � m � j=0 � m � n=j rnγn,n−j � xj � , and the proof now proceeds exactly as in the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' fv,x = fv,ax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Take an arbitrary element p(x) ∈ R[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Then fv,x(p(x)) ≤ fv,x(p(a−1x)) applying Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='6 to a−1 = fv,ax(p(x)) changing variables x �→ ax throughout ≤ fv,ax(p(ax)) applying Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='6 to a = fv,x(p(x)) changing variables x �→ a−1x throughout, from which we can conclude that fv,x(p(x)) = fv,ax(p(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Finally, we fix any y ∈ R[x] such that R[x] = R[y] and fv,x = fv,y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Denote this common filtration by f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' It follows that: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' R[[x]] = R[[y]] as filtered left R-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Proof of Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' In case (i), set y = ax, so that f := fv,x = fv,y by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' in case (ii), set y = x − t, and use Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' In both cases, R[x] = R[y].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Now Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='8 implies that R[[y]] and R[[x]] can be identified as filtered modules, and the multiplication data has already been calculated in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='2, so the conclusion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' 4 Skew derivations on semisimple artinian rings 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='1 Reducing to orbits For now, we do not assume any of the Hypotheses (H1–3), and we let (R, v) be an arbitrary filtered ring such that R admits a decomposition R ∼= B × C (as unfiltered rings).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' We will abuse notation and write R = B × C (as unfiltered rings).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' We will also write B (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' C) for the ideal B × 0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' 0 × C) of R, so that there are inclusion maps jB : B → R and jC : C → R and projection maps πB : R → B and πC : R → C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Suppose further that the filtration on R is complete and positive, and that R admits a skew derivation (σ, δ) which restricts to skew derivations on B and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' (That is, setting σB = πBσjB and δB = πBδjB, we have that (σB, δB) is a skew derivation on B, and likewise for C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=') 11 Write vB and vC for the restrictions of v to B and C respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' In general, even if (σ, δ) is compatible with v, it may not be true that (σB, δB) is compatible with vB, and so we must restrict to the case in which the decomposition R ∼= B × C and the filtration v interact nicely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' The following lemma is an immediate consequence of the definition of the product filtration, as in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' If v = vB × vC, then (σB, δB) is compatible with vB, and (σC, δC) is compatible with vC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Hence, under the assumption v = vB × vC, we may define the filtered rings B[[xB;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σB, δB]], with filtration fB, satisfying fB|B = vB and fB(xB) = 1, C[[xC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σC, δC]], with filtration fC, satisfying fC|C = vC and fC(xC) = 1, as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Suppose that v = vB × vC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Then there is an isomorphism of filtered rings ϕ : R[[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ]] → B[[xB;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σB, δB]] × C[[xC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σC, δC]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' It is straightforward to check that the maps ϕ : R[[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ]] → B[[xB;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σB, δB]] × C[[xC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σC, δC]] � i≥0 rixi �→ �� i≥0 πB(ri)xi B, � i≥0 πC(ri)xi C � and θ : B[[xB;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σB, δB]] × C[[xC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σC, δC]] → R[[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ]] �� i≥0 bixi B, � i≥0 cixi C � �→ � i≥0 (jB(bi), jC(ci))xi are the mutually inverse filtered isomorphisms as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Let Q′ be an arbitrary semisimple artinian filtered ring admitting a skew derivation (σ, δ), and write the minimal nonzero ideals of Q′ as A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' , Ae, so that Q′ = �e i=1 Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' In the case where the Ai fall into several σ-orbits, write ρ for the permutation of the indexing set {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' , e} induced on the set {A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' , Ae} by σ as in Property 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Let S be a union of (some) orbits of ρ and S′ = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' , e}\\S, and suppose (to avoid trivial cases) that both S and S′ are nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Now set B′ = � i∈S Ai and C′ = � i∈S′ Ai: it follows from Property 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='1 that (σB′, δB′) restricts to a skew derivation on B′, and likewise for C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Assume now that each Ai ∼= Mni(Fi), where ni ≥ 1 is some positive integer and Fi is the Goldie ring of quotients of a complete discrete valuation ring Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Set Oi to be the preimage in Ai of Mni(Di), so that O′ = O1 × · · · × Oe is a maximal order in Q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Moreover, if j = ρ(i) then Aj = σ(Ai) so Mni(Fi) ∼= Mnj(Fj), which implies that ni = nj and Fi ∼= Fj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Suppose that all of these rings are given their natural filtrations: that is, each Di retains its discrete valuation, and Fi inherits the J(Di)-adic valuation (see §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='4), Mni(Di) and Mni(Fi) are given the corresponding matrix filtrations (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='2), Oi and Ai inherit their filtrations from Mni(Di) and Mni(Fi) under the above isomor- phisms, and 12 O′ and Q′ are given the product filtrations (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='1) from the Oi and Ai respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Then O′, B′ and C′ as defined above will satisfy the hypotheses of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Moreover, if S is taken to be a single orbit of ρ with |S| = d, then (after renumbering so that S = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' , d} and writing n = n1 = · · · = nd) the ring O := O′ ∩ B′ as defined above, its Goldie ring of quotients Q := B′, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' will satisfy Hypotheses (H1–3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='2 Untwisting inner automorphisms In this subsection, we assume the full force of Hypotheses (H1–3) and adopt their notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Without loss of generality, reordering the Ai if necessary, we will set σ(Ai) = Ai+1 for 1 ≤ i ≤ d−1 and σ(Ad) = A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' We may now invoke Property 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' In particular, there is a decomposition σ = η ◦ Mn(τ)ι, where η is an inner automorphism of Q, say η = ca for some a ∈ Q×, and τ is an automorphism of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Both η and Mn(τ)ι preserve O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Since η is an inner automorphism of Q, it will preserve each Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' But σ(Ai) = Ai+1 (with indices interpreted modulo d), so Mn(τ)ι must send Ai to Ai+1, and hence τ sends Fi to Fi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' However, since σ and η are continuous, it follows that τ is continuous, and so Di+1 = Oi+1 ∩ Fi+1 = τ(Di), hence Mn(τ)ι preserves O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Now it follows that η = σ ◦ Mn(τ −1)ι preserves O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Our aim in this subsection is to “untwist” η by making a change of variables x �→ x′, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' find an element x′ ∈ O[[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ]] such that O[[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ]] = O[[x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Mn(τ)ι, δ′]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' By Theorem B(i), it would suffice if v(a) = v(a−1) = 0: this would imply that a, a−1 ∈ O, and we could then set x′ = a−1x, giving δ′ = a−1δ as in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Of course, in general, a will not necessarily have this property: for instance, if O is a complete discrete valuation ring with central uniformiser π, then ca = cπra for all r ∈ Z, and v(πra) will usually not be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Surprisingly, this naive obstruction is the only kind of obstruction that occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' There exist an element b ∈ Q×, an inner automorphism η′ = cb of Q, and an automorphism τ ′ of F such that σ = η′ ◦ Mn(τ ′)ι and v(b) = v(b−1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Suppose a = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' , ad) ∈ Q×: then, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='3, we have aiOia−1 i = Oi for each i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Let ki = v(ai), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' ai ∈ J(Oi)ki \\ J(Oi)ki+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' So, if πj is a uniformiser of Dj, Ij is the identity matrix of Mn(Dj), and ˜πj := ι−1(πjIj), then we have aj = bj˜π kj j for some bj ∈ Oj with v(bj) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Set b = (b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' , bd), so that v(b) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' By Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='10(ii), ˜πj is normal in Oj, so the right ideal bjOj is in fact a two-sided ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Moreover, by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='9(ii) and Morita equivalence, as the ideal bjOj contains the element bj of value 0, it must be equal to Oj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Hence bj is a unit in Oj, and we have v(bj) = v(b−1 j ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Now set η′(r) = brb−1 for all r ∈ Q and τ ′(s) = Πτ(s)Π−1, where Π = (πk1 1 I1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' , πkd d Id).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' The claim now follows from a short calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' 13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='3 Untwisting inner σ-derivations In this subsection, let (R, f) be an arbitrary Z-filtered ring admitting a compatible skew deriva- tion (σ, δ), and write the f-level sets of R as FnR for n ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' We will also suppose that R = Mn(A) for some ring A, f = Mn(g) for some filtration g on A, σ = Mn(τ) for some automorphism τ of A, and δ = Mn(θ) + ε, where θ is a τ-derivation of A and ε is an inner σ-derivation of R, say ε = dσ,u for some u ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' (Compare Property 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=') We will write the standard set of matrix units in R as {eij}1≤i,j≤n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Our aim in this subsection is to “untwist” ε by making a change of variables x �→ x′, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' find an element x′ ∈ R[[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Mn(τ), δ]] such that R≥0[[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Mn(τ), δ]] = R≥0[[x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Mn(τ), Mn(θ)]] where of course, R≥0 is the positively filtered subring of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' As before, we would be done by Theorem B(ii) if we had f(u) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' This is also unreasonable to expect, albeit this time for slightly less obvious reasons: for instance, if A is the division ring of fractions of a complete discrete valuation ring with uniformiser π, then Mn(θ)+dMn(τ),u = Mn(θ+dτ,πr)+dMn(τ),u−πrI for all r ∈ Z, and v(u − πrI) can be less than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' However, again, under mild conditions this is the only obstruction that occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' In the above setup, there exist an element u′ ∈ R, a τ-derivation θ′ of A, and an inner σ-derivation ε′ = inn(u′) of R such that δ = Mn(θ′) + ε′ and f(u′) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Write u = � i,j uijeij for some coefficients uij ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Let 1 ≤ p, q ≤ n be arbitrary, and consider the matrix unit epq ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' By assumption, δ(epq) ∈ F1R, and so ε(epq) ≡ −Mn(θ)(epq) mod F1R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='1) But we can calculate the left-hand side of this congruence (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='1) explicitly as ε(epq) = � i,j (uijeijepq − epquijeij) = � i uipeiq − � j uqjepj, and the right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='1) is just −θ(1A)epq, which is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' So we may rewrite (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='1) as � i uipeiq − � j uqjepj ≡ 0 mod F1R, and equate corresponding entries, to get \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 uip ∈ F1A i ̸= p uqj ∈ F1A j ̸= q upp − uqq ∈ F1A, and so, as p and q were arbitrary, we get u ≡ u111R mod F1R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Now setting u′ := u − u111R, and defining θ′(a) := θ(a) + u11a − τ(a)u11 and ε′ := dMn(τ),u′, we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Upshot: in this case, using Theorem B(i) we can pass to the case when δ = Mn(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Proof of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Firstly, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='4 shows that there exist some τ ∈ Aut(F) and some b ∈ Q× satisfying v(b) = v(b−1) = 0 such that σ = cb ◦ Mn(τ)ι, and both of these preserve O 14 by the same argument as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' So by Theorem B(i), we may set x′ = b−1x to get O[[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ]] = O[[x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Mn(τ)ι, δ′]] for the Mn(τ)ι-derivation δ′ := b−1δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Secondly, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='5 shows that there exist some τ-derivation θ of F and some u ∈ Q satisfying v(u) ≥ 1 such that δ′ = Mn(θ)ι + dMn(τ)ι,u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' So by Theorem B(ii), we may set x′′ = x′ − u to get O[[x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Mn(τ)ι, δ′]] = O[[x′′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Mn(τ)ι, Mn(θ)ι]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Finally, the maps O[[x′′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Mn(τ)ι, Mn(θ)ι]] → Mn(D)[[y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Mn(τ), Mn(θ)]] � i≥0 qi(x′′)i �→ � i≥0 ι(qi)yi and Mn(D)[[y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Mn(τ), Mn(θ)]] → Mn(D[[z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' τ, θ]]) � i≥0 � n � j,k=1 cijkejk � yi �→ n � j,k=1 �� i≥0 cijkzi � ejk can now be checked to be filtered ring isomorphisms, and vO = Mn(vD) ◦ ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' 5 Applications Throughout this section, we assume our data (Q, O, vQ) satisfies Hypotheses (H1–3) + (S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='1 Polynomial elements Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' A polynomial element of R[[x]] (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' R[[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ]]) is an element of R[x] (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' R[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' We first recall the following important generalisation of the Weierstrass preparation theorem for skew power series rings, essentially due to Venjakob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Every nonzero right ideal of D[[y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' τ, θ]] contains a nonzero polynomial element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Let v be the J(D)-adic filtration and π a uniformiser for D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Then every nonzero ele- ment r ∈ D[[y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' τ, θ]] can be written as r = (s0 + s1y + s2y2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' )πm, where all si ∈ D and infi≥0{v(si)} = 0, by Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='10(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Hence s = s0 + s1y + s2y2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' satisfies the hypotheses of [19, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='1], and so a right-hand version of [19, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='2] tells us that s can be expressed uniquely as s = Pu, where u ∈ D[[y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' τ, θ]]× and P ∈ D[y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' τ, θ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' In particular, if A is a nonzero right ideal of D[[y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' τ, θ]], then given any nonzero r ∈ A, we can write it as r = Puπm as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Since π is normal in D[[y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' τ, θ]] by Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='10(i), this is just r = Pπmu′ for some unit u′ ∈ D[[y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' τ, θ]], and hence the polynomial element Pπm is also an element of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='10(i) also implies a similar result for F ⊗D D[[y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' τ, θ]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Every nonzero (two-sided) ideal of O[[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ]] contains a nonzero polynomial element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' 15 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Theorem A tells us that there exists an isomorphism ϕ : O[[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ]] → Mn(D[[y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' τ, θ]]) extending ι, such that y = ϕ(ax − t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' In particular, if P ∈ D[[y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' τ, θ]] is polynomial in the variable y, then ϕ−1(PI) (where I is the identity matrix) is polynomial in the variable x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' The result now follows from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='2 and by Morita equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Proof of Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' The first statement is simply restating Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' For the second statement, take a nonzero ideal I of Q ⊗O O[[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Then J := I ∩ O[[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ]] is clearly an ideal of O[[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' By Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='10(ii), multiplying any nonzero element of I by an appropriate power of the regular element π will give an element of J, so J ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Hence, by the first statement, we know that J ∩ O[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ] ̸= 0, and so I ∩ Q[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ] ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' But as we are assuming that Q[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ] is simple, it must follow that I ∩ Q[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ] = Q[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' In particular, 1 ∈ I, and so I = Q ⊗O O[[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='2 Uniform dimension This subsection continues the work of [14] on uniform dimensions (Goldie ranks) of skew power series rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' As we have already remarked in the Introduction, Theorem B sometimes allows us to reduce directly to the results of [14] when the derivation is inner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' However, in the special case of Hypotheses (H1–3) + (S), we can now prove similar results about skew power series rings over O for arbitrary derivations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Proof of Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' By Theorem A, we know that O[[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ]] ∼= Mn(D[[y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' τ, θ]]) for some ap- propriate skew derivation (τ, θ), and hence that r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='udim(O[[x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' σ, δ]]) = n(r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='udim(D[[y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' τ, θ]])) [17, Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='11(iii)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' In the same way, r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='udim(O) = n(r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='udim(D)), and of course r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='udim(D) = 1 as D is a noetherian integral domain [17, Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='11(i)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' It remains to show that r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='udim(D[[y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' τ, θ]]) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' So let U be a uniform right ideal of D[[y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' τ, θ]], and let I be an arbitrary nonzero right ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' By Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='2, U ∩ D[y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' τ, θ] ̸= 0 and I ∩ D[y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' τ, θ] ̸= 0, and so as D[y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' τ, θ] is a prime ring [17, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='9(iii)], their inter- section (U ∩ I) ∩ D[y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' τ, θ] is also nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' In particular, this shows that U is an essential right ideal of D[[y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' τ, θ]], and so r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='udim(D[[y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' τ, θ]]) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' References [1] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Ardakov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Prime ideals in nilpotent Iwasawa algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Inventiones mathematicae, 190(2):439–503, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' [2] H.' metadata={'source': 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J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Algebra Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=', 12(4), 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' [19] Otmar Venjakob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' A noncommutative Weierstrass preparation theorem and applications to Iwasawa theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Reine Angew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=', 559:153–191, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' [20] William Woods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Dimension theory in iterated local skew power series rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Algebr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Represent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' Published online: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content='1007/s10468-022-10144-3, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} +page_content=' 17' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE0T4oBgHgl3EQfxAEX/content/2301.02639v1.pdf'} diff --git a/.gitattributes b/.gitattributes index 722488bca9ec175937846abe85e8bbdbdbabbe1e..887a3ccd12ad5cfe41cc48e1e9a3c453c26fa408 100644 --- 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+Communication +George Giakkoupis +Inria, Rennes, France +george.giakkoupis@inria.fr +Isabella Ziccardi +Bocconi University, Milan, Italy +isabella.ziccardi@unibocconi.it +Abstract +We study a simple random process that computes a maximal independent set (MIS) on a +general n-vertex graph. Each vertex has a binary state, black or white, where black indicates +inclusion into the MIS. The vertex states are arbitrary initially, and are updated in parallel: +In each round, every vertex whose state is “inconsistent” with its neighbors’, i.e., it is black +and has a black neighbor, or it is white and all neighbors are white, changes its state with +probability 1/2. The process stabilizes with probability 1 on any graph, and the resulting set of +black vertices is an MIS. It is also easy to see that the expected stabilization time is O(log n) +on certain graph families, such as cliques and trees. However, analyzing the process on graphs +beyond these simple cases seems challenging. +Our main result is that the process stabilizes in poly(log n) rounds w.h.p. on Gn,p random +graphs, for 0 ≤ p ≤ poly(log n) · n−1/2 and p ≥ 1/ poly(log n). Further, an extension of this +process, with larger but still constant vertex state space, stabilizes in poly(log n) rounds on Gn,p +w.h.p., for all 1 ≤ p ≤ 1. We conjecture that this improved bound holds for the original process +as well. In fact, we believe that the original process stabilizes in poly(log n) rounds on any given +n-vertex graph w.h.p. Both processes readily translate into distributed/parallel MIS algorithms, +which are self-stabilizing, use constant space (and constant random bits per round), and assume +restricted communication as in the beeping or the synchronous stone age models. To the best of +our knowledge, no previously known MIS algorithm is self-stabilizing, uses constant space and +constant randomness, and stabilizes in poly(log n) rounds in general or random graphs. +1 +Introduction +Finding a maximal independent set (MIS) is a fundamental problem in parallel and distributed +computing. Given a graph G = (V, E), the objective is to identify a set of vertices S ⊆ V such +that no two vertices u, v ∈ S are adjacent to each other (independence property), and no vertex +u ∈ V \S can be added to S without violating independence (maximality property). The significance +of the problem in parallel computing was first recognised in the early 80s [32, 8], due to its various +applications in symmetry breaking [24], and it has been studied extensively every since (see [7] for +a review of work until 2015, and [4, 17] for state of the art results). +In this paper we explore simple distributed random processes on graphs that find an MIS +starting from arbitrary initial states of the vertices. These processes immediately translate into self- +stabilizing [10, 11] synchronous distributed algorithms for network systems with severely restricted +computation and communication capabilities, such as wireless sensor networks. The processes we +consider are also relevant to certain biological cellular networks. For example, it is known that a +biological process occurring during the development of the nervous system of a fly is equivalent to +computing an MIS [2, 23]. +1 + +The main random process we consider, which we call the 2-state MIS process, is as follows. +Each vertex has a binary state, black or white, where black indicates inclusion into the MIS. The +vertex states are arbitrary initially and are updated in synchronous rounds. In each round, every +vertex u whose state violates the independence or maximality properties, i.e., u is black and has +a black neighbor, or it is white and has no black neighbor, changes its state to the opposite state +with probability 1/2. It is easy to see that the state of a vertex stabilizes as soon as it is black +and has no black neighbors, or it is white and has a stabilized black neighbor; and when all vertices +have stabilized, the set of black vertices is an MIS. It is also immediate that, on any graph G, the +process stabilizes eventually with probability 1 (due to the randomization) .1 +The 2-state MIS process can be viewed as a natural parallelization (with the addition of ran- +domness) of a simple self-stabilizing sequential deterministic algorithm, proposed in [28, 20], where +in each step a single node updates its state (from black to white, if the node has a black neighbor, +and from white to black if it has no black neighbors). [28] also observed that by randomizing the +transitions of the sequential algorithm we obtain an algorithm that stabilizes with probability 1 on +a general adversarial scheduler model, which includes the synchronous model. A similar observa- +tion follows from a general transformation framework proposed in [31]. The sequential algorithm is +know to stabilize after each process has taken at most 2 steps (regardless of the scheduling order). +However, analyzing the stabilization time of the parallel process seems a much more challenging +problem, and has not been studied until now. +The 2-state MIS process directly translates into a self-stabilizing MIS algorithm for the harsh +beeping communication model [9]. In that model, in every synchronous round, each node either +listens or beeps, and a listening node can only differentiate between none of its neighbors’ beeping, +or at least one beeping. In our case, we can let black nodes beep in each round, while white nodes +listen. Black nodes must be able to detect collisions (otherwise they cannot tell if they have a +black neighbor), thus we assume the beeping model version with sender collision detection (a.k.a. +full-duplex model) [1, 16]. +We also propose a simple variant of the 2-state MIS process, called the 3-state MIS process, +which has an additional state and does not require collision detection (see Definition 5). +This +variant is suitable for the synchronous stone age model [13, 12]. The synchronous stone age model +can be viewed as an extension of the beeping model over a constant number of channels (without +collision detection): each node beeps in at most one channel and listens to the other channels. +Overall, the algorithms obtained from the 2-state and 3-state MIS processes have several at- +tractive properties: they use a constant number of states (2 or 3) and one random bit per round, +they do not require node IDs or any global graph information (such as the number of vertices n +or the maximum degree ∆), assume very week communication (the beeping or stone age models), +they are self-stabilizing, and are extremely simple. We will prove that, on some families of graphs, +these algorithms are also fast, i.e., they stabilize (from an arbitrary initial state) in a number of +rounds that is poly-logarithmic in n, w.h.p.2 Moreover, despite that we were not able to prove such +as strong result here, we believe that these algorithms are fast in all graphs. +Several self-stabilizing distributed MIS algorithms have been proposed in the literature, but as +far as we know, none possesses all the above properties. Known self-stabilizing MIS algorithms for +the beeping model require (approximate) knowledge of n, use space that is a super-constant function +of n, and require a super-constant number of random bits [1, 23, 16]. In the stone age model, an +MIS algorithm proposed in [13] has similar properties as our algorithms (and is provably fast for all +1We could have defined the process so that the transition from white to black (when the white vertex has no black +neighbors) occurs with probability 1, but we opted for a randomized transition because it simplifies our analysis. +2In this paper, we do not analyze the 3-state MIS process, but we expect that it behaves similarly (or better than) +the 2-state MIS process. +2 + +graphs) but is not self-stabilizing; while a self-stabilizing algorithm for the model proposed recently +in [12] is fast only on graphs whose diameter is bounded by a known constant D. Other randomized +self-stabilizing MIS algorithms required super constant state and communication [30]. +Finally, +known deterministic self-stabilizing MIS algorithms require distinct node IDS, super constant state +and communication, and are in general much slower than the randomized algorithms, stabilizing +in time linear in n or in the maximum degree ∆ [22, 18, 29, 5]. +1.1 +Our Contribution +We first analyze the stabilization time of the 2-state MIS process on complete graphs and on graphs +with bounded arboricity.3 We also provide an upper bound in terms of the maximum degree for +general graphs. The proof of these results is mostly straightforward. +Theorem 1. The stabilization time of the 2-state MIS process on n-vertex graph G is +• O(log n) in expectation and O(log2 n) w.h.p., if G is the complete graph Kn. +• O(log n) w.h.p., if G has bounded arboricity. +• at most O(∆ log n) w.h.p., if the maximum degree of G is ∆. +A main technical contribution of the paper is the analysis of the 2-state MIS process on Erd˝os- +R´enyi Gn,p random graphs. We show a poly-logarithmic upper bound for Gn,p random graphs when +the average degree np is at most poly(log n) · √n. The same bound is easily obtained also when +the average degree is at least n/ poly(log n). +Theorem 2. The stabilization time of the 2-state MIS process on a Gn,p random graph, such that +0 ≤ p ≤ poly(log n) · n−1/2 or p ≥ 1/ poly(log n), is at most poly(log n) w.h.p. +Our proof techniques do not yield a poly-logarithmic upper bound for the 2-state MIS process +on Gn,p for the complete range of p. Our second technical contribution is an extension of the 2-state +MIS process that provably stabilizes in poly-logarithmic time w.h.p. on Gn,p for all 0 ≤ p ≤ 1. The +extended process uses a phase clock sub-process proposed in [12]. Interestingly, unlike [12], we do +not use the phase clock for synchronization, but rather as a local non-synchronized counter (see +Section 1.2 for a more detailed discussion). +Theorem 3. There is an extension of the 2-state process, with 18 states, such that the stabilization +time of the process on a Gn,p random graph, for any 0 ≤ p ≤ 1, is at most poly(log n) w.h.p. +We believe that the bound of Theorem 3 holds for the 2-state MIS process, as well. In fact, +we conjecture that the stabilization time of the 2-state MIS process is poly(log n) w.h.p. on any +given n-vertex graph. We also conjecture that the same is true for the 3-state MIS process. For the +2-state process, the best general upper bound we can hope for is O(log2 n), as the process requires +Θ(log2 n) rounds to stabilize on the complete graph Kn w.h.p.4 For the 3-state process, we have +no example of a graph where the stabilization time is larger than O(log n). +3The arboricity of a graph is the minimum number of forests into which we can partition its edges. +4It also requires Θ(log2 n) rounds in expectation to stabilize on a graph consisting of √n disjoint cliques K√n. +3 + +1.2 +Analysis Overview and Techniques +Below we give an overview of the analysis of the 2-state MIS process and its extension, on Gn,p +random graphs. +To avoid having to deal simultaneously with the randomness of the graph and the arbitrary +initialization of vertex states, we deal with graph randomness first. We define a family of good +graphs, containing those graphs that satisfy all structural properties that we will need for the +analysis, e.g., bounds on the average degree of any induced subgraph, and bounds on the number +of common neighbors of any two vertices (see Definition 17). We then show that a Gn,p random +graph is good w.h.p., and assume an arbitrary good graph in the analysis. +The analysis proceeds by showing that starting from any vertex states, the process makes +sufficient progress after O(log n) rounds, where progress is measured by the expected number of +vertices that stabilize. +In the 2-state MIS process, we call a vertex active if it is black and has a black neighbor, or it is +white and has no black neighbors. Thus, active vertices change their state to a uniformly random +state in the next step. A vertex is k-active if it is active and has at most k active neighbors. +An elementary property of the 2-state MIS process is that if a vertex is k-active, then it becomes +stabilized black in O(log k) rounds with probability Ω(1/k). +We also use an extension of this +property to sets of active vertices.5 These two properties, combined with structural properties of +good graphs, suffice to show the desired expected progress in the case in which the number of +non-stabilized vertices or the number of active vertices is large enough. +The more difficult case is when the number of non-stabilized vertices is relatively small, namely +O(p−1 log2 n), and a smaller than 1/ poly(log n) fraction of them are active. One may expect this +to be an easy case, since the induced subgraph on a random subset of O(p−1 log2 n) vertices has +maximum degree ∆ = O(log2 n) w.h.p. (and Theorem 1 gives an O(log3 n) bound for that ∆). +However, the above bound on ∆ does not apply to an induced subgraph on an arbitrary subset of +O(p−1 log2 n) vertices. Nevertheless, it is true that the average degree is O(log2 n), thus a constant +fraction of vertices have degree O(log2 n). +Let u be one such vertex, i.e., of degree d = O(log2 n) in the induced subgraph of non-stabilized +vertices. To prevent u from becoming active (and thus d-active) or becoming stabilized, in each +round at least one neighbor of u must be non-stabilized black. We show that, roughly speaking, +if a vertex v has probability b of being non-stabilized black at some point during an interval of r +rounds (ignoring the first few rounds, e.g., if v is black initially) then v has probability poly(b/r) of +becoming θ-active in that interval. For the purposes of the analysis, it suffices to set r = O(log log n). +Then θ is, roughly, bounded by the maximum number of common neighbors two nodes may have, +thus θ ≤ poly(log n) if p ≤ poly(log n) · n−1/2 (see Section 4.1 for the relevant lemmas). +If each of the d neighbors of u has probability less than 1/(2d) of becoming non-stabilized +black in the next r rounds, then u has a constant probability of becoming active (or stabilize). +On the other hand, if there is some neighbor v that has probability b ≥ 1/(2d) of becoming non- +stabilized black in the next r rounds, we saw above that v becomes θ-active with probability at +least poly(b/r) = 1/ poly(log n). We conclude that, with probability 1/ poly(log n), u is poly(log n)- +active or has some poly(log n)-active neighbor at some point in the next r = O(log log n) rounds. +It follows that u stabilizes with probability 1/ poly(log n) in the next O(log n) rounds.6 +When p > poly(log n)·n−1/2, the last case of the analysis above does not give a poly-logarithmic +bound. A way to overcome this problem is to control how often a vertex can change its state from +white to black. We extend the 2-state MIS process by incorporating such a control mechanism. +5Similar properties are commonly used in the analysis of distributed MIS algorithms in the literature. +6We suspect that a refinement of this argument may be useful for a broader class of graphs. +4 + +We call the new process the 3-color MIS process. It consists of two sub-processes running in +parallel: The first is similar to the 2-state MIS process with the addition of a third color, grey; a +black vertex now becomes gray instead of white, a gray vertex becomes white after a while, and +other vertices treat gray vertices as white. The transition from gray to white is controlled by the +second sub-process, called the logarithmic switch. +In the logarithmic switch, each vertex has an on/off binary variable, and a gray vertex changes +to white if the switch variable of the vertex is on. We would like that the logarithmic switch satisfy +two basic properties: (i) a vertex switches from off to on every Θ(log n) rounds; and (ii) it switches +from on to off every O(1) rounds.7 However, we do not know how to implement property (i) using +constant states. We observe that it suffices if property (i) is satisfied only when p > poly(log n) · +n−1/2; for smaller p, a weaker property suffices: (i′) a vertex switches from off to on after at most +O(log n) rounds. It is not immediately obvious how to implement this distinction, because we want +the process to work for all 0 ≤ p ≤ 1 without knowing p (or anything else about the graph topology). +We achieve that as follows. +We exploit the fact that if p > poly(log n) · n−1/2 then the graph has constant diameter (in +fact diameter 2). The logarithmic switch process we devise is similar to the phase clock process +RandPhase proposed in [12]. RandPhase assumes that an upper bound D on the graph diameter +is available to the process and uses D + 3 states. The core mechanism of the logarithmic switch is +identical to that of RandPhase for D = 3 (not 2!), but the underlying graph may have arbitrary +(and unknown) diameter. The logarithmic switch includes also a mapping of the states to the on/off +values of the switch. Unlike RandPhase which is used for sychronization (it achieve synchronous +phases of length D + Θ(log n)), the purpose of the logarithmic switch is not synchronization, as it +is not required that the switch variables of different vertices change simultaneously. +Roadmap. +The rest of the paper is organized as follows. Section 2 contains the definition and +some basic properties of the 2-state and 3-state MIS processes. +Section 3 provides a proof of +Theorem 1. Section 4 proves Theorem 2. Section 5 defines the 3-color MIS process and proves +Theorem 3. And Appendix B reviews related work. +Notation. +Let G = (V, E) be a graph on n vertices. For each vertex u ∈ V , N(u) = {v: (u, v) ∈ +E} is the set of neighbors of u, and N +(u) = N(u) ∪ {u}. Similarly, for a set of vertices S ⊆ V , +we define N(S) = � +u∈S N(u) \ S and N +(S) = � +u∈S N +(u) = N(S) ∪ S. For two (not necessarily +disjoint) sets S, T ⊆ V , we let E(S, T) = {(u, v) ∈ E : u ∈ S, v ∈ T} be the set of edges with one +endpoint in S and the other in T. We also define E(S) = E(S, S). By G[S] we denote the induced +subgraph of G on S ⊆ V , i.e., G[S] = (S, E(S)). +2 +The 2-State and 3-State MIS Processes +We define two self-stabilizing distributed graph processes that compute a maximal independent set +when applied on any given graph. +Definition 4 (2-State MIS Process). In the 2-state MIS process on graph G = (V, E), each vertex +u ∈ V has a binary state from the set {black, white}, and all states are updated in parallel +7The reason why a logarithmic switch suffices, rather than a ‘double-logarithmic’ switch is that, in the induced +subgraph on O(p−1 log2 n) vertices consider in the last case of the analysis of the 2-state MIS process, a constant +fraction of vertices have at most O(log n) neighbors of degree Ω(log3 n). +5 + +rounds. The initial state c0(u) of vertex u can be arbitrary, and in each round t = 1, 2, . . . , u’s +state is updated from ct−1(u) to ct(u) according to the following rule. +let NC t(u) = {ct−1(v): v ∈ N(u)} +if +� +ct−1(u) = black and NC t(u) ∋ black +� +or +� +ct−1(u) = white and NC t(u) ̸∋ black +� +then +let ct(u) be a uniformly random state from {black, white} +else set ct(u) = ct−1(u) +We say that vertex u is black or white if its state is black or white, respectively. We say that +u is active if it is black and has some black neighbor, or it is white and has no black neighbors. +We say that vertex u is stable, if either it is black and has no black neighbors, or it is white and +has a neighbor that is black and stable. It is immediate from the update rule that once a vertex +becomes stable, it remains stable thereafter, and its state no longer changes. The stabilization +time of vertex u is the earliest round at the end of which u is stable. The stabilization time +of the process is the earliest round at the end of which all vertices are stable. It is easy to verify +that after the stabilization time of the process, the set of black vertices is an MIS of G. +We let Bt = {u ∈ V : ct(u) = black} be the set of black vertices at the end of round t ≥ 0, and +let Wt = V \ Bt be the set of white vertices. We let +At = {u ∈ Bt : N(u) ∩ Bt ̸= ∅} ∪ {u ∈ Wt : N(u) ∩ Bt = ∅} +denote the set of active vertices at the end of round t. We let It = {u ∈ Bt : N(u) ∩ Bt = ∅} be the +set of stable black vertices at the and of round t (note that It is an independent set and is a subset +of the final MIS). Finally, we let Vt = V \ N +(It) be the set of vertices that are not stable at the +end of round t. +Definition 5 (3-State MIS Process). In the 3-state MIS process on G = (V, E), each vertex u ∈ V +has a state from set {black1, black0, white}, and the states are updated in parallel rounds. The +initial state c0(u) of u is arbitrary, and in each round t ≥ 1, u’s state is updated as follows. +let NC t(u) = {ct−1(v): v ∈ N(u)} +if ct−1(u) = black1 or +� +ct−1(u) = black0 and NC t(u) ̸∋ black1 +� +or +� +ct−1(u) = white +and NC t(u) = {white} +� +then +let ct(u) be a uniformly random state from {black1, black0} +else if ct−1(u) = black0 then +set ct(u) = white +else set ct(u) = ct−1(u) +In the 3-state MIS process, we say that a vertex u is black when its state is black1 or black0. +Then the stable vertices and the stabilization times are defined as before. Note that the state of a +stable black vertex alternates perpetually between states black1 and black0. +In this paper we focus on the 2-state MIS process, but we expect that all our upper bound +results should carry over to the 3-state MIS process. +2.1 +Basic Properties of the 2-State MIS Process +We show some elementary properties of the 2-state MIS process. In the analysis, it will be conve- +nient to assume that at the beginning of each round t ≥ 1, we flip for each vertex u an independent +coin φt(u) such that P[φt(u) = black] = P[φt(u) = white] = 1/2. Then if u must update its state +to a random state in that round, i.e., if u ∈ At−1, we set ct(u) = φt(u); while if u /∈ At−1, then +φt(u) is not used by the algorithm. +The lemmas below apply for any graph G = (V, E), and the probabilistic statements assume +that we know the states of vertices at the end of round t (i.e., Bt or Wt is given). The first lemma +6 + +says than an active vertex u with k active neighbors has probability Ω(1/k) to become stable black +in the next O(log k) rounds. +Lemma 6. If u ∈ At and |N(u) ∩ At| = k ≥ 1, then the probability that u ∈ It+log(k+1) is at least +(2ek)−1. +Proof. Let r = ⌈log(k + 1)⌉. The probability that u ∈ It+r is lower bounded by the probability +that φt+1(v) = · · · = φt+r(v) = black holds for v = u and does not hold for any v ∈ N(u) ∩ At, +which is +(1/2)r · (1 − (1/2)r)k ≥ (1/2)r · e−k/(2r−1) ≥ (1/2k) · (1/e). +(1) +For the first inequality we used the fact (1 − 1/n)n−1 ≥ e−1, and for the second we used that +log(k + 1) ≤ r ≤ log(k) + 1. +The next statement is a generalization of Lemma 6 to multiple active vertices u1, . . . , uℓ. We +will apply this result to the set of active neighbors of a vertex u, to lower bound the probability +that u is stable after a logarithmic number of rounds (because a neighbors becomes stable black). +The proof can be found in Appendix A.1. +Lemma 7. Suppose that u1, . . . , uℓ ∈ At, and |N(ui) ∩ At| = ki > 0, for each 1 ≤ i ≤ ℓ. Then the +probability that {u1, . . . , uℓ} ∩ It+log(maxi ki+1) ̸= ∅ is at least (1/5) · min +� +1, � +i(2ki)−1� +. +3 +Simple Bounds for the 2-State MIS Process +We show some simple bounds on the stabilization time of the 2-state MIS process on certain graph +families, namely, the complete graph and trees (or more generally, graphs of bounded arboricity). +We also show a basic upper bound in terms of the maximum degree on a general graph. +Theorem 8. The stabilization time of the 2-state MIS process on the complete graph Kn = (V, E) +is O(log n) in expectation and O(log2 n) w.h.p. More concretely, for any k > 0, the stabilization +time is at least k · log n with probability 2−Θ(k). +Proof. We call round t critical if |Bt| ≤ 1, and we call it stable if |Bt| = 1. Let pa be the probability +that the next critical round is stable, given that |At| = a ≥ 2. Note that in graph Kn, At = Bt if +|Bt| > 1, At = ∅ if |Bt| = 1, and At = V if Bt = ∅; thus |At| ̸= 1. We argue that for any a ≥ 2, +2/3 ≤ pa ≤ 17/21. +The lower bound follows from the observation that, for any i ≥ 2 and j ≥ 1, the conditional +probability that round j is stable, given that it is critical and that |Aj−1| = i, is +(i +1)2−i +(i +1)2−i+2−i = +i +i+1 ≥ +2/3, since i ≥ 2. For the upper bound we observe that, for any i ≥ 3 and j ≥ 1, the conditional +probability of |Bj| ∈ {2, 0}, given that |Bj| ≤ 2 and that |Aj−1| = i, is +(i +2)2−i+2−i +(i +2)2−i+(i +1)2−i+2−i = i2−i+2 +i2+i+2 ≥ +4/7. Also, p2 = 2/3 < 17/21. Then, for any a ≥ 3, we have 1 − pa ≥ (4/7) · (1 − p2), which implies +pa ≤ 17/21. +Next, consider the number of rounds r from a non-stable critical round (when all nodes are +white) until the next critical round. The probability that r > k is lower and upper bounded by +1 − e−n2−k ≤ 1 − (1 − 2−k)n ≤ n2−k. +7 + +Combining the above we obtain that (i) from any given non-stable round, the probability that +a stable round is reached in at most k = log n + 1 rounds is at least 2/3 − n2−k ≥ 1/6; (ii) from +any given non-stable critical round, the probability that the next critical round is non-stable and is +reached in more than k = log n − 2 rounds is at least 1 − 17/24 − e−n2−k > 1/6; and (iii) assuming +round t = 0 is not critical, the probability that the first critical round is non-stable is at least +1 − 17/24. These statements, together, imply that the stabilization time is at least k log n with +probability 2−Θ(k). And from that, the expectation and high-probability bounds follow. +Remark 9. From Theorem 8, it is immediate that the expected stabilization time of the 2-state MIS +process is Θ(log2 n) on a graph G that is the disjoint union of √n cliques K√n. The same bound +holds also w.h.p. +Remark 10. A similar analysis as for Theorem 8 gives an upper bound of O(log n) on the stabi- +lization time of the 3-state MIS process on Kn, both in expectation and w.h.p. The reason is that +once Bt ̸= ∅ then Bt′ ̸= ∅ for all t′ ≥ t (thus the next critical round is stable). +Theorem 11. The stabilization time of the 2-state MIS process on any graph G = (V, E) of bounded +arboricity (e.g., G is a tree) is O(log n) w.h.p. +Proof. Recall that the arboricity λ of G is the minimum number of forests into which its edges can +be partitioned, and is equal up to a factor of 2 to the maximum average degree in any subgraph [26]. +Suppose that the average degree of any subgraph of G is at most d ≤ 2λ. Let St be the subset of +Vt consisting of of all vertices u ∈ Vt with |N(u) ∩ Vt| ≤ d. Then |St| ≥ |Vt|/(d + 1). If u ∈ St \ At +and |N(u) ∩ Vt| = du, the probability that N(u) ⊆ Wt+1 is 2−du ≥ 2−d. Thus, for each u ∈ St, +the probability that u ∈ At ∪ At+1 is at least 2−d. And if u ∈ At ∪ At+1, Lemma 6 gives that +u ∈ It+log(d+1)+1 with probability at least (2ed)−1. It follows +E +� +|Vt+log(d+1)+1| +�� |Vt| +� +≤ |Vt| − (2ed)−1 · 2−d · |Vt|/(d − 1) ≤ (1 − ǫ) · |Vt|, +for some constant ǫ = ǫ(d). Let r = log(d + 1) + 1. Applying the above inequality iteratively, +we obtain E[|Vrt|] ≤ (1 − ǫ)rn ≤ e−ǫrn. Thus for t = 3ǫ−1 ln n, E[|Vrt|] ≤ n−2, and by Markov’s +inequality, P[|Vrt| ≥ 1] ≤ n−2, which implies the lemma. +Theorem 12. The stabilization time of the 2-state MIS process on any graph G = (V, E) of +maximum degree ∆ is at most O(∆ log n) w.h.p. +Proof. We observe that if u ∈ Vt then N +(u)∩At ̸= ∅. Let u ∈ V0, and let (v1, t1), (v2, t2), (v3, t3), . . . +be a random sequence of vertex-round pairs defined as follows: Let t0 = 0. For each i ≥ 1, if +u ∈ Vti−1, then vi is an arbitrary vertex from the set N(u)∩Ati−1, and ti = min{j > ti−1 : vi /∈ Aj}; +while if u /∈ Vti−1, then (vi, ti) = (u, ti−1). +We focus on the first r = 6e∆ log n elements of the sequence above. We bound the probability +that u ∈ Vtr. For each 1 ≤ i ≤ r, the conditional probability that vi ∈ Iti+1 (and thus u /∈ Vti+1), +given vi and Bti, is at least 1/(2e∆), from Lemma 6. It follows that +P[u ∈ Vtr] ≤ (1 − 1/(2e∆))r ≤ e−r/(2e∆) = n−3. +Next, we bound the value of tr. For each 1 ≤ i ≤ r and t ≥ ti−1, if vi ∈ At then the conditional +probability that vi /∈ At+1, given (vi, ti) and Bt, is exactly 1/2 (in all cases). It follows that the +probability of tr > 4r is upper bound by the probability that a sequence of 4r fair coin tosses +contains fewer than r heads. Thus, by a Chernoff bound, +P[tr > 4r] ≤ e−(1/2)22r/2 = e−e∆ log n < n−3. +Combining the above results, we obtain that P[u /∈ V4r] ≥ P[{u /∈ Vtr} ∩ {tr ≤ 4r}] ≥ 1 − 2n−3. +(Recall that r = 6e∆ log n.) Finally, a union bound over all u ∈ V competes the proof. +8 + +4 +The 2-State MIS Process on Random Graphs +We first show some additional properties of the 2-state MIS process, which hold for any graph but +are useful only when adjacent vertices do not have many common neighbors. Then we show some +structural properties of Gn,p random graphs. Finally, we use these properties to show a poly(log n) +upper bound on the stabilization time of the 2-state MIS process on Gn,p random graphs. +4.1 +Refined Properties of the 2-State MIS Process +We call a vertex k-active if it is active and has at most k active neighbors. Let +Ak +t = {u ∈ At : |N(u) ∩ At| ≤ k} +be the set of k-active vertices at the end of round t. From Lemma 6, a k-active vertex has probability +at least Ω(1/k) to become stable black in the next O(log k) rounds. It is thus desirable to have +k-active vertices for small values k. +In this section we establish lower bounds on the probability that a given vertex u becomes +k-active at some point in the next r rounds, as a function of the probability that u is active (but +has possibly more than k active neighbors) at a point in a certain subinterval of those r rounds. +The next key lemma is the base of all the other results in the section. It lower bounds the +probability q of a white vertex u, which is non-active and non-stable, to become k-active after a +single round. The lower bound is expressed in terms of the probability p that u is active white after +two rounds. The value of k depends on the number of active neighbors of u, and, crucially, on the +number of their common neighbors with u. +Lemma 13. Suppose that u ∈ Vt \ At,8 and let θ = |N(u) ∩ N +(At ∩ N(u))| be the number of u’s +neighbors that are active or adjacent to an active neighbor of u at the end of round t. Let p be the +probability that u ∈ At+2 ∩ Wt+2, and q the probability that u ∈ Ak +t+1 where k = θ + ⌈log(1/p)⌉. +Then q ≥ pα, where α = 1/log(4/3) ≤ 2.41. +Proof. Let D = N(u) ∩ At. In round t + 1, each v ∈ D updates its state to a random state, while +each v ∈ N(u) \ D remains white. Let Z = N(u) ∩ At+1 \ N +(D) be the set of active neighbors of +u at the end of round t + 1 that are at distance at least two away from set D. Clearly, Z does not +depend on the random choices of vertices v ∈ D in round t + 1. +We have that u ∈ At+1 if and only if all v ∈ D update their state to white in round t + 1, i.e., +φt+1(v) = white.9 Also |N(u) ∩ At+1| ≤ |N(u) ∩ N +(D)| + |Z| = θ + |Z|. It follows +q ≥ (1/2)d · P[|Z| ≤ λ], +where d = |D| and λ = ⌈log(1/p)⌉. +We have that u ∈ At+2 ∩ Wt+2 only if φt+1(v) or φt+2(v) = white for every v ∈ D, and +φt+2(v) = white for every v ∈ Z. It follows that +p ≤ (3/4)d · +� +i≥0 +P[|Z| = i]/2i. +(2) +Let ε = P[|Z| ≤ λ]. Then +p ≤ (3/4)d · +� +ε + (1 − ε)/2λ+1� +≤ (3/4)d · (ε + (1 − ε) · p/2) . +8Note that u ∈ Vt \ At implies u ∈ Wt ∩ Wt+1. +9Recall the discussion about coin flips φt(v) at the beginning of Section 2.1. +9 + +This implies that p ≤ ε + (1 − ε)· p/2, thus p ≤ 2ε/(1 + ε), and substituting that above yields +p ≤ (3/4)d · (ε + (1 − ε) · ε/(1 + ε)) = (3/4)d · +2ε +1 + ε. +Finally, since (3/4)dα = (1/2)d, and for all x ∈ [0, 1], +� +2x +1+x +�α +≤ +� +2x +1+x +�2 += x · +4x +(1+x)2 ≤ x, +pα ≤ (3/4)dα · +� 2ε +1 + ε +�α +≤ (1/2)d · ε ≤ q. +Next, we use the above Lemma 13 to prove a similar result over a sequence of r rounds. For +any vertex u ∈ V and i ≥ 1, let +θu(i) = max{|N(u) ∩ N +(S)|: S ⊆ N(u), |S| ≤ i}. +(3) +Lemma 14. Suppose that u ∈ Vt \ At and let d = |N(u) ∩ At|. Let pr be the probability that +u ∈ At+1 ∪ · · · ∪ At+r, and let qr be the probability that u ∈ Ak +t+1 ∪ · · · ∪ Ak +t+r−1, where +k = θu +� +α log +� +4r +pr−2−d +�� ++ +� +log +� +4r +pr−2−d +�� +, +and α = 1/log(4/3). Then, for any r ≥ 2, qr ≥ r1−α · +� +pr−2−d +2 +�α +. +Proof. For i ≥ 0, let di = |N(u) ∩ At+i|, and define the following events: Wi is the event that +u ∈ Wt+i; Ai is the event that u ∈ At+i; Ak +i is the event that u ∈ Ak +t+i; and Hi = ¯ +A0 ∩ ¯ +A1 ∩· · ·∩ ¯ +Ai. +Let also Xi be the event that the states of the vertices at the end of round t + i are such that the +conditional probability of Ai+2 ∩ Wi+2 is at least pr−p1 +4r +. Let r ≥ 2 and λ = ⌊α log +� +4r +pr−p1 +� +⌋. Then +pr = +� +1≤i≤r +P[Ai ∩ Hi−1] += p1 + +� +2≤i≤r +P[Ai ∩ Hi−1] += p1 + +� +2≤i≤r +P[Ai ∩ Wi ∩ Hi−1] +(since Ai ∩ Hi−1 implies Wi) +≤ p1 + +� +2≤i≤r +P[Ai ∩ Wi ∩ Hi−2] +(since Hi−1 implies Hi−2) +≤ p1 + +� +2≤i≤r +P[Ai ∩ Wi ∩ Hi−2 ∩ {di−2 ≤ λ} ∩ Xi−2] ++ +� +2≤i≤r +P[Ai ∩ Wi ∩ Hi−2 ∩ {di−2 > λ}] + +� +2≤i≤r +P[Ai ∩ Wi ∩ ¯ +Xi−2]. +Each of the last two sums above is at most pr−p1 +4 +, because for each non-zero sum term, we have +P[Ai ∩ Wi ∩ Hi−2 ∩ {di−2 > λ}] ≤ P[Ai ∩ Wi | Hi−2, di−2 > λ] ≤ +�3 +4 +�λ+1 +≤ pr − p1 +4r +, +similarly to (2), and P[Ai ∩ Wi ∩ ¯ +Xi−2] ≤ P[Ai ∩ Wi | ¯ +Xi−2] ≤ pr−p1 +4r +. Applying these above gives +pr − p1 +2 +≤ +� +2≤i≤r +P[Ai ∩ Wi ∩ Hi−2 ∩ {di−2 ≤ λ} ∩ Xi−2] += +� +2≤i≤r +P[Ai ∩ Wi | Hi−2, di−2 ≤ λ, Xi−2] · P[Hi−2 ∩ {di−2 ≤ λ} ∩ Xi−2]. +10 + +Next we lower bound qr. We have +qr ≥ +� +1≤i≤r−1 +P[Ak +i ∩ Hi−1] += +� +2≤i≤r +P[Ak +i−1 ∩ Hi−2] +≥ +� +2≤i≤r +P[Ak +i−1 ∩ Hi−2 ∩ {di−2 ≤ λ} ∩ Xi−2] += +� +2≤i≤r +P[Ak +i−1 | Hi−2, di−2 ≤ λ, Xi−2] · P[Hi−2 ∩ {di−2 ≤ λ} ∩ Xi−2]. +From Lemma 13, applied for round t + i − 2, using p ≥ pr−p1 +4 +and θ ≤ θu(λ), and observing that +p1 = 2−d, we obtain +P[Ak +i−1 | Hi−2, di−2 ≤ λ, Xi−2] ≥ (P[Ai ∩ Wi | Hi−2, di−2 ≤ λ, Xi−2])α . +We substitute this to the previous equation above, and then use Jensen’s inequality to complete +the proof: Let ν = � +2≤i≤r P[Hi−2 ∩ {di−2 ≤ λ} ∩ Xi−2] ≤ r. +qr ≥ +� +2≤i≤r +� +P[Ak +i | Hi−2, di−2 ≤ λ, Xi−2] +�α +· P[Hi−2 ∩ {di−2 ≤ λ} ∩ Xi−2] +≥ ν · + + � +2≤i≤r +P[Ak +i | Hi−2, di−2 ≤ λ, Xi−2] · P[Hi−2 ∩ {di−2 ≤ λ} ∩ Xi−2]/ν + + +α +≥ ν · +�pr − p1 +2ν +�α +≥ r · +�pr − 2−d +2r +�α +. +Lemma 14 assumes that vertex u is initially not active. The next lemma shows a similar result +for the case where u is active initially. In this case, in place of the probability pr that u becomes +active at some point in the interval {t+1, . . . , t+r}, we use the probability br that u becomes black +at some point of a subinterval {t + ℓ, . . . , t + r}. The proof proceeds by considering the first round +after t when either u has at most k black neighbors, or u is white. If the first condition holds, then +u has probability 1/2 of being black, and thus of being k-active. If only the second condition holds +then we are in the case of Lemma 14. The proof can be found in Appendix A.2. +Lemma 15. Suppose that u ∈ At. Let ℓ ≥ 2 and r ≥ ℓ + 2, let br be the probability that u ∈ +Bt+ℓ ∪· · ·∪Bt+r, and suppose that br ≥ 1/2ℓ−2. Let qr be the probability that u ∈ Ak +t ∪· · ·∪Ak +t+r−1, +where +k = θu +� +α log (32r/br) +� ++ log (32r/br) + log(1/br) + 3. +Then qr ≥ r1−α · (br/16)α, where α = 1/log(4/3). +In the last lemma of this section, we consider the case in which Lemma 14 does not give a large +enough lower bound for qr, even though pr is large, because the difference pr − 2−d is small. We +proceed by essentially reducing this case to the case of Lemma 15, after a single round. The proof +is in Appendix A.3. +Lemma 16. Suppose that u ∈ Vt \ At, and let d = |N(u) ∩ At|. Let ℓ ≥ 5 and r ≥ ℓ + 2, let pr be +the probability that u ∈ At+1 ∪ · · · ∪ At+r−1, let br be the probability that u ∈ Bt+ℓ ∪ · · · ∪ Bt+r, and +11 + +suppose that br ≥ 1/2ℓ−4 and br ≥ 2(pr − 2−d). Let qr be the probability that u ∈ Ak +t ∪ · · · ∪ Ak +t+r−1, +where +k = θu +� +α log (128r/br) +� ++ log (128r/br) + log(4/br) + 3. +Then qr ≥ r1−α · (br/64)α, where α = 1/log(4/3). +4.2 +Structural Properties of Gn,p and Good Graphs +We describe some structural properties that a graph must possess in order for the analysis given in +the following sections to carry through. A graph satisfying these properties is called a good graph. +Then we show that a random Gn,p graph is a good graph w.h.p. +Definition 17 (Good Graphs). Let n be a positive integer and 0 < p < 1. A graph G = (V, E) +with n vertices is (n, p)-good if it satisfies all the following properties: +(P1) For any set S ⊆ V , the average degree of induced subgraph G[S] is at most max{8p|S|, 4 ln n}. +(P2) For any set S ⊆ V of size |S| ≥ 40 ln(n)/p, +|{u ∈ V \ S : |N(u) ∩ S| < p|S|/2}| ≤ |S|/2. +(P3) For any three disjoint sets S, T, I ⊆ V such that |S| ≥ 2|T| and (S ∪ T) ∩ N(I) = ∅, +|N(T) \ N +(S ∪ I)| ≤ |N(S) \ N +(I)| + 8 ln2(n)/p. +(P4) For any two disjoint sets S, T ⊆ V such that |S| ≥ |T| and |T| ≤ ln(n)/p, |E(S, T)| ≤ 6|S| ln n. +(P5) No two vertices u, v ∈ V have more than max{6np2, 4 ln n} common neighbors. +(P6) If p ≥ 2(ln(n)/n)1/2 then diam(G) ≤ 2. +Lemma 18. A random graph G = (V, E) drawn from Gn,p is (n, p)-good with probability 1−O(n−2). +The proof of Lemma 18 can be found in Appendix A.4. +4.3 +Analysis of the 2-State MIS Process on Gn,p +In this section, we prove the following bound on the stabilization time of the 2-state MIS process +on a random Gn,p graph. +Theorem 19. The stabilization time of the 2-state MIS process on a random graph drawn from +Gn,p, where p = O( +� +log(n)/n) or p = Ω(1/ log2.5 n), is O(log5.5 n) with probability 1 − O(n−2). +The theorem follows by combining Lemma 18 and the next lemma, which analyzes the 2-state +MIS process on a good graph. +Lemma 20. The stabilization time of the 2-state MIS process on any (n, p)-good graph G = (V, E), +where p = O( +� +log(n)/n) or p = Ω(1/ log2.5 n), is O(log5.5 n) with probability 1 − O(n−2). +It is straightforward to extend the above statements so that p ≤ poly(log n) · n−1/2 or p ≥ +1/ poly(log n), for any desired poly(log n) term, by adjusting the exponent of log n in the stabiliza- +tion time bound. +12 + +4.3.1 +Proof of Lemma 20 +We show that starting from any vector of vertex states, the process makes sufficient progress after +poly(log n) rounds, where progress is measured by the expected number of vertices that become +stable. All lemmas below assume G = (V, E) is an arbitrary (n, p)-good graph, and the probabilistic +statements assume we know the states of the vertices at the end of round t. +The first lemma +considers the case in which the number of active vertices is large, namely, |At| = Ω(log(n)/p). +Lemma 21. If |At| ≥ 80 ln(n)/p then there is a constant ǫ > 0 such that E[|Vt+log n|] ≤ (1 − ǫ)·|Vt|. +Proof. From property (P1) in Definition 17 of good graphs, the average degree of the induced +subgraph G[At] is at most k = max{8p|At|, 4 ln n} = 8p|At|. +Let S be a subset of At consisting of the |At|/2 vertices u ∈ At with the smallest degree in +G[At], i.e., for any two vertices u ∈ S and u′ ∈ At \ S, |N(u) ∩ At| ≤ |N(u′) ∩ At|. It follows that +for all u ∈ S, |N(u) ∩ At| ≤ 2k; thus S ⊆ A2k +t . Let R = {u ∈ V \ S : |N(u) ∩ S| < p|S|/2}. Since +|S| = |At|/2 ≥ 40 ln(n)/p, property (P2) in Definition 17 yields |R| ≤ |S|/2. Then the number of +vertices u ∈ Vt with |N(u) ∩ S| ≥ p|S|/2 is at least +|Vt \ (S ∪ R)| ≥ |Vt| − (|S| + |R|) ≥ |Vt| − 3|S|/2 = |Vt| − 3|At|/4 ≥ |Vt|/4. +Since each of those vertices u has at least p|S|/2 neighbors in S ⊆ A2k +t , Lemma 7 gives that the +probability at least one neighbor of u is stable black (and thus u is also stable) at the end of round +t + log n is at least +(1/5) · min +� +1, (p|S|/2) · (4k)−1� += (1/5) · min +� +1, (p|At|/4) · (32p|At|)−1� += 1/640. +Then the expected number of vertices that are not stable at the end of round t + log n is +E[|Vt+log n|] ≤ |Vt| − (|Vt|/4) · 1/640 ≤ |Vt| − |Vt|/2560. +The next lemma considers the case in which the number of vertices that are not stable is large, +namely |Vt| = Ω(ln2(n)/p), and |At| = O(ln(n)/p). +Lemma 22. If |Vt| ≥ 10 ln2(n)/p and |At| ≤ 80 ln(n)/p then there is a constant ǫ > 0 such that +E[|Vt+log n|] ≤ (1 − ǫ/ ln n) · |Vt|. +Proof. From property (P1) in Definition 17, the average degree of graph G[At] is at most +k = max{8p|At|, 4 ln n} ≤ 640 ln n. +Let S be a subset of At consisting of the 2|At|/3 vertices u ∈ At with the smallest degree in G[At], +and let T = At \ S. Then for all u ∈ S, |N(u) ∩ At| ≤ 3k; thus S ⊆ A3k +t . +The set Vt consist of (i) all the active vertices, u ∈ At = S ∪ T, and (ii) all the non-active +vertices that are not in N +(It) (these vertices are white and have at least one active neighbor). We +can thus partition Vt into the four distinct sets: S, N(S)\N(It), T \N(S), and N(T)\N +(S ∪It). +For the sizes of these sets, we have |T \ N(S)| ≤ |T| < |S| and, by property (P3) in Definition 17, +|N(T) \ N +(S ∪ It)| ≤ |N(S) \ N(It)| + 8 ln2(n)/p. +Using these two inequalities, the fact that the sizes of the four sets above sum to |Vt|, and the +assumption |Vt| ≥ 10 ln2(n)/p, we obtain +|S| + |N(S) \ N(It)| ≥ (|Vt| − 8 ln2(n)/p)/2 ≥ |Vt|/10. +13 + +Therefore, at least |Vt|/10 vertices u ∈ Vt are in S or adjacent to a vertex from S. From Lemma 6, +each u ∈ S ⊆ A3k +t +is stable black (and all its neighbors are stable white) at the end of round +t + log n, with probability at least 1/(6ek). It follows that +E[|Vt+log n|] ≤ |Vt| − (|Vt|/10) · 1/(6ek) ≤ |Vt| − |Vt|/(1.1 · 105 ln n). +In the next lemma we analyze the remaining case, in which |Vt| = O(ln2(n)/p) and |At| = +O(ln(n)/p). In fact, the lemma does not require a bound on |At|. Unlike the previous lemmas, +however, it requires that p = O( +� +log(n)/n). +Lemma 23. If |Vt| ≤ 10 ln2(n)/p and p ≤ c +� +log(n)/n, for some constant c > 0, then there is a +constant ǫ = ǫ(c) > 0 such that E[|Vt+2 log n|] ≤ +� +1 − ǫ/ ln3.5 n +� +· |Vt|.10 +Proof. From property (P1) in Definition 17, the average degree of graph G[Vt] is at most +k = max{8p|Vt|, 4 ln n} ≤ 80 ln2 n. +Let T be a subset of Vt consisting of the min{ln(n)/p, |Vt|/2} vertices u ∈ Vt with the largest degree +in G[Vt], and let S = Vt \ T. Then |S| ≥ |T|, and for all u ∈ S, |N(u) ∩ Vt| is at most +d = k|Vt|/|T| ≤ k · max{p|Vt|/ ln n, 2} ≤ 800 ln3 n. +From property (P4) in Definition 17, the number of edges between S and T is |E(S, T)| ≤ 6|S| ln n. +Let R = {u ∈ S : |N(u)∩T| ≤ 12 ln n}. Then |R| ≥ |S|/2 ≥ |Vt|/4. We will show for some constant +ǫ′ = ǫ′(c) that +P[u /∈ Vt+2 log n] ≥ ǫ′ ln−α−1 n · (ln ln n)−α, +for all u ∈ R. +(4) +It follows that E[|Vt+2 log n|] ≤ |Vt| − (|Vt|/4) · ǫ′ ln−α−1 n · (ln ln n)−α. Since α = 1/log(4/3) ≤ 2.41, +the above implies the lemma. To complete the proof it remains to show (4). +Let u ∈ R. We partition the neighbors of u in G[Vt] into sets N(u) ∩ S and N(u) ∩ T, and let +x = P[N(u) ∩ S ∩ A ̸= ∅] +and +y = P[N(u) ∩ T ∩ B ̸= ∅], +where A = At ∪ · · · ∪ At+r−2, B = Bt+r−2 ∪ Bt+r−1 ∪ Bt+r, and r = log(48 ln n) + 6. We distinguish +the following three cases: x + y ≤ 1/2, x ≥ 1/4, and y ≥ 1/4. +Case x + y ≤ 1/2: With probability at least 1 − (x + y) ≥ 1/2, we have N(u) ∩ S ∩ A = ∅ +and N(u) ∩ T ∩ B = ∅. If N(u) ∩ S ∩ A = ∅ then N(u) ∩ S ⊆ Wt+r−2 (it is easy to see that +N(u) ∩ S ∩ It+r−2 = ∅). Similarly, if N(u) ∩ T ∩ B = ∅, it is immediate that N(u) ∩ T ⊆ Wt+r−2. +Thus, with probability at least 1/2, we have that N(u) ⊆ Wt+r−2. If N(u) ⊆ Wt+r−2, then either +u ∈ At+r−2 ∩ Wt−r−2 or u ∈ It+r−2. Therefore, with probability at least 1/2, either u /∈ Vt+r−2 +or u ∈ At+r−2. If u ∈ At+r−2, then u ∈ Ad +t+r−2 since |N(u) ∩ Vt| ≤ d, and from Lemma 6, the +probability that u ∈ It+r−2+log n is at least (2ed)−1. Combining the last two statements yields that +the probability of u /∈ Vt+r−2+log n is at least (1/2) · (2ed)−1 ≥ (8700 ln3 n)−1, which implies (4). +Case x ≥ 1/4: With probability at least 1/4, there is a pair v, j such that v ∈ N(u) ∩ S, +0 ≤ j ≤ r − 2, and v ∈ At+j. And if v ∈ At+j then v ∈ Ad +t+j since |N(v) ∩ Vt| ≤ d, and from +Lemma 6, the probability that v ∈ It+j+log n is at least (2ed)−1. We conclude that the probability +that u ∈ N +(It+r−2+log n) is at least (1/4) · (2ed)−1 ≥ (17400 ln3 n)−1, which implies (4). +Case y ≥ 1/4: There exists some v∗ ∈ N(u) ∩ T such that +P[v∗ ∈ B] ≥ y/|N(u) ∩ T| ≥ (4 · 12 ln n)−1 = (48 ln n)−1. +10If c is super constant, then the proof gives E[|Vt+2 log n|] ≤ +� +1 − ǫ/(c2 ln3.5 n) +� +· |Vt|. +14 + +If v∗ ∈ At then we can apply Lemma 15, for ℓ = r − 2 ≥ log(48 ln n) + 2 and br ≥ (48 ln n)−1, to +obtain that v∗ ∈ Aλ +t ∪ · · · ∪ Aλ +t+r−1 with probability at least q = r1−α · (16 · 48 ln n)−α, where +λ = θv∗� +α log (32r · 48 ln n) +� ++ log (32r · 48 ln n) + log(48 ln n) + 3. +Suppose now that v∗ ∈ Vt \ At, and let +p∗ = P[v∗ ∈ At+1 ∪ · · · ∪ At+r−1] − 2−|N(v∗)∩At|. +If p∗ ≥ P[v∗ ∈ B]/2 ≥ (96 ln n)−1, then we can apply Lemma 14 (using r − 1 in place of r), to +obtain that v∗ ∈ Aλ′ +t ∪· · ·∪Aλ′ +t+r−2 with probability at least q′ = r1−α ·(p∗/2)α ≥ r1−α·(192 ln n)−α, +where +λ′ = θv∗� +α log (4r · 96 ln n) +� ++ log (4r · 96 ln n) . +If p∗ < P[v∗ ∈ B]/2, then we can apply Lemma 16, for ℓ = r − 2 ≥ log(48 ln n) + 4 and br = +P[v∗ ∈ B] ≥ (48 ln n)−1, to obtain that that v∗ ∈ Aλ′′ +t +∪ · · · ∪ Aλ′′ +t+r−1 with probability at least +q′′ = r1−α · (64 · 48 ln n)−α, where +λ′′ = θv∗� +α log (128r · 48 ln n) +� ++ log (128r · 48 ln n) + log(4 · 48 ln n) + 3. +In all the settings above, we have q, q′, q′′ ≥ ε ln−α n · (ln ln n)1−α and λ, λ′, λ′′ ≤ β ln n · ln ln n, for +some constants ε, β > 0, where the bound on λ, λ′, λ′′ holds because property (P5) and assumption +p ≤ c +� +log(n)/n imply that for any v ∈ V , θv(i) ≤ i · (6c2 + 4) log n (recall the definition of θv +from (3)). Therefore, the probability that v∗ is (β · ln n · ln ln n)-active at the end of some round +in {t, . . . , t + r − 1} is at least ε · ln−α n · (ln ln n)1−α, and from Lemma 6, the probability that +v∗ ∈ It+r−1+log n is at least ε · ln−α n · (ln ln n)1−α · (2eβ · ln n · ln ln n)−1. Thus with at least that +probability we have u /∈ Vt+r−1+log n. This completes the proof of (4). +Putting the Pieces Together. +First, suppose that p ≤ c +� +log(n)/n for some constant c > 0. +From Lemmas 21 to 23, E[|Vt+2 log n|] ≤ +� +1 − ǫ/ ln3.5 n +� +· E[|Vt|], for any t ≥ 0. Iteratively applying +this inequality, we obtain that for any i ≥ 0, +E[|V2i log n|] ≤ +� +1 − ǫ/ ln3.5 n +�i · n. +Substituting i = 3 ln4.5 n/ǫ yields E +� +|V(6/ǫ) log n·ln4.5 n| +� +≤ n−2, and by Markov’s inequality, it follows +P[|V(6/ǫ) log n·ln4.5 n| ≥ 1] ≤ n−2. +If p ≥ ε/ ln2.5 n for some constant ε > 0, then we use Lemmas 21 and 22 as above to obtain that +P[|Vt| ≥ 10 ln2(n)/p] ≤ n−2 for some t = O(log3 n). We also observe that if |Vt| < 10 ln2(n)/p then +the maximum degree of graph (V, E(Vt)) is ∆ < |Vt| ≤ 10 ln2(n)/p ≤ 10 ln4.5(n)/ε, and Theorem 12 +yields a bound of O(∆ log n) = O(log5.5 n). Combining the two completes the proof of Lemma 20. +Remark 24. Some of the logarithmic factors can be shaved off with a more careful analysis. For +example, using a “pipelining” argument, one could improve the bound on halving |Vt| obtained +from Lemma 23, from O(log n · ln3.5 n) to O(log n + ln3.5 n), thus saving one logarithmic factor. +5 +Logarithmic Switch and the 3-Color MIS Process +We present an extension of the 2-state MIS process, called 3-color MIS process, which uses one +additional color, grey, and includes also a sub-process, called logarithmic switch, which runs in +parallel to the main process. Then we analyze the 3-color MIS process on Gn,p random graphs. +15 + +5.1 +The Logarithmic Switch Process +We first introduce an abstract logarithmic switch process, by specifying its properties. Then we +describe an actual randomized graph process that satisfies these properties with high probability +and in a self-stabilizing manner, using 6 states per vertex. +Definition 25 (Logarithmic Switch Process). An (a, b)-logarithmic switch process on G = (V, E) +generates for each vertex u ∈ V a binary sequence σ0(u), σ1(u), . . . , where σt(u) ∈ {on, off} for +each t ≥ 0, such that the following properties hold for all u ∈ V . +(S1) Every run of consecutive off values in sequence σ0(u), σ1(u), . . . has length at most a ln n. +(S2) If diam(G) ≤ 2 then every run of consecutive off values in sequence σt(u), σt+1(u), . . . has +length at least a +6 ln n, where t = min{i ≥ a +6 ln n: σi(u) = on}. +(S3) If diam(G) ≤ 2 then every run of consecutive on values in sequence σt(u), σt+1(u), . . . has +length at most b, where t is some constant independent of n. +Definition 26 (Randomized Logarithmic Switch). In the randomized logarithmic switch process on +G = (V, E), each vertex u ∈ V has a state, called level, that takes on values in the set {0, 1, . . . , 5}. +The initial value level0(u) of u can be arbitrary, and in each round t ≥ 1 the level of u is updated +according to the following rule, which uses a global parameter 0 < ζ < 1. +if level t−1(u) = 5 then +choose a random bit bt(u) such that P[bt(u) = 0] = ζ +end +if (level t−1(u) = 5 and bt(u) = 1) or level t−1(u) = 0 then +set level t(u) = 5 +else set level t(u) = max{level t−1(v): v ∈ N +(u)} − 1 +Finally, we define the following mapping of the levels to the binary on/off values of Definition 25. +For each u ∈ V and t ≥ 0, +σt(u) = +� +on +if levelt(u) ≤ 2 +off +if levelt(u) ≥ 3. +Lemma 27. For any graph G = (V, E), the randomized logarithmic switch process with parameter +0 < ζ ≤ 1/2 satisfies properties (S1) to (S3) for a = 4/ζ and b = 3, with probability 1 − O(n−2), +during the first n rounds. +Proof. Let u ∈ V , and let Sv ⊆ V be the set of vertices at distance at most 2 from u. If u has level +at least 3 in all rounds t, . . . , t+a ln n, then no vertex v ∈ Su has level 0 in rounds t+2, . . . , t+a ln n; +and at least one vertex v ∈ Su must be at level 5 in all rounds t + 2, . . . , t + a ln n − 2. It follows +that the probability there is some u ∈ V and t ≤ n such that u has level at least 3 in all rounds +t, . . . , t + a ln n is at most +n2(1 − ζ)a ln n−4 ≤ n2−aζ/(1 − ζ)4 ≤ 16 · n−2, +when aζ = 4. Thus, property (S1) holds with probability at least 1 − O(n−2). +Next we assume diam(G) ≤ 2. The rest of the proof is similar to that in [12]. Observe that +there must be a vertex v and a round t∗ ≤ 5 such that levelt∗(u) = 5. And from the end of round +t∗ + 2, all vertices “synchronize” in the sense that once a vertex reaches level 2 in a round, all +vertices reach level 2 in that round, then the they all reach level 1 in the next round, then level +0, and then 5. It follows that property (S3) holds for b = 3, starting from round t∗ + 2 ≤ 7. The +property holds with probability 1, and for all rounds after round t∗ + 2, not just for the first n. +16 + +As mentioned above, after vertices have synchronized, all n vertices move from level 0 to level +5 simultaneously, each time. When that happens, the number of rounds until there are no vertices +left at level 5 is greater than a ln n − 6 with probability at most +n(1 − ζ)a ln n−6 ≤ 64 · n−3, +as before; and is smaller than r = a +6 ln n with probability at most +(1 − (1 − ζ)r)n ≤ e−n(1−ζ)r ≤ e−n4−ζr = e−n4−(aζ/6) ln n ≤ e−n0.07 = O(n−3). +Combining the above, using a union bound, we obtain that property (S2) holds with probability +1 − O(n−2). +5.2 +The 3-Color MIS Process +We now define the 3-color MIS process, which is an extensions of the 2-state MIS process. +Definition 28 (3-Color MIS Process). The process consists of two (sub-)processes that run in +parallel on G = (V, E). The first is an (a, 3)-logarithmic switch process, where a = 512, which gen- +erates a value σt(u) ∈ {on, off} for each vertex u ∈ V in each round t ≥ 0. The second is a variant +of the 2-state MIS process, where each vertex u ∈ V has a state ct(u) ∈ {black, white, gray}, c0(u) +can be arbitrary, and in each round t ≥ 1, u’s state is updated as follows. +let NC t(u) = {ct−1(v): v ∈ N(u)} +if ct−1(u) = black and NC t(u) ∋ black then +let ct(u) be a uniformly random state from {black, gray} +else if ct−1(u) = white and NC t(u) ̸∋ black then +let ct(u) be a uniformly random state from {black, white} +else if ct−1(u) = gray and σt−1(u) = on then +set ct(u) = white +else set ct(u) = ct−1(u) +There are precisely two differences in the update rule above compared to that for the 2-state +MIS process: a black vertex with a black neighbor changes to gray with probability 1/2, rather +than to white; and a gray vertex changes to white if its switch value is on. Note that a gray vertex +is treated similarly to a non-active white vertex. +A vertex is stable, if it is black and has no black neighbors, or it is not black and has a neighbor +that is stable black. Other than that, the remaining definitions and notations are the same as in +the 2-state MIS process, namely, of active vertices, stabilization times, Bt, Wt, At, Ak +t , It, and +Vt. We also let Γt = V \ (Bt ∪ Wt) denote the set of gray vertices at the end of round t. +The definition of the 3-color MIS process above assumes an arbitrary logarithmic switch process. +We can use the randomized logarithmic switch from Definition 26, which uses 6 states per vertex, +to obtain a 3-color MIS process that uses 6 · 3 = 18 states in total. The probability parameter of +the randomized switch is ζ = 4/a = 27, thus at most 7 random bits are required per round for each +vertex (plus one more for each active vertex). +We note that Lemmas 6 and 7 and their proofs carry over to the 3-color MIS process, without +changes. We will use also the two simple lemmas below that are specific to the 3-color MIS process. +Recall that a = 512 is a parameter of the logarithmic switch. +Lemma 29. If t ≥ a ln n and u ∈ Γt then u ∈ At−a ln n ∪ · · · ∪ At−1. +17 + +Proof. By property (S1) in Definition 25 of the logarithmic switch, a vertex is gray for at most +a ln n consecutive rounds. Also if a vertex becomes gray in round j > 0, it must be active black at +the end of round j − 1. Combining these two facts implies the lemma. +Lemma 30. If diam(G) ≤ 2, u ∈ V , t ≥ a +6 ln n, and t′ = t + a +6 ln n, then the expected number of +times u is active black between rounds t and t′ is E [|{j : u ∈ Bj ∩ Vj} ∩ {t, . . . , t′}| | Bt, Wt] ≤ 4. +Proof. From properties (S2) and (S3) in Definition 25, it is easy to see that sequence ct(u), . . . , ct′(u) +contains at most two runs of consecutive black states. Moreover, the expected length of the prefix +of each black run until u becomes stable black or the run finishes (and u becomes gray) is 2. It +follows that u is non-stable black in at most 4 rounds in expectation. +Lemmas 13 and 14 hold also for the 3-color MIS process, when u ∈ Vt \ (At ∪ Γt) and thus +u ∈ Wt.11 The next simple lemma will be used together with Lemma 14. +Lemma 31. Let t ≥ 0, u ∈ V , and d > 0. Let t′ ≥ t be the first round when either u is white and +has at least d black neighbors, or u is stable. The expected number of rounds t < j < t′ at which u +is black and has at least d black neighbors is at most 3. +Proof. The lemma is obtained using the observations that: each time u’s state changes from white +to black, it is equally likely that it remained white; and, when u is active black, it becomes gray in +the next step with probability 1/2. +5.3 +Analysis of the 3-Color MIS Process on Gn,p +We show that the stabilization time of the 3-color MIS process on Gn,p random graphs is poly(log n), +for the complete range of values of p. +Theorem 32. The stabilization time of the 3-color MIS process on a random graph drawn from +Gn,p is O(log6 n) with probability 1 − O(n−2). +As before, it suffices to show that the above bound holds for good graphs, and apply Lemma 18. +Lemma 33. The stabilization time of the 3-color MIS process on any (n, p)-good graph G = (V, E) +is O(log6 n) with probability 1 − O(n−2). +5.3.1 +Proof of Lemma 33 +The proof strategy is similar to Lemma 20’s: From any vector of vertex states at the end of round +t, we show that the process makes sufficient progress in expectation in poly(log n) rounds. The +main difference is that now we show that this is also true even in the case of |Vt| = O(log2(n)/p) +when diam(G) ≤ 2, which corresponds to the case of p = Ω( +� +log(n)/n), by property (P6) in +Definition 17. This is precisely the case that we could not handle in the analysis of the 2 state MIS +process. The relevant lemma is Lemma 36. +We first observe that Lemma 21, which considers that case of |At| = Ω(log(n)/p), carries over +to the 3-color MIS process, without any changes in the proof. +Next we consider the case where |At| = O(ln(n)/p), |Vt| = Ω(ln2(n)/p), and |Γt| = O(ln2(n)/p). +The following lemma is very similar to Lemma 22, except that it requires also a bound on |Γt|. +Recall that a = 512 is a parameter of the logarithmic switch. +11The proofs require just minor modifications, mostly replacing some occurrences of “white” by “not black” or +“gray”. +18 + +Lemma 34. If |Vt| ≥ 82a ln2(n)/p, |At| ≤ 80 ln(n)/p, and |Γt| ≤ 80a ln2(n)/p, then there is a +constant ǫ > 0 such that E[|Vt+log n|] ≤ (1 − ǫ/ ln n) · |Vt|. +Proof. The proof is very similar to Lemma 22’s. As before, from property (P1) in Definition 17, +the average degree of G[At] is at most k = max{8p|At|, 4 ln n} ≤ 640 ln n. We let S be a subset of +At consisting of the 2|At|/3 vertices u ∈ At with the smallest degree in G[At], and let T = At \ S. +Then for all u ∈ S, |N(u) ∩ At| ≤ 3k, thus S ⊆ A3k +t . +The set Vt consist of (i) all active vertices, u ∈ At = S ∪T, (ii) all non-active non-stable vertices +that have some active neighbor, and (iii) all non-active non-stable vertices have no active neighbors +(these vertices are gray). We can thus partition Vt into the five distinct sets: S, N(S) \ N(It), +T \ N(S), N(T) \ N +(S ∪ It), and Vt \ N +(T ∪ Sf) ⊆ Γt. We have that |Vt \ N +(T ∪ S)| ≤ |Γt| ≤ +80a ln2(n)/p, |T \ N(S)| ≤ |T| < |S|, and, by property (P3) in Definition 17, +|N(T) \ N +(S ∪ It)| ≤ |N(S) \ N(It)| + 8 ln2(n)/p. +Using these three inequalities, the fact that the sizes of the five sets above sum to |Vt|, the assump- +tion |Vt| ≥ 82a ln2(n)/p, and that a ≥ 8, we obtain +|S| + |N(S) \ N(It)| ≥ (|Vt| − (80a + 8) ln2(n)/p)/2 ≥ (|Vt| − 81a ln2(n)/p)/2 ≥ |Vt|/82a. +Therefore, at least |Vt|/82a vertices u ∈ Vt are in S or adjacent to S. And, from Lemma 6, each +u ∈ S ⊆ A3k +t +is in It+log n, with probability at least 1/(6ek). It follows +E[|Vt+log n|] ≤ |Vt| − (|Vt|/82a) · 1/(6ek) ≤ |Vt| − |Vt|/(1.1 · 82a · 104 ln n). +Next we assume |At| = O(ln(n)/p) and |Vt| = Ω(ln2(n)/p), as in the previous lemma, but now +|Γt| = Ω(ln2(n)/p). We reduce this case to the previous cases using Lemma 29. +Lemma 35. If |Vt| ≥ 83a ln2(n)/p, |At| ≤ 80 ln(n)/p, and |Γt| > 80a ln2(n)/p, then there is a +constant ǫ > 0 such that E[|Vt+a ln n+log n|] ≤ (1 − ǫ/ ln n) · |Vt|. +Proof. Let τ = min{j ≥ t: |Vj| ≤ 82a ln2(n)/p or |Aj| ≥ 80 ln(n)/p or |Γj| ≤ 80a ln2(n)/p}. We +have τ ≤ t + a ln n, because if |Γt+a ln n| > 80a ln2(n)/p, then Lemma 29 implies there is some +j ∈ {t, . . . , t + a ln n − 1} such that |Aj| ≥ |Γt+a ln n|/(a ln n) ≥ 80 ln(n)/p. We distinguish three +cases depending on which condition in the definition of τ is satisfied first. If |Vτ| ≤ 82a ln2(n)/p, +then +|Vt+a ln n| ≤ |Vτ| ≤ 82a ln2(n)/p ≤ (1 − 1/83) · |Vt|. +If |Aτ| ≥ 80 ln(n)/p, then Lemma 21 yields E[|Vt+a ln n+log n|] ≤ E[|Vt+τ+log n|] ≤ (1 − ǫ) · |Vt|. +Last, if |Γτ| ≤ 80a ln2(n)/p and the other two conditions do not hold, then Lemma 34 gives +E[|Vt+a ln n+log n|] ≤ (1 − ǫ/ ln n) · |Vt|. +The next two lemmas deal with the case of |Vt| = O(ln2(n)/p). The first one assumes diam(G) ≤ +2, and thus covers the case of p = Ω( +� +log(n)/n), by property (P6) in Definition 17; while the second +lemma assumes p = O( +� +log(n)/n) and is similar to Lemma 23. +Lemma 36. For any t ≥ a +6 ln n, if |Vt| ≤ 83a ln2(n)/p and diam(G) ≤ 2 then there is a constant +ǫ > 0 such that E[|Vt+ 7 +6 a log n+log n|] ≤ +� +1 − ǫ/ ln3 n +� +· |Vt|. +Proof. From property (P1) in Definition 17, the average degree of induced subgraph G[Vt] is at most +k = max{8p|Vt|, 4 ln n} ≤ 664a ln2 n. Let T be a subset of Vt consisting of the min{ln(n)/p, |Vt|/2} +19 + +vertices u ∈ Vt with the largest degree in G[Vt], and let S = Vt \ T. Then |S| ≥ |T|, and all u ∈ S, +|N(u) ∩ Vt| is at most +d = k|Vt|/|T| ≤ k · max{p|Vt|/ ln n, 2} ≤ 55112 ln3 n. +From property (P4) in Definition 17, |E(S, T)| ≤ 6|S| ln n. Let R = {u ∈ S : |N(u) ∩ T| ≤ 12 ln n}. +Then |R| ≥ |S|/2 ≥ |Vt|/4. We will show that, for some constant ǫ′ > 0, +P[u /∈ Vt+ 7 +6 a ln n+log n] ≥ ǫ′ ln−3 n, +for all u ∈ R. +(5) +From this, it follows that E[|Vt+ 7 +6 a ln n+log n|] ≤ |Vt| − (|Vt|/4) · ǫ′ ln−3 n. To complete the proof of +the lemma it remains to prove (5). +Let u ∈ R, and suppose that u /∈ Γt (we deal with the case u ∈ Γt at the end). From Lemma 30, +the expected value of � +t≤j≤t+ a +6 ln n |(N(u) ∩ T) ∩ (Bj ∩ Vj)|, that is, the total number of times that +vertices v ∈ N(u)∩T are active black between rounds t and a +6 ln n, is at most 4·|N(u)∩T| ≤ 4·12 ln n. +Then, by Markov’s inequality, that number is at most 5 · 12 ln n with probability at least 1/5. And +since a +6 > 5 · 12, it follows that, with probability at least 1/5, there is some j ∈ {t, . . . , t + a +6 ln n} +such that (N(u) ∩ T) ∩ (Bj ∩ Vj) = ∅. +Next we claim that, if (N(u) ∩ T) ∩ (Bj ∩ Vj) = ∅ for some j ≥ t, then (i) u /∈ Vj, or (ii) u ∈ Aj′ +for some t ≤ j′ < j, or (iii) (N +(u) ∩ S) ∩ Aj ̸= ∅. Indeed, suppose that (i) and (ii) do not +hold, i.e., u ∈ Vj and u /∈ At ∪ · · · ∪ Aj−1. +From u ∈ Vj, it follows N +(u) ∩ Ij = ∅. +From +u /∈ At ∪ · · · ∪ Aj−1 and the assumption u /∈ Γt, it follows u ∈ Wj. Then, if (N(u) ∩ S) ∩ Bj ̸= ∅, +each vertex v ∈ (N(u) ∩ S) ∩ Bj is in Aj; while if (N(u) ∩ S) ∩ Bj = ∅, then N(u) ∩ Bj = ∅ and +u ∈ Aj. Therefore (iii) holds. +From the above, it follows that with probability at least 1/5, there is some t ≤ j ≤ t + a +6 ln n +such that u /∈ Vj or (N +(u) ∩ S) ∩ Aj ̸= ∅. And if v ∈ (N +(u) ∩ S) ∩ Aj, then v ∈ Ad +j, and from +Lemma 6, the probability that v ∈ Ij+log n is at least 1/(6ed). We conclude that +P[u /∈ Vt+ a +6 ln n+log n] ≥ (1/5) · 1/(6ed) ≥ (4.5 · 106 ln3 n)−1, +which implies (5). +Finally, if u /∈ Γt, we consider the first round j > t such that u /∈ Γj. From property (S1), +j ≤ t + a ln n. Then we apply the result for the previous case to complete the proof of (5). +Lemma 37. If |Vt| ≤ 83a ln2(n)/p and p ≤ c +� +log(n)/n for some constant c > 0, then there is a +constant ǫ = ǫ(c) > 0 such that E[|Vt+log1.1 n|] ≤ +� +1 − ǫ/ ln3.9 n +� +· |Vt|. +Proof Sketch. We define the set S, T, R and the degree thresholds k, d as in the proof of Lemma 36, +and we show +P[u /∈ Vt+log1.1 n] ≥ ǫ′ ln−3.9 n, +for all u ∈ R, +(6) +which implies the lemma. Next we prove (6). +Let u ∈ R, and suppose that u /∈ Γt (we deal with case u ∈ Γt at the end). For each v ∈ N(u)∩T, +let tv ≥ t be the first round when either v is white and has at least ℓ = ln n black neighbors, or is +stable; and let xv be the number of rounds t ≤ j ≤ min{tv, t+r} at which v is black and has at least +ℓ black neighbors, where r = 12 ln n · ln2 ln n. From Lemma 31, the probability that xv ≤ ln2 ln n +for all v is at least 1 − |N(u) ∩ T| · e−Ω(ln2 ln n) = 1 − e−ω(ln ln n). For each v ∈ N(u) ∩ T let pv be +the conditional probability that v ∈ Btv+1 ∪ · · · ∪ Bt+r, given Bt, Wt. +If � +v∈N(u)∩T pv ≤ 1/2, then with probability at least 1/2 − e−ω(ln ln n) > 1/3, the total number +of rounds in which at least one v ∈ N(u) ∩ T is black and has at least ℓ black neighbors is at +20 + +most |N(u) ∩ T| · ln2 ln n ≤ 12 ln n · ln2 ln n ≤ r, thus there is some j ∈ {t, . . . , t + r} such that no +v ∈ N(u) ∩ T is black and has at least ℓ black neighbors. Then we can infer that with probability +at least 1/3 some vertex in N +(u) is stable black or is d-active at some round in {t, . . . , t + r}, in +the same way as in the proof of Lemma 36, and then obtain (6) using Lemma 6. +If � +v∈N(u)∩T pv > 1/2, then there is some v∗ ∈ N(u) ∩ T such that pv∗ ≥ (2|N(u) ∩ T|)−1 ≥ +(24 ln n)−1. We can then apply Lemma 14 to v∗ at round tv∗ to show that the probability vertex +v∗ is z-active at some round in {t, . . . , t + r}, where z = θu +� +α log +4r +pv∗−2−ℓ +� ++ log +4r +pv∗−2−ℓ = O(log n · +log log n), is at least r1−α· +� +pv∗−2−ℓ +2 +�α += Ω(ln3.9 n), as α ≤ 2.41 Again we obtain (6) using Lemma 6. +Finally, as before, if u /∈ Γt, we consider the first round j > t such that u /∈ Γj, and apply the +result for the previous case to complete the proof of (6). +We can now conclude the proof of Lemma 33, as we did for Lemma 20. From Lemmas 21 and 34 +to 37, we have that for any t ≥ a +6 ln n, E[|Vt+log1.1 n|] ≤ +� +1 − ǫ/ ln3.9 n +� +· E[|Vt|]. Iteratively applying +this inequality, and using by Markov’s inequality, we obtain as before P[|Vc′ ln6 n| ≥ 1] ≤ n−2, for a +large enough constant c′ > 0. This completes the proof of Lemma 33. +APPENDIX +A +Omitted Proofs +A.1 +Proof of Lemma 7 +We assume k1 ≤ k2 ≤ · · · ≤ kℓ. For 1 ≤ i ≤ ℓ, let ri = ⌈log(ki + 1)⌉, let Ei be the event that +φt+1(ui) = · · · = φt+ri(ui) = black, and let E = � +i Ei. Then +P[E] = 1 − +� +i +� +1 − 1 +2ri +� +≥ 1 − +� +i +� +1 − 1 +2ki +� +≥ +� +1 − e− � +i +1 +2ki +� +≥ +� +1 − e−1� +· min +� +1, +� +i +1 +2ki +� +. +Suppose that E occurs and let j be the smallest index such that Ej occurs, i.e., ¯E1 ∩ · · · ∩ ¯Ej−1 ∩ Ej +occurs. If gj = |N(uj) ∩ {u1, . . . , uj−1}|, then the probability that none of the kj vertices v ∈ +N(uj) ∩ At satisfies φt+1(v) = · · · = φt+rj(v) = black is +(1 − 2−rj)kj−gj ≥ (1 − 2−rj)kj ≥ e−1, +similarly to (1). Combining this with the previous inequality we obtain that the probability that +ui ∈ It+ri for at least one vertex ui ∈ {u1, . . . , uℓ} is at least e−1 · +� +1 − e−1� +· min +� +1, � +i +1 +2ki +� +≥ +1 +5 · min +� +1, � +i +1 +2ki +� +. +A.2 +Proof of Lemma 15 +Let B be the event u ∈ Bt+ℓ ∪ · · · ∪ Bt+r; then P[B] = br. Let +τ = min{j > t: u ∈ Wj or |N(u) ∩ Bj| ≤ k} +be the first round j > t at the end of which u is white or has at most k black neighbors. We +have P[τ > t + ℓ] ≤ 2ℓ ≤ br/4, since τ > t + ℓ implies φt+1(u) = · · · = φt+ℓ(u) = black. Thus +P[τ ≤ t + ℓ] ≥ 1 − br/4. Let +x = P[|N(u) ∩ Bτ| ≤ k | τ ≤ t + ℓ]. +21 + +We distinguish two cases, x ≥ br/4 and x ≤ br/4. +First suppose that x ≥ br/4. For any given j > t, +P[u ∈ Aj | τ = j, |N(u) ∩ Bτ| ≤ k] = 1/2. +The reason is that u ∈ Bj−1 and |N(u) ∩ Bj−1| > k > 0 if τ = j > t + 1, and u ∈ At = Aj−1 if +τ = j = t+1. In either case u ∈ Aj−1, thus the state of u at the end of round j is chosen uniformly +at random, independently of the remaining choices in round j. +In particular, u is black with +probability 1/2 when 0 < |N(u)∩Bj| ≤ k, and is white with probability 1/2 when |N(u)∩Bj| = 0. +It follows that +P[{u ∈ Aτ} ∩ {|N(u) ∩ Bτ| ≤ k} ∩ {τ ≤ t + ℓ}] ≥ (1/2) · x · (1 − br/4) ≥ 3br/32. +Since the event on the left side implies that u is k-active at the end of round τ ≤ t + ℓ < t + r, and +3br/32 is greater than the desired lower bound for qr, the lemma holds in this case. +Suppose now that x ≤ br/4. Then +P[B ∩ {|N(u) ∩ Bτ| > k} ∩ {τ ≤ t + ℓ}] ≥ P[B] − P[τ > t + ℓ] − x ≥ br − br/4 − br/4 = br/2. +If τ ≤ t+ℓ and |N(u)∩Bτ| > k (and thus u ∈ Wτ by τ’s definition), we define the following events: +Ak is the event that u ∈ Ak +τ+1 ∪ · · · ∪ Ak +t+r−1; A is the event that u ∈ Aτ+1 ∪ · · · ∪ At+r−1; and X is +the event that the states of vertices at the end of round τ are such that the conditional probability +of A, given these states and τ, is at least br/4. +If τ ≤ t + ℓ and |N(u) ∩ Bτ| > k, then event B implies A, because vertex u, which is non-active +white at the end of round τ, cannot become black before becoming active first. Thus, from the last +inequality above, it follows +P[A ∩ {|N(u) ∩ Bτ| > k} ∩ {τ ≤ t + ℓ}] ≥ br/2. +Also +P[A ∩ X ∩ {|N(u) ∩ Bτ| > k} ∩ {τ ≤ t + ℓ}] ≥ br/2 − br/4 = br/4. +We can now apply Lemma 14, starting from round τ ≤ t + ℓ, using d > k ≥ log(1/br) + 3 and +pr ≥ br/4, to obtain +P[Ak ∩ X ∩ {|N(u) ∩ Bτ| > k} ∩ {τ ≤ t + ℓ}] ≥ r1−α · +�br/4 − 2k +2 +�α +≥ r1−α · +�br/4 − br/8 +2 +�α +. +It follows that qr = P[Ak] ≥ r1−α · +� +br/4−br/8 +2 +�α +, which concludes the proof of this case. +A.3 +Proof of Lemma 16 +Proof. We have P[u ∈ At+1] = 2−d and P[u ∈ (At+2 ∪ · · · ∪ At+r−1) \ At+1] = pr − 2−d. We also +note that if u /∈ At+1 then u ∈ Wt+1, and u may become black in a subsequent round only after it +becomes active. It follows that +P[{u ∈ (Bt+ℓ ∪ · · · ∪ Bt+r) ∩ At+1] = br − P[u ∈ (Bt+ℓ ∪ · · · ∪ Bt+r) \ At+1] +≥ br − P[u ∈ (At+2 ∪ · · · ∪ At+r−1) \ At+1] +≥ br − (pr − 2−d) +≥ br/2. +22 + +Let X be the event that the states of vertices at the end of round t+1 are such that the conditional +probability of u ∈ Bt+ℓ ∪ · · · ∪ Bt+r is at least br/4. Then +P[{u ∈ (Bt+ℓ ∪ · · · ∪ Bt+r) ∩ At+1} ∩ X] ≥ br/2 − P[{u ∈ (Bt+ℓ ∪ · · · ∪ Bt+r) ∩ At+1} ∩ ¯ +X ] +≥ br/4. +We can now apply Lemma 15, starting from round t + 1 and using br/4 in place of br, to obtain +P[{u ∈ (Ak +t+1 ∪ · · · ∪ Ak +t+r−1) ∩ At+1} ∩ X] ≥ r1−α · (br/64)α . +This implies the lemma. +A.4 +Proof of Lemma 18 +The proof of consists of a series of lemmas. In all these lemmas, the graph G = (V, E) considered +is a random graph drawn from Gn,p. +Property (P1) holds trivially for sets S of size k ≤ 4 ln n. The next lemma (applied for all +k > 4 ln n, and then combining the results using a union bound) shows that G satisfies the property +for all larger sets, with probability at least 1 − n−Ω(log n). +Lemma 38. Let G = (V, E) be a random graph drawn from Gn,p, and let k ≥ 1. With probability +at least 1 − n−k, all subgraphs of G on k vertices have at most max{4pk2, 2k ln n} edges. +Proof. The probability there is a subgraph with k vertices and at least r = max{2k ln n, 4pk2} +edges is at most +�n +k +� +· +�k2/2 +r +� +· pr ≤ nk · +�ek2 +2r +�r +· pr = ek ln n−r ln +2r +epk2 ≤ ek ln n−2k ln n·ln 8pk2 +epk2 ≤ n−k. +The next lemma shows that G satisfies property (P2) with probability 1 − n−Ω(log n/p) +Lemma 39. Let G = (V, E) be a random graph drawn from Gn,p, and let k ≥ 40 ln(n)/p. With +probability at least 1 − n−k, every set S ⊆ V of size |S| = k satisfies +|{u ∈ V : |N(u) ∩ S)| < pk/2}| ≤ k/2. +Proof. For any set S of size k, and any vertex u ∈ V \ S, the expected number of neighbors of u in +S is pk. By a Chernoff bound, the probability that u has fewer than pk/2 neighbors in S is at most +e−pk/8. Then the probability there is some set S of size k such that at least k/2 vertices u ∈ V \ S +have fewer than pk/2 neighbors in S, is at most +�n +k +� +· +�n − k +k/2 +� +· e−(k/2)·pk/8 ≤ nk · nk/2 · e−pk2/16 = e(3/2)k ln n−pk2/16 ≤ n−k. +Lemma 40. Let G = (V, E) be a random graph drawn from Gn,p, and let k = 3 ln(n)/p. With +probability at least 1 − n−k, every set S ⊆ V of size |S| ≥ k satisfies |V \ N +(S)| ≤ k. +Proof. The probability there is a set S of size k with |V \ N +(S)| ≥ k is at most +�n +k +� +· +�n − k +k +� +· (1 − p)k2 ≤ nk · nk · e−pk2 = e2k ln n−pk2 = n−k. +The next lemma shows that G satisfies property (P3) with probability 1 − n−Ω(log n/p). +23 + +Lemma 41. Let G = (V, E) be a random graph drawn from Gn,p. With probability at least 1 − +n− ln(n)/p, every triplet of disjoint sets S, T, I ⊆ V , such that |S| ≥ 2|T| and (S ∪ T) ∩ N(I) = ∅, +satisfies +|N(T) \ N +(S ∪ I)| − |N(S) \ N +(I))| ≤ 8 ln2(n)/p. +(7) +Proof. From Lemma 40, with probability at least 1−n−3 ln(n)/p, all sets S, I ⊆ V such that |S∪I| ≥ +3 ln(n)/p satisfy |V \ N +(S ∪ I)| ≤ 3 ln(n)/p, and thus +|N(T) \ N +(S ∪ I)| ≤ |V \ N +(S ∪ I)| ≤ 3 ln(n)/p, +which implies (7). +Next we assume that |S ∪ I| ≤ 3 ln(n)/p. Since |S| ≥ 2|S|, we have |S ∪ T ∪ I| ≤ 4.5 ln(n)/p, +thus there are at most n4.5 ln(n)/p different triplets S, T, I. Choose one such triplet S, T, I, before +revealing the edges of G. Then reveal the edges incident to vertices u ∈ I; this determines N(I). +Let U = V \ (S ∪ T ∪ N +(I)). +The two sets on the left side of (7) can then be expressed as +N(T) \ N +(S ∪ I) = U ∩ N(T) \ N(S), and N(S) \ N +(I) = U ∩ N(S). For every u ∈ U, the +probability that u ∈ N(T) \ N(S) is +p1 = P[u ∈ N(T) \ N(S)] = +� +1 − (1 − p)|T|� +· (1 − p)|S|, +and the probability that u ∈ N(S) is +p2 = P[u ∈ N(S)] = 1 − (1 − p)|S|. +Letting ε = (1 − p)|T| and using that |S| ≥ 2|T|, we obtain p1 ≤ (1 − ε) · ε2 and p2 ≥ 1 − ε2. Thus +p2 +p1 +≥ +1 − ε2 +(1 − ε) · ε2 = 1 + ε +ε2 +≥ 2. +It follows that, by considering all vertices u ∈ U one after the other, and revealing all edges incident +to each u at the moment u is considered, we can analyze the difference +D = |U ∩ N(T) \ N(S)| − |U ∩ N(S)| = |N(T) \ N +(S ∪ I)| − |N(S) \ N +(I)| +as a biased random walk on the integers starting at 0, and moving to the right with probability +p1 and to the left with probability p2. +The probability that the (infinite) random walk every +reaches value i ≥ 1 is know to be (p1/p2)i ≤ 2−i. Thus, P[D ≥ i] ≤ 2−i. And the probability that +D ≥ 8 ln2(n)/p for at least one possible triplet S, T, I is then at most +n4.5 ln(n)/p · 2−8 ln2(n)/p ≤ n−ln n/p. +Property (P4) holds trivially for sets S of size k ≤ 6 ln n, since |S| ≥ |T|. The next lemma +(applied for all k > 6 ln n) shows that G satisfies the property with probability at least 1−n−Ω(log n) +for all larger sets. +Lemma 42. Let G = (V, E) be a random graph drawn from Gn,p, and let k ≥ 1. With probability +at least 1 − n−2k, every pair of disjoint sets S, T ⊆ V , such that |S| = k ≥ |T| and |T| ≤ ln(n)/p, +satisfies |E(S, T)| ≤ 6k ln n. +Proof. For any given pair S, T, the expected value of |E(S, T)| is p · |S| · |T| ≤ k ln n, and by a +Chernoff bound, the probability that |E(S, T)| ≥ 6k ln n is at most 2−6k ln n. Then the probability +there is at least one pair S, T such that |E(S, T)| ≥ 6k ln n is at most +n|S| · n|T| · 2−6k ln n ≤ n2k · 2−6k ln n ≤ n−2k. +24 + +Our last lemma implies that properties (P5) and (P6) hold with probability 1 − O(n−2). +Lemma 43. In a random graph G drawn from Gn,p, the probability that no two vertices have k +common neighbors is at least 1 − n2 · (ep2n/k)k. And the probability that diam(G) ≤ 2 is at least +1 − n2 · e−p2(n−1). +Proof. The probability there is a pair of vertices that have at least k common neighbors is at most +�n +2 +� +· +�n−2 +k +� +· p2k ≤ n2 · +� +ep2n +k +�k +. And the probability there is a pair of vertices with no common +neighbors and no adjacent to each other is +�n +2 +� +· (1 − p) · (1 − p2)n−2 ≤ n2 · e−p2(n−1). +B +Other Related Work +In 1985, Luby [24] proposed a simple distributed randomized algorithm that finds an MIS in +time O(log n) w.h.p. Simultaneously, Alon et al. [3] proposed a similar algorithm with the same +performance. Both algorithms work with O(log n)-bit messages and need access to O(log n) random +bits at each round. +Due to various applications in radio sensor networks, restricted distributed models of communi- +cation were introduced, in which the MIS problem has been widely studied. In the beeping model, +introduced by Cornejo and Kuhn [9], nodes have no knowledge of the local or global structure of +the network, do not have access to synchronized clocks and the communication among nodes relies +completely on carrier sensing (as described in the introduction). Afek et al. [1] show that in the +version of the beeping model where nodes are initially asleep and are woken up by an adversary, +it is not possible to locally converge to an MIS in sub-polynomial time. Therefore, they consider +various relaxations on the model, providing algorithms converging to an MIS in a polylogarithmic +number of rounds. In detail, if the nodes know an upper bound on the size of the network, or if +the beeping nodes are awakened by the neighbor’s beep, the MIS can be found in time O(log3 n) +w.h.p. If the nodes have synchronous clocks, an MIS can be found in time O(log2 n) w.h.p. We +remark that the authors provide a self-stabilizing algorithm just in the first setting, i.e. when an +upper bound on the size of the network is known by the nodes and that. In all algorithms, the +nodes have super-constnt state and have access to a super-constant number of random bits. +In the version of the beeping model with synchronized clocks, collision detection, and simul- +taneous wakeup, Afek et al. [2] had earlier shown that the MIS problem is solved by a biological +process in time O(log2 n) w.h.p. [1] showed that this bound is also achievable without knowledge of +an upper bound on the size of the network. Jeavons et al. [23] improved these results, showing that +an MIS can be found in time O(log n) w.h.p. An improved analyisis of the local complexity of this +algorithm was provided by Ghaffari [16]. In the same version of the beeping model without collision +detection, Holzer and Lynch in [21], proposed a variant of the algorithm of [15], and showed that it +converges locally in time O((log ∆ + log 1/ε) · log 1/ε) with probability at least 1 − ε on a network +with maximum degree ∆. All these algorithms require super constant space and random bits per +round. +Emek et al. [13] introduced the stone age model, inspired by biological cellular networks or +networks of microprocessor devices. In this model, the nodes can communicate by transmitting +messages belonging to a finite communication alphabet. +The nodes communicate in an asyn- +chronous environment, where the pattern is decided by an adversary, and they have no knowledge +about the size of the network. In the stone age model, the MIS problem was considered by [13, 12]. +In [13], is provided an algorithm that compute a MIS in O(log2 n) rounds. However, it assumes +that all the nodes have the same initial state, and therefore is not self-stabilizing. In [12], they +25 + +provided a self-stabilizing algorithm that stabilizes in time O((D + log n) log n) w.h.p., and the +possible number of states of each node is O(D), where D is the diameter of the graph. +In [25], the authors introduced a randomized distributed algorithm that finds an MIS in time +O(log n) w.h.p. In particular, the algorithm is an adaptation of Luby’s algorithm so that messages +of just 1 bit are used. They consider an anonymous network, but in their setting, the vertices can +distinguish between their neighbors, and each vertex needs a number of states that depends on n +and the node degree. +MIS algorithm has also received a lot of attention from the Self-Stabilization community. For +a survey of those algorithms see [19]. +We first cite here the self-stabilizing algorithm for non-anonymous networks, i.e. where vertices +have IDs. In [18], the authors provide a simple deterministic distributed algorithm that stabilizes on +an MIS in O(n) time and O(n2) moves (i.e. total number of state changed), in a synchronous model. +In [22], the authors proposed a deterministic two-state algorithm that works under distributed +scheduler (an adversary that, at each time, selects arbitrarily a set of processes to execute). Both +algorithms stabilize in time O(n2). In [29], Turau introduces a 3-state self-stabilizing algorithm that +stabilizes in O(n) moves, under a distributed scheduler. A breakthrough was achieved by Barenboim +et al. [5], who proposed a self-stabilizing algorithm for the MIS and other related problems, in the +synchronous model. They prove that the algorithm stabilizes after O(∆ + log∗ n) rounds. +Assuming anonymous networks Shukla et al. [28] proposed two deterministic two-state self- +stabilizing algorithms, that work under a centralized scheduler (an adversary that selects one process +to execute at each round) and stabilizes in O(n) rounds. In [30], Turau introduced a synchronous +randomized self-stabilizing algorithm for MIS that stabilizes w.h.p. in O(log n) rounds w.h.p. The +possible states of the nodes are O(log n). +Next, we briefly summarize the best known upper bounds to compute an MIS in the distributed +LOCAL model on arbitrary graphs. Barenboim et al. [6] proved that an MIS can be computed +with a distributed deterministic algorithm in O(∆ + log∗ n) rounds and Ghaffari et al. [17] provide +an upper bound of O(log5 n). Regarding distributed randomized algorithms, Ghaffari [15] provides +an upper bound of O(log ∆) + 2O(√log log n) w.h.p., which, thanks to [27, 17], was improved to +O(log ∆ + log5 log n) w.h.p. See also [14]. +The current best-known lower bound for finding an MIS is proved by Balliu et al. [4], who +show that computing an MIS in the LOCAL model requires Ω(min{∆, log n/ log log n}) rounds +deterministically, and Ω(min{∆, log log n/ log log log n}) rounds with a randomized algorithm. +References +[1] Yehuda Afek, Noga Alon, Ziv Bar-Joseph, Alejandro Cornejo, Bernhard Haeupler, and Fabian Kuhn. +Beeping a maximal independent set. Distributed Comput., 26(4):195–208, 2013. +[2] Yehuda Afek, Noga Alon, Omer Barad, Eran Hornstein, Naama Barkai, and Ziv Bar-Joseph. A biological +solution to a fundamental distributed computing problem. Science, 331(6014):183—185, January 2011. +[3] Noga Alon, L´aszl´o Babai, and Alon Itai. +A fast and simple randomized parallel algorithm for the +maximal independent set problem. J. Algorithms, 7(4):567–583, 1986. +[4] Alkida Balliu, Sebastian Brandt, Juho Hirvonen, Dennis Olivetti, Mika¨el Rabie, and Jukka Suomela. +Lower bounds for maximal matchings and maximal independent sets. J. ACM, 68(5):39:1–39:30, 2021. +[5] Leonid Barenboim, Michael Elkin, and Uri Goldenberg. Locally-Iterative distributed (∆ + 1)-coloring +and applications. J. ACM, 69(1):5:1–5:26, 2022. +[6] Leonid Barenboim, Michael Elkin, and Fabian Kuhn. Distributed (∆+1)-coloring in linear (in ∆) time. +SIAM J. Comput., 43(1):72–95, 2014. +26 + +[7] Leonid Barenboim, Michael Elkin, Seth Pettie, and Johannes Schneider. The locality of distributed +symmetry breaking. J. ACM, 63(3):20:1–20:45, 2016. +[8] Stephen A. Cook. An overview of computational complexity. Commun. ACM, 26(6):400–408, 1983. +[9] Alejandro Cornejo and Fabian Kuhn. Deploying wireless networks with beeps. In Proc. Distributed +Computing, 24th International Symposium, DISC, pages 148–162, 2010. +[10] Edsger W. Dijkstra. Self-stabilizing systems in spite of distributed control. Commun. ACM, 17(11):643– +644, 1974. +[11] Shlomi Dolev. Self-Stabilization. MIT Press, 2000. +[12] Yuval Emek and Eyal Keren. A thin self-stabilizing asynchronous unison algorithm with applications to +fault tolerant biological networks. In Proc. ACM Symposium on Principles of Distributed Computing, +PODC, pages 93–102, 2021. +[13] Yuval Emek and Roger Wattenhofer. Stone age distributed computing. In Proc. ACM Symposium on +Principles of Distributed Computing, PODC, pages 137–146, 2013. +[14] Salwa Faour, Mohsen Ghaffari, Christoph Grunau, Fabian Kuhn, and V´aclav Rozhon. Local distributed +rounding: Generalized to mis, matching, set cover, and beyond. CoRR, abs/2209.11651, 2022. +[15] Mohsen Ghaffari. An improved distributed algorithm for maximal independent set. In Proc. Twenty- +Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, pages 270–277, 2016. +[16] Mohsen Ghaffari. Distributed MIS via all-to-all communication. In Proc. ACM Symposium on Principles +of Distributed Computing, PODC, pages 141–149, 2017. +[17] Mohsen Ghaffari, Christoph Grunau, and V´aclav Rozhon. Improved deterministic network decomposi- +tion. In Proc. ACM-SIAM Symposium on Discrete Algorithms, SODA, pages 2904–2923, 2021. +[18] Wayne Goddard, Stephen T. Hedetniemi, David Pokrass Jacobs, and Pradip K. Srimani. Self-stabilizing +protocols for maximal matching and maximal independent sets for ad hoc networks. In Proc. 17th +International Parallel and Distributed Processing Symposium (IPDPS 2003), page 162, 2003. +[19] Nabil Guellati and Hamamache Kheddouci. A survey on self-stabilizing algorithms for independence, +domination, coloring, and matching in graphs. J. Parallel Distributed Comput., 70(4):406–415, 2010. +[20] S.M. Hedetniemi, S.T. Hedetniemi, D.P. Jacobs, and P.K. Srimani. +Self-stabilizing algorithms for +minimal dominating sets and maximal independent sets. Computers & Mathematics with Applications, +46(5):805–811, 2003. +[21] Stephan Holzer and Nancy A. Lynch. Beeping a maximal independent set fast. CoRR, abs/1704.07133, +2017. +[22] Michiyo Ikeda, Sayaka Kamei, and Hirotsugu Kakugawa. A space-optimal self-stabilizing algorithm for +the maximal independent set problem. In Proc. 3rd International Conference on Parallel and Distributed +Computing, Applications and Technologies, PDCAT, pages 70–74, 2002. +[23] Peter Jeavons, Alex Scott, and Lei Xu. Feedback from nature: Simple randomised distributed algorithms +for maximal independent set selection and greedy colouring. Distributed Comput., 29(5):377–393, 2016. +[24] Michael Luby. A simple parallel algorithm for the maximal independent set problem. SIAM J. Comput., +15(4):1036–1053, 1986. +[25] Yves M´etivier, John Michael Robson, Nasser Saheb-Djahromi, and Akka Zemmari. An optimal bit +complexity randomized distributed MIS algorithm. Distributed Comput., 23(5-6):331–340, 2011. +[26] C. St.J. A. Nash-Williams. Decomposition of finite graphs into forests. J. Lond. Math. Soc., s1-39(1):12– +12, 1964. +[27] V´aclav Rozhon and Mohsen Ghaffari. Polylogarithmic-time deterministic network decomposition and +distributed derandomization. In Proc. 52nd Annual ACM SIGACT Symposium on Theory of Comput- +ing, STOC, pages 350–363, 2020. +27 + +[28] Sandeep K. Shukla, Daniel J. Rosenkrantz, and Sekharipuram S. Ravi. Observations on self-stabilizing +graph algorithms for anonymous networks. In Proc. 2nd Workshop on Self-Stabilizing Systems, SSS, +1995. +[29] Volker Turau. Linear self-stabilizing algorithms for the independent and dominating set problems using +an unfair distributed scheduler. Inf. Process. Lett., 103(3):88–93, 2007. +[30] Volker Turau. +Making randomized algorithms self-stabilizing. +In Proc. Structural Information and +Communication Complexity - 26th International Colloquium, SIROCCO, pages 309–324, 2019. +[31] Volker Turau and Christoph Weyer. Randomized self-stabilizing algorithms for wireless sensor net- +works. In Proc. Self-Organizing Systems, First International Workshop, IWSOS, and Third Interna- +tional Workshop on New Trends in Network Architectures and Services, EuroNGI, pages 74–89, 2006. +[32] Leslie G. Valiant. Parallel computation. In Proc. 7th IBM Symposium on Mathematical Foundations of +Computer Science, 1982. +28 + diff --git a/09E4T4oBgHgl3EQfZwye/content/tmp_files/load_file.txt b/09E4T4oBgHgl3EQfZwye/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f0f3444a30c9f09fae36bab761a825167d5ecb13 --- /dev/null +++ b/09E4T4oBgHgl3EQfZwye/content/tmp_files/load_file.txt @@ -0,0 +1,1132 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf,len=1131 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='05059v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='DC] 12 Jan 2023 Distributed Self-Stabilizing MIS with Few States and Weak Communication George Giakkoupis Inria, Rennes, France george.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='giakkoupis@inria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='fr Isabella Ziccardi Bocconi University, Milan, Italy isabella.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='ziccardi@unibocconi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='it Abstract We study a simple random process that computes a maximal independent set (MIS) on a general n-vertex graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Each vertex has a binary state, black or white, where black indicates inclusion into the MIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The vertex states are arbitrary initially, and are updated in parallel: In each round, every vertex whose state is “inconsistent” with its neighbors’, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=', it is black and has a black neighbor, or it is white and all neighbors are white, changes its state with probability 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The process stabilizes with probability 1 on any graph, and the resulting set of black vertices is an MIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' It is also easy to see that the expected stabilization time is O(log n) on certain graph families, such as cliques and trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' However, analyzing the process on graphs beyond these simple cases seems challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Our main result is that the process stabilizes in poly(log n) rounds w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' on Gn,p random graphs, for 0 ≤ p ≤ poly(log n) · n−1/2 and p ≥ 1/ poly(log n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Further, an extension of this process, with larger but still constant vertex state space, stabilizes in poly(log n) rounds on Gn,p w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=', for all 1 ≤ p ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We conjecture that this improved bound holds for the original process as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' In fact, we believe that the original process stabilizes in poly(log n) rounds on any given n-vertex graph w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Both processes readily translate into distributed/parallel MIS algorithms, which are self-stabilizing, use constant space (and constant random bits per round), and assume restricted communication as in the beeping or the synchronous stone age models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' To the best of our knowledge, no previously known MIS algorithm is self-stabilizing, uses constant space and constant randomness, and stabilizes in poly(log n) rounds in general or random graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' 1 Introduction Finding a maximal independent set (MIS) is a fundamental problem in parallel and distributed computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Given a graph G = (V, E), the objective is to identify a set of vertices S ⊆ V such that no two vertices u, v ∈ S are adjacent to each other (independence property), and no vertex u ∈ V \\S can be added to S without violating independence (maximality property).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The significance of the problem in parallel computing was first recognised in the early 80s [32, 8], due to its various applications in symmetry breaking [24], and it has been studied extensively every since (see [7] for a review of work until 2015, and [4, 17] for state of the art results).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' In this paper we explore simple distributed random processes on graphs that find an MIS starting from arbitrary initial states of the vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' These processes immediately translate into self- stabilizing [10, 11] synchronous distributed algorithms for network systems with severely restricted computation and communication capabilities, such as wireless sensor networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The processes we consider are also relevant to certain biological cellular networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' For example, it is known that a biological process occurring during the development of the nervous system of a fly is equivalent to computing an MIS [2, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' 1 The main random process we consider, which we call the 2-state MIS process, is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Each vertex has a binary state, black or white, where black indicates inclusion into the MIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The vertex states are arbitrary initially and are updated in synchronous rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' In each round, every vertex u whose state violates the independence or maximality properties, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=', u is black and has a black neighbor, or it is white and has no black neighbor, changes its state to the opposite state with probability 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' It is easy to see that the state of a vertex stabilizes as soon as it is black and has no black neighbors, or it is white and has a stabilized black neighbor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' and when all vertices have stabilized, the set of black vertices is an MIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' It is also immediate that, on any graph G, the process stabilizes eventually with probability 1 (due to the randomization) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='1 The 2-state MIS process can be viewed as a natural parallelization (with the addition of ran- domness) of a simple self-stabilizing sequential deterministic algorithm, proposed in [28, 20], where in each step a single node updates its state (from black to white, if the node has a black neighbor, and from white to black if it has no black neighbors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' [28] also observed that by randomizing the transitions of the sequential algorithm we obtain an algorithm that stabilizes with probability 1 on a general adversarial scheduler model, which includes the synchronous model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' A similar observa- tion follows from a general transformation framework proposed in [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The sequential algorithm is know to stabilize after each process has taken at most 2 steps (regardless of the scheduling order).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' However, analyzing the stabilization time of the parallel process seems a much more challenging problem, and has not been studied until now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The 2-state MIS process directly translates into a self-stabilizing MIS algorithm for the harsh beeping communication model [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' In that model, in every synchronous round, each node either listens or beeps, and a listening node can only differentiate between none of its neighbors’ beeping, or at least one beeping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' In our case, we can let black nodes beep in each round, while white nodes listen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Black nodes must be able to detect collisions (otherwise they cannot tell if they have a black neighbor), thus we assume the beeping model version with sender collision detection (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' full-duplex model) [1, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We also propose a simple variant of the 2-state MIS process, called the 3-state MIS process, which has an additional state and does not require collision detection (see Definition 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' This variant is suitable for the synchronous stone age model [13, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The synchronous stone age model can be viewed as an extension of the beeping model over a constant number of channels (without collision detection): each node beeps in at most one channel and listens to the other channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Overall, the algorithms obtained from the 2-state and 3-state MIS processes have several at- tractive properties: they use a constant number of states (2 or 3) and one random bit per round, they do not require node IDs or any global graph information (such as the number of vertices n or the maximum degree ∆), assume very week communication (the beeping or stone age models), they are self-stabilizing, and are extremely simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We will prove that, on some families of graphs, these algorithms are also fast, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=', they stabilize (from an arbitrary initial state) in a number of rounds that is poly-logarithmic in n, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='2 Moreover, despite that we were not able to prove such as strong result here, we believe that these algorithms are fast in all graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Several self-stabilizing distributed MIS algorithms have been proposed in the literature, but as far as we know, none possesses all the above properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Known self-stabilizing MIS algorithms for the beeping model require (approximate) knowledge of n, use space that is a super-constant function of n, and require a super-constant number of random bits [1, 23, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' In the stone age model, an MIS algorithm proposed in [13] has similar properties as our algorithms (and is provably fast for all 1We could have defined the process so that the transition from white to black (when the white vertex has no black neighbors) occurs with probability 1, but we opted for a randomized transition because it simplifies our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' 2In this paper, we do not analyze the 3-state MIS process, but we expect that it behaves similarly (or better than) the 2-state MIS process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' 2 graphs) but is not self-stabilizing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' while a self-stabilizing algorithm for the model proposed recently in [12] is fast only on graphs whose diameter is bounded by a known constant D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Other randomized self-stabilizing MIS algorithms required super constant state and communication [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Finally, known deterministic self-stabilizing MIS algorithms require distinct node IDS, super constant state and communication, and are in general much slower than the randomized algorithms, stabilizing in time linear in n or in the maximum degree ∆ [22, 18, 29, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='1 Our Contribution We first analyze the stabilization time of the 2-state MIS process on complete graphs and on graphs with bounded arboricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='3 We also provide an upper bound in terms of the maximum degree for general graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The proof of these results is mostly straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The stabilization time of the 2-state MIS process on n-vertex graph G is O(log n) in expectation and O(log2 n) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=', if G is the complete graph Kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' O(log n) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=', if G has bounded arboricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' at most O(∆ log n) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=', if the maximum degree of G is ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' A main technical contribution of the paper is the analysis of the 2-state MIS process on Erd˝os- R´enyi Gn,p random graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We show a poly-logarithmic upper bound for Gn,p random graphs when the average degree np is at most poly(log n) · √n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The same bound is easily obtained also when the average degree is at least n/ poly(log n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The stabilization time of the 2-state MIS process on a Gn,p random graph, such that 0 ≤ p ≤ poly(log n) · n−1/2 or p ≥ 1/ poly(log n), is at most poly(log n) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Our proof techniques do not yield a poly-logarithmic upper bound for the 2-state MIS process on Gn,p for the complete range of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Our second technical contribution is an extension of the 2-state MIS process that provably stabilizes in poly-logarithmic time w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' on Gn,p for all 0 ≤ p ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The extended process uses a phase clock sub-process proposed in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Interestingly, unlike [12], we do not use the phase clock for synchronization, but rather as a local non-synchronized counter (see Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='2 for a more detailed discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' There is an extension of the 2-state process, with 18 states, such that the stabilization time of the process on a Gn,p random graph, for any 0 ≤ p ≤ 1, is at most poly(log n) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We believe that the bound of Theorem 3 holds for the 2-state MIS process, as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' In fact, we conjecture that the stabilization time of the 2-state MIS process is poly(log n) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' on any given n-vertex graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We also conjecture that the same is true for the 3-state MIS process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' For the 2-state process, the best general upper bound we can hope for is O(log2 n), as the process requires Θ(log2 n) rounds to stabilize on the complete graph Kn w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='4 For the 3-state process, we have no example of a graph where the stabilization time is larger than O(log n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' 3The arboricity of a graph is the minimum number of forests into which we can partition its edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' 4It also requires Θ(log2 n) rounds in expectation to stabilize on a graph consisting of √n disjoint cliques K√n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='2 Analysis Overview and Techniques Below we give an overview of the analysis of the 2-state MIS process and its extension, on Gn,p random graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' To avoid having to deal simultaneously with the randomness of the graph and the arbitrary initialization of vertex states, we deal with graph randomness first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We define a family of good graphs, containing those graphs that satisfy all structural properties that we will need for the analysis, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=', bounds on the average degree of any induced subgraph, and bounds on the number of common neighbors of any two vertices (see Definition 17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We then show that a Gn,p random graph is good w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=', and assume an arbitrary good graph in the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The analysis proceeds by showing that starting from any vertex states, the process makes sufficient progress after O(log n) rounds, where progress is measured by the expected number of vertices that stabilize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' In the 2-state MIS process, we call a vertex active if it is black and has a black neighbor, or it is white and has no black neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Thus, active vertices change their state to a uniformly random state in the next step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' A vertex is k-active if it is active and has at most k active neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' An elementary property of the 2-state MIS process is that if a vertex is k-active, then it becomes stabilized black in O(log k) rounds with probability Ω(1/k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We also use an extension of this property to sets of active vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='5 These two properties, combined with structural properties of good graphs, suffice to show the desired expected progress in the case in which the number of non-stabilized vertices or the number of active vertices is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The more difficult case is when the number of non-stabilized vertices is relatively small, namely O(p−1 log2 n), and a smaller than 1/ poly(log n) fraction of them are active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' One may expect this to be an easy case, since the induced subgraph on a random subset of O(p−1 log2 n) vertices has maximum degree ∆ = O(log2 n) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' (and Theorem 1 gives an O(log3 n) bound for that ∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' However, the above bound on ∆ does not apply to an induced subgraph on an arbitrary subset of O(p−1 log2 n) vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Nevertheless, it is true that the average degree is O(log2 n), thus a constant fraction of vertices have degree O(log2 n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Let u be one such vertex, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=', of degree d = O(log2 n) in the induced subgraph of non-stabilized vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' To prevent u from becoming active (and thus d-active) or becoming stabilized, in each round at least one neighbor of u must be non-stabilized black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We show that, roughly speaking, if a vertex v has probability b of being non-stabilized black at some point during an interval of r rounds (ignoring the first few rounds, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=', if v is black initially) then v has probability poly(b/r) of becoming θ-active in that interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' For the purposes of the analysis, it suffices to set r = O(log log n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Then θ is, roughly, bounded by the maximum number of common neighbors two nodes may have, thus θ ≤ poly(log n) if p ≤ poly(log n) · n−1/2 (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='1 for the relevant lemmas).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' If each of the d neighbors of u has probability less than 1/(2d) of becoming non-stabilized black in the next r rounds, then u has a constant probability of becoming active (or stabilize).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' On the other hand, if there is some neighbor v that has probability b ≥ 1/(2d) of becoming non- stabilized black in the next r rounds, we saw above that v becomes θ-active with probability at least poly(b/r) = 1/ poly(log n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We conclude that, with probability 1/ poly(log n), u is poly(log n)- active or has some poly(log n)-active neighbor at some point in the next r = O(log log n) rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' It follows that u stabilizes with probability 1/ poly(log n) in the next O(log n) rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='6 When p > poly(log n)·n−1/2, the last case of the analysis above does not give a poly-logarithmic bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' A way to overcome this problem is to control how often a vertex can change its state from white to black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We extend the 2-state MIS process by incorporating such a control mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' 5Similar properties are commonly used in the analysis of distributed MIS algorithms in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' 6We suspect that a refinement of this argument may be useful for a broader class of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' 4 We call the new process the 3-color MIS process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' It consists of two sub-processes running in parallel: The first is similar to the 2-state MIS process with the addition of a third color, grey;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' a black vertex now becomes gray instead of white, a gray vertex becomes white after a while, and other vertices treat gray vertices as white.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The transition from gray to white is controlled by the second sub-process, called the logarithmic switch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' In the logarithmic switch, each vertex has an on/off binary variable, and a gray vertex changes to white if the switch variable of the vertex is on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We would like that the logarithmic switch satisfy two basic properties: (i) a vertex switches from off to on every Θ(log n) rounds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' and (ii) it switches from on to off every O(1) rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='7 However, we do not know how to implement property (i) using constant states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We observe that it suffices if property (i) is satisfied only when p > poly(log n) · n−1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' for smaller p, a weaker property suffices: (i′) a vertex switches from off to on after at most O(log n) rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' It is not immediately obvious how to implement this distinction, because we want the process to work for all 0 ≤ p ≤ 1 without knowing p (or anything else about the graph topology).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We achieve that as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We exploit the fact that if p > poly(log n) · n−1/2 then the graph has constant diameter (in fact diameter 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The logarithmic switch process we devise is similar to the phase clock process RandPhase proposed in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' RandPhase assumes that an upper bound D on the graph diameter is available to the process and uses D + 3 states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The core mechanism of the logarithmic switch is identical to that of RandPhase for D = 3 (not 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' ), but the underlying graph may have arbitrary (and unknown) diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The logarithmic switch includes also a mapping of the states to the on/off values of the switch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Unlike RandPhase which is used for sychronization (it achieve synchronous phases of length D + Θ(log n)), the purpose of the logarithmic switch is not synchronization, as it is not required that the switch variables of different vertices change simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Roadmap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Section 2 contains the definition and some basic properties of the 2-state and 3-state MIS processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Section 3 provides a proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Section 4 proves Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Section 5 defines the 3-color MIS process and proves Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' And Appendix B reviews related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Let G = (V, E) be a graph on n vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' For each vertex u ∈ V , N(u) = {v: (u, v) ∈ E} is the set of neighbors of u, and N +(u) = N(u) ∪ {u}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Similarly, for a set of vertices S ⊆ V , we define N(S) = � u∈S N(u) \\ S and N +(S) = � u∈S N +(u) = N(S) ∪ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' For two (not necessarily disjoint) sets S, T ⊆ V , we let E(S, T) = {(u, v) ∈ E : u ∈ S, v ∈ T} be the set of edges with one endpoint in S and the other in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We also define E(S) = E(S, S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' By G[S] we denote the induced subgraph of G on S ⊆ V , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=', G[S] = (S, E(S)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' 2 The 2-State and 3-State MIS Processes We define two self-stabilizing distributed graph processes that compute a maximal independent set when applied on any given graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Definition 4 (2-State MIS Process).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' In the 2-state MIS process on graph G = (V, E), each vertex u ∈ V has a binary state from the set {black, white}, and all states are updated in parallel 7The reason why a logarithmic switch suffices, rather than a ‘double-logarithmic’ switch is that, in the induced subgraph on O(p−1 log2 n) vertices consider in the last case of the analysis of the 2-state MIS process, a constant fraction of vertices have at most O(log n) neighbors of degree Ω(log3 n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' 5 rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The initial state c0(u) of vertex u can be arbitrary, and in each round t = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' , u’s state is updated from ct−1(u) to ct(u) according to the following rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' let NC t(u) = {ct−1(v): v ∈ N(u)} if � ct−1(u) = black and NC t(u) ∋ black � or � ct−1(u) = white and NC t(u) ̸∋ black � then let ct(u) be a uniformly random state from {black, white} else set ct(u) = ct−1(u) We say that vertex u is black or white if its state is black or white, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We say that u is active if it is black and has some black neighbor, or it is white and has no black neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We say that vertex u is stable, if either it is black and has no black neighbors, or it is white and has a neighbor that is black and stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' It is immediate from the update rule that once a vertex becomes stable, it remains stable thereafter, and its state no longer changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The stabilization time of vertex u is the earliest round at the end of which u is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The stabilization time of the process is the earliest round at the end of which all vertices are stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' It is easy to verify that after the stabilization time of the process, the set of black vertices is an MIS of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We let Bt = {u ∈ V : ct(u) = black} be the set of black vertices at the end of round t ≥ 0, and let Wt = V \\ Bt be the set of white vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We let At = {u ∈ Bt : N(u) ∩ Bt ̸= ∅} ∪ {u ∈ Wt : N(u) ∩ Bt = ∅} denote the set of active vertices at the end of round t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We let It = {u ∈ Bt : N(u) ∩ Bt = ∅} be the set of stable black vertices at the and of round t (note that It is an independent set and is a subset of the final MIS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Finally, we let Vt = V \\ N +(It) be the set of vertices that are not stable at the end of round t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Definition 5 (3-State MIS Process).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' In the 3-state MIS process on G = (V, E), each vertex u ∈ V has a state from set {black1, black0, white}, and the states are updated in parallel rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The initial state c0(u) of u is arbitrary, and in each round t ≥ 1, u’s state is updated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' let NC t(u) = {ct−1(v): v ∈ N(u)} if ct−1(u) = black1 or � ct−1(u) = black0 and NC t(u) ̸∋ black1 � or � ct−1(u) = white and NC t(u) = {white} � then let ct(u) be a uniformly random state from {black1, black0} else if ct−1(u) = black0 then set ct(u) = white else set ct(u) = ct−1(u) In the 3-state MIS process, we say that a vertex u is black when its state is black1 or black0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Then the stable vertices and the stabilization times are defined as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Note that the state of a stable black vertex alternates perpetually between states black1 and black0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' In this paper we focus on the 2-state MIS process, but we expect that all our upper bound results should carry over to the 3-state MIS process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='1 Basic Properties of the 2-State MIS Process We show some elementary properties of the 2-state MIS process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' In the analysis, it will be conve- nient to assume that at the beginning of each round t ≥ 1, we flip for each vertex u an independent coin φt(u) such that P[φt(u) = black] = P[φt(u) = white] = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Then if u must update its state to a random state in that round, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=', if u ∈ At−1, we set ct(u) = φt(u);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' while if u /∈ At−1, then φt(u) is not used by the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The lemmas below apply for any graph G = (V, E), and the probabilistic statements assume that we know the states of vertices at the end of round t (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=', Bt or Wt is given).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The first lemma 6 says than an active vertex u with k active neighbors has probability Ω(1/k) to become stable black in the next O(log k) rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' If u ∈ At and |N(u) ∩ At| = k ≥ 1, then the probability that u ∈ It+log(k+1) is at least (2ek)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Let r = ⌈log(k + 1)⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The probability that u ∈ It+r is lower bounded by the probability that φt+1(v) = · · · = φt+r(v) = black holds for v = u and does not hold for any v ∈ N(u) ∩ At, which is (1/2)r · (1 − (1/2)r)k ≥ (1/2)r · e−k/(2r−1) ≥ (1/2k) · (1/e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' (1) For the first inequality we used the fact (1 − 1/n)n−1 ≥ e−1, and for the second we used that log(k + 1) ≤ r ≤ log(k) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The next statement is a generalization of Lemma 6 to multiple active vertices u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' , uℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We will apply this result to the set of active neighbors of a vertex u, to lower bound the probability that u is stable after a logarithmic number of rounds (because a neighbors becomes stable black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The proof can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Suppose that u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' , uℓ ∈ At, and |N(ui) ∩ At| = ki > 0, for each 1 ≤ i ≤ ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Then the probability that {u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' , uℓ} ∩ It+log(maxi ki+1) ̸= ∅ is at least (1/5) · min � 1, � i(2ki)−1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' 3 Simple Bounds for the 2-State MIS Process We show some simple bounds on the stabilization time of the 2-state MIS process on certain graph families, namely, the complete graph and trees (or more generally, graphs of bounded arboricity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We also show a basic upper bound in terms of the maximum degree on a general graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The stabilization time of the 2-state MIS process on the complete graph Kn = (V, E) is O(log n) in expectation and O(log2 n) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' More concretely, for any k > 0, the stabilization time is at least k · log n with probability 2−Θ(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We call round t critical if |Bt| ≤ 1, and we call it stable if |Bt| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Let pa be the probability that the next critical round is stable, given that |At| = a ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Note that in graph Kn, At = Bt if |Bt| > 1, At = ∅ if |Bt| = 1, and At = V if Bt = ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' thus |At| ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We argue that for any a ≥ 2, 2/3 ≤ pa ≤ 17/21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The lower bound follows from the observation that, for any i ≥ 2 and j ≥ 1, the conditional probability that round j is stable, given that it is critical and that |Aj−1| = i, is (i 1)2−i (i 1)2−i+2−i = i i+1 ≥ 2/3, since i ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' For the upper bound we observe that, for any i ≥ 3 and j ≥ 1, the conditional probability of |Bj| ∈ {2, 0}, given that |Bj| ≤ 2 and that |Aj−1| = i, is (i 2)2−i+2−i (i 2)2−i+(i 1)2−i+2−i = i2−i+2 i2+i+2 ≥ 4/7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Also, p2 = 2/3 < 17/21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Then, for any a ≥ 3, we have 1 − pa ≥ (4/7) · (1 − p2), which implies pa ≤ 17/21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Next, consider the number of rounds r from a non-stable critical round (when all nodes are white) until the next critical round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The probability that r > k is lower and upper bounded by 1 − e−n2−k ≤ 1 − (1 − 2−k)n ≤ n2−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' 7 Combining the above we obtain that (i) from any given non-stable round, the probability that a stable round is reached in at most k = log n + 1 rounds is at least 2/3 − n2−k ≥ 1/6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' (ii) from any given non-stable critical round, the probability that the next critical round is non-stable and is reached in more than k = log n − 2 rounds is at least 1 − 17/24 − e−n2−k > 1/6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' and (iii) assuming round t = 0 is not critical, the probability that the first critical round is non-stable is at least 1 − 17/24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' These statements, together, imply that the stabilization time is at least k log n with probability 2−Θ(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' And from that, the expectation and high-probability bounds follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Remark 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' From Theorem 8, it is immediate that the expected stabilization time of the 2-state MIS process is Θ(log2 n) on a graph G that is the disjoint union of √n cliques K√n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The same bound holds also w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Remark 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' A similar analysis as for Theorem 8 gives an upper bound of O(log n) on the stabi- lization time of the 3-state MIS process on Kn, both in expectation and w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The reason is that once Bt ̸= ∅ then Bt′ ̸= ∅ for all t′ ≥ t (thus the next critical round is stable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The stabilization time of the 2-state MIS process on any graph G = (V, E) of bounded arboricity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=', G is a tree) is O(log n) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Recall that the arboricity λ of G is the minimum number of forests into which its edges can be partitioned, and is equal up to a factor of 2 to the maximum average degree in any subgraph [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Suppose that the average degree of any subgraph of G is at most d ≤ 2λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Let St be the subset of Vt consisting of of all vertices u ∈ Vt with |N(u) ∩ Vt| ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Then |St| ≥ |Vt|/(d + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' If u ∈ St \\ At and |N(u) ∩ Vt| = du, the probability that N(u) ⊆ Wt+1 is 2−du ≥ 2−d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Thus, for each u ∈ St, the probability that u ∈ At ∪ At+1 is at least 2−d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' And if u ∈ At ∪ At+1, Lemma 6 gives that u ∈ It+log(d+1)+1 with probability at least (2ed)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' It follows E � |Vt+log(d+1)+1| �� |Vt| � ≤ |Vt| − (2ed)−1 · 2−d · |Vt|/(d − 1) ≤ (1 − ǫ) · |Vt|, for some constant ǫ = ǫ(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Let r = log(d + 1) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Applying the above inequality iteratively, we obtain E[|Vrt|] ≤ (1 − ǫ)rn ≤ e−ǫrn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Thus for t = 3ǫ−1 ln n, E[|Vrt|] ≤ n−2, and by Markov’s inequality, P[|Vrt| ≥ 1] ≤ n−2, which implies the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The stabilization time of the 2-state MIS process on any graph G = (V, E) of maximum degree ∆ is at most O(∆ log n) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We observe that if u ∈ Vt then N +(u)∩At ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Let u ∈ V0, and let (v1, t1), (v2, t2), (v3, t3), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' be a random sequence of vertex-round pairs defined as follows: Let t0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' For each i ≥ 1, if u ∈ Vti−1, then vi is an arbitrary vertex from the set N(u)∩Ati−1, and ti = min{j > ti−1 : vi /∈ Aj};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' while if u /∈ Vti−1, then (vi, ti) = (u, ti−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We focus on the first r = 6e∆ log n elements of the sequence above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We bound the probability that u ∈ Vtr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' For each 1 ≤ i ≤ r, the conditional probability that vi ∈ Iti+1 (and thus u /∈ Vti+1), given vi and Bti, is at least 1/(2e∆), from Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' It follows that P[u ∈ Vtr] ≤ (1 − 1/(2e∆))r ≤ e−r/(2e∆) = n−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Next, we bound the value of tr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' For each 1 ≤ i ≤ r and t ≥ ti−1, if vi ∈ At then the conditional probability that vi /∈ At+1, given (vi, ti) and Bt, is exactly 1/2 (in all cases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' It follows that the probability of tr > 4r is upper bound by the probability that a sequence of 4r fair coin tosses contains fewer than r heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Thus, by a Chernoff bound, P[tr > 4r] ≤ e−(1/2)22r/2 = e−e∆ log n < n−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Combining the above results, we obtain that P[u /∈ V4r] ≥ P[{u /∈ Vtr} ∩ {tr ≤ 4r}] ≥ 1 − 2n−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' (Recall that r = 6e∆ log n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=') Finally, a union bound over all u ∈ V competes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' 8 4 The 2-State MIS Process on Random Graphs We first show some additional properties of the 2-state MIS process, which hold for any graph but are useful only when adjacent vertices do not have many common neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Then we show some structural properties of Gn,p random graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Finally, we use these properties to show a poly(log n) upper bound on the stabilization time of the 2-state MIS process on Gn,p random graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='1 Refined Properties of the 2-State MIS Process We call a vertex k-active if it is active and has at most k active neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Let Ak t = {u ∈ At : |N(u) ∩ At| ≤ k} be the set of k-active vertices at the end of round t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' From Lemma 6, a k-active vertex has probability at least Ω(1/k) to become stable black in the next O(log k) rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' It is thus desirable to have k-active vertices for small values k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' In this section we establish lower bounds on the probability that a given vertex u becomes k-active at some point in the next r rounds, as a function of the probability that u is active (but has possibly more than k active neighbors) at a point in a certain subinterval of those r rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The next key lemma is the base of all the other results in the section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' It lower bounds the probability q of a white vertex u, which is non-active and non-stable, to become k-active after a single round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The lower bound is expressed in terms of the probability p that u is active white after two rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The value of k depends on the number of active neighbors of u, and, crucially, on the number of their common neighbors with u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Suppose that u ∈ Vt \\ At,8 and let θ = |N(u) ∩ N +(At ∩ N(u))| be the number of u’s neighbors that are active or adjacent to an active neighbor of u at the end of round t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Let p be the probability that u ∈ At+2 ∩ Wt+2, and q the probability that u ∈ Ak t+1 where k = θ + ⌈log(1/p)⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Then q ≥ pα, where α = 1/log(4/3) ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Let D = N(u) ∩ At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' In round t + 1, each v ∈ D updates its state to a random state, while each v ∈ N(u) \\ D remains white.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Let Z = N(u) ∩ At+1 \\ N +(D) be the set of active neighbors of u at the end of round t + 1 that are at distance at least two away from set D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Clearly, Z does not depend on the random choices of vertices v ∈ D in round t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We have that u ∈ At+1 if and only if all v ∈ D update their state to white in round t + 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=', φt+1(v) = white.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='9 Also |N(u) ∩ At+1| ≤ |N(u) ∩ N +(D)| + |Z| = θ + |Z|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' It follows q ≥ (1/2)d · P[|Z| ≤ λ], where d = |D| and λ = ⌈log(1/p)⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We have that u ∈ At+2 ∩ Wt+2 only if φt+1(v) or φt+2(v) = white for every v ∈ D, and φt+2(v) = white for every v ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' It follows that p ≤ (3/4)d · � i≥0 P[|Z| = i]/2i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' (2) Let ε = P[|Z| ≤ λ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Then p ≤ (3/4)d · � ε + (1 − ε)/2λ+1� ≤ (3/4)d · (ε + (1 − ε) · p/2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' 8Note that u ∈ Vt \\ At implies u ∈ Wt ∩ Wt+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' 9Recall the discussion about coin flips φt(v) at the beginning of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' 9 This implies that p ≤ ε + (1 − ε)· p/2, thus p ≤ 2ε/(1 + ε), and substituting that above yields p ≤ (3/4)d · (ε + (1 − ε) · ε/(1 + ε)) = (3/4)d · 2ε 1 + ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Finally, since (3/4)dα = (1/2)d, and for all x ∈ [0, 1], � 2x 1+x �α ≤ � 2x 1+x �2 = x · 4x (1+x)2 ≤ x, pα ≤ (3/4)dα · � 2ε 1 + ε �α ≤ (1/2)d · ε ≤ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Next, we use the above Lemma 13 to prove a similar result over a sequence of r rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' For any vertex u ∈ V and i ≥ 1, let θu(i) = max{|N(u) ∩ N +(S)|: S ⊆ N(u), |S| ≤ i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' (3) Lemma 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Suppose that u ∈ Vt \\ At and let d = |N(u) ∩ At|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Let pr be the probability that u ∈ At+1 ∪ · · · ∪ At+r, and let qr be the probability that u ∈ Ak t+1 ∪ · · · ∪ Ak t+r−1, where k = θu � α log � 4r pr−2−d �� + � log � 4r pr−2−d �� , and α = 1/log(4/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Then, for any r ≥ 2, qr ≥ r1−α · � pr−2−d 2 �α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' For i ≥ 0, let di = |N(u) ∩ At+i|, and define the following events: Wi is the event that u ∈ Wt+i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Ai is the event that u ∈ At+i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Ak i is the event that u ∈ Ak t+i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' and Hi = ¯ A0 ∩ ¯ A1 ∩· · ·∩ ¯ Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Let also Xi be the event that the states of the vertices at the end of round t + i are such that the conditional probability of Ai+2 ∩ Wi+2 is at least pr−p1 4r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Let r ≥ 2 and λ = ⌊α log � 4r pr−p1 � ⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Then pr = � 1≤i≤r P[Ai ∩ Hi−1] = p1 + � 2≤i≤r P[Ai ∩ Hi−1] = p1 + � 2≤i≤r P[Ai ∩ Wi ∩ Hi−1] (since Ai ∩ Hi−1 implies Wi) ≤ p1 + � 2≤i≤r P[Ai ∩ Wi ∩ Hi−2] (since Hi−1 implies Hi−2) ≤ p1 + � 2≤i≤r P[Ai ∩ Wi ∩ Hi−2 ∩ {di−2 ≤ λ} ∩ Xi−2] + � 2≤i≤r P[Ai ∩ Wi ∩ Hi−2 ∩ {di−2 > λ}] + � 2≤i≤r P[Ai ∩ Wi ∩ ¯ Xi−2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Each of the last two sums above is at most pr−p1 4 , because for each non-zero sum term, we have P[Ai ∩ Wi ∩ Hi−2 ∩ {di−2 > λ}] ≤ P[Ai ∩ Wi | Hi−2, di−2 > λ] ≤ �3 4 �λ+1 ≤ pr − p1 4r , similarly to (2), and P[Ai ∩ Wi ∩ ¯ Xi−2] ≤ P[Ai ∩ Wi | ¯ Xi−2] ≤ pr−p1 4r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Applying these above gives pr − p1 2 ≤ � 2≤i≤r P[Ai ∩ Wi ∩ Hi−2 ∩ {di−2 ≤ λ} ∩ Xi−2] = � 2≤i≤r P[Ai ∩ Wi | Hi−2, di−2 ≤ λ, Xi−2] · P[Hi−2 ∩ {di−2 ≤ λ} ∩ Xi−2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' 10 Next we lower bound qr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We have qr ≥ � 1≤i≤r−1 P[Ak i ∩ Hi−1] = � 2≤i≤r P[Ak i−1 ∩ Hi−2] ≥ � 2≤i≤r P[Ak i−1 ∩ Hi−2 ∩ {di−2 ≤ λ} ∩ Xi−2] = � 2≤i≤r P[Ak i−1 | Hi−2, di−2 ≤ λ, Xi−2] · P[Hi−2 ∩ {di−2 ≤ λ} ∩ Xi−2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' From Lemma 13, applied for round t + i − 2, using p ≥ pr−p1 4 and θ ≤ θu(λ), and observing that p1 = 2−d, we obtain P[Ak i−1 | Hi−2, di−2 ≤ λ, Xi−2] ≥ (P[Ai ∩ Wi | Hi−2, di−2 ≤ λ, Xi−2])α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We substitute this to the previous equation above, and then use Jensen’s inequality to complete the proof: Let ν = � 2≤i≤r P[Hi−2 ∩ {di−2 ≤ λ} ∩ Xi−2] ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' qr ≥ � 2≤i≤r � P[Ak i | Hi−2, di−2 ≤ λ, Xi−2] �α P[Hi−2 ∩ {di−2 ≤ λ} ∩ Xi−2] ≥ ν · \uf8eb \uf8ed � 2≤i≤r P[Ak i | Hi−2, di−2 ≤ λ, Xi−2] · P[Hi−2 ∩ {di−2 ≤ λ} ∩ Xi−2]/ν \uf8f6 \uf8f8 α ≥ ν · �pr − p1 2ν �α ≥ r · �pr − 2−d 2r �α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Lemma 14 assumes that vertex u is initially not active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The next lemma shows a similar result for the case where u is active initially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' In this case, in place of the probability pr that u becomes active at some point in the interval {t+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' , t+r}, we use the probability br that u becomes black at some point of a subinterval {t + ℓ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' , t + r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The proof proceeds by considering the first round after t when either u has at most k black neighbors, or u is white.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' If the first condition holds, then u has probability 1/2 of being black, and thus of being k-active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' If only the second condition holds then we are in the case of Lemma 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The proof can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Lemma 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Suppose that u ∈ At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Let ℓ ≥ 2 and r ≥ ℓ + 2, let br be the probability that u ∈ Bt+ℓ ∪· · ·∪Bt+r, and suppose that br ≥ 1/2ℓ−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Let qr be the probability that u ∈ Ak t ∪· · ·∪Ak t+r−1, where k = θu � α log (32r/br) � + log (32r/br) + log(1/br) + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Then qr ≥ r1−α · (br/16)α, where α = 1/log(4/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' In the last lemma of this section, we consider the case in which Lemma 14 does not give a large enough lower bound for qr, even though pr is large, because the difference pr − 2−d is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We proceed by essentially reducing this case to the case of Lemma 15, after a single round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The proof is in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Lemma 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Suppose that u ∈ Vt \\ At, and let d = |N(u) ∩ At|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Let ℓ ≥ 5 and r ≥ ℓ + 2, let pr be the probability that u ∈ At+1 ∪ · · · ∪ At+r−1, let br be the probability that u ∈ Bt+ℓ ∪ · · · ∪ Bt+r, and 11 suppose that br ≥ 1/2ℓ−4 and br ≥ 2(pr − 2−d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Let qr be the probability that u ∈ Ak t ∪ · · · ∪ Ak t+r−1, where k = θu � α log (128r/br) � + log (128r/br) + log(4/br) + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Then qr ≥ r1−α · (br/64)α, where α = 1/log(4/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='2 Structural Properties of Gn,p and Good Graphs We describe some structural properties that a graph must possess in order for the analysis given in the following sections to carry through.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' A graph satisfying these properties is called a good graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Then we show that a random Gn,p graph is a good graph w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Definition 17 (Good Graphs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Let n be a positive integer and 0 < p < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' A graph G = (V, E) with n vertices is (n, p)-good if it satisfies all the following properties: (P1) For any set S ⊆ V , the average degree of induced subgraph G[S] is at most max{8p|S|, 4 ln n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' (P2) For any set S ⊆ V of size |S| ≥ 40 ln(n)/p, |{u ∈ V \\ S : |N(u) ∩ S| < p|S|/2}| ≤ |S|/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' (P3) For any three disjoint sets S, T, I ⊆ V such that |S| ≥ 2|T| and (S ∪ T) ∩ N(I) = ∅, |N(T) \\ N +(S ∪ I)| ≤ |N(S) \\ N +(I)| + 8 ln2(n)/p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' (P4) For any two disjoint sets S, T ⊆ V such that |S| ≥ |T| and |T| ≤ ln(n)/p, |E(S, T)| ≤ 6|S| ln n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' (P5) No two vertices u, v ∈ V have more than max{6np2, 4 ln n} common neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' (P6) If p ≥ 2(ln(n)/n)1/2 then diam(G) ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Lemma 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' A random graph G = (V, E) drawn from Gn,p is (n, p)-good with probability 1−O(n−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The proof of Lemma 18 can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='3 Analysis of the 2-State MIS Process on Gn,p In this section, we prove the following bound on the stabilization time of the 2-state MIS process on a random Gn,p graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Theorem 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The stabilization time of the 2-state MIS process on a random graph drawn from Gn,p, where p = O( � log(n)/n) or p = Ω(1/ log2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='5 n), is O(log5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='5 n) with probability 1 − O(n−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The theorem follows by combining Lemma 18 and the next lemma, which analyzes the 2-state MIS process on a good graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Lemma 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The stabilization time of the 2-state MIS process on any (n, p)-good graph G = (V, E), where p = O( � log(n)/n) or p = Ω(1/ log2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='5 n), is O(log5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='5 n) with probability 1 − O(n−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' It is straightforward to extend the above statements so that p ≤ poly(log n) · n−1/2 or p ≥ 1/ poly(log n), for any desired poly(log n) term, by adjusting the exponent of log n in the stabiliza- tion time bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' 12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='1 Proof of Lemma 20 We show that starting from any vector of vertex states, the process makes sufficient progress after poly(log n) rounds, where progress is measured by the expected number of vertices that become stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' All lemmas below assume G = (V, E) is an arbitrary (n, p)-good graph, and the probabilistic statements assume we know the states of the vertices at the end of round t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The first lemma considers the case in which the number of active vertices is large, namely, |At| = Ω(log(n)/p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Lemma 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' If |At| ≥ 80 ln(n)/p then there is a constant ǫ > 0 such that E[|Vt+log n|] ≤ (1 − ǫ)·|Vt|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' From property (P1) in Definition 17 of good graphs, the average degree of the induced subgraph G[At] is at most k = max{8p|At|, 4 ln n} = 8p|At|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Let S be a subset of At consisting of the |At|/2 vertices u ∈ At with the smallest degree in G[At], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=', for any two vertices u ∈ S and u′ ∈ At \\ S, |N(u) ∩ At| ≤ |N(u′) ∩ At|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' It follows that for all u ∈ S, |N(u) ∩ At| ≤ 2k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' thus S ⊆ A2k t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Let R = {u ∈ V \\ S : |N(u) ∩ S| < p|S|/2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Since |S| = |At|/2 ≥ 40 ln(n)/p, property (P2) in Definition 17 yields |R| ≤ |S|/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Then the number of vertices u ∈ Vt with |N(u) ∩ S| ≥ p|S|/2 is at least |Vt \\ (S ∪ R)| ≥ |Vt| − (|S| + |R|) ≥ |Vt| − 3|S|/2 = |Vt| − 3|At|/4 ≥ |Vt|/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Since each of those vertices u has at least p|S|/2 neighbors in S ⊆ A2k t , Lemma 7 gives that the probability at least one neighbor of u is stable black (and thus u is also stable) at the end of round t + log n is at least (1/5) · min � 1, (p|S|/2) · (4k)−1� = (1/5) · min � 1, (p|At|/4) · (32p|At|)−1� = 1/640.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Then the expected number of vertices that are not stable at the end of round t + log n is E[|Vt+log n|] ≤ |Vt| − (|Vt|/4) · 1/640 ≤ |Vt| − |Vt|/2560.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The next lemma considers the case in which the number of vertices that are not stable is large, namely |Vt| = Ω(ln2(n)/p), and |At| = O(ln(n)/p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Lemma 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' If |Vt| ≥ 10 ln2(n)/p and |At| ≤ 80 ln(n)/p then there is a constant ǫ > 0 such that E[|Vt+log n|] ≤ (1 − ǫ/ ln n) · |Vt|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' From property (P1) in Definition 17, the average degree of graph G[At] is at most k = max{8p|At|, 4 ln n} ≤ 640 ln n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Let S be a subset of At consisting of the 2|At|/3 vertices u ∈ At with the smallest degree in G[At], and let T = At \\ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Then for all u ∈ S, |N(u) ∩ At| ≤ 3k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' thus S ⊆ A3k t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The set Vt consist of (i) all the active vertices, u ∈ At = S ∪ T, and (ii) all the non-active vertices that are not in N +(It) (these vertices are white and have at least one active neighbor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We can thus partition Vt into the four distinct sets: S, N(S)\\N(It), T \\N(S), and N(T)\\N +(S ∪It).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' For the sizes of these sets, we have |T \\ N(S)| ≤ |T| < |S| and, by property (P3) in Definition 17, |N(T) \\ N +(S ∪ It)| ≤ |N(S) \\ N(It)| + 8 ln2(n)/p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Using these two inequalities, the fact that the sizes of the four sets above sum to |Vt|, and the assumption |Vt| ≥ 10 ln2(n)/p, we obtain |S| + |N(S) \\ N(It)| ≥ (|Vt| − 8 ln2(n)/p)/2 ≥ |Vt|/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' 13 Therefore, at least |Vt|/10 vertices u ∈ Vt are in S or adjacent to a vertex from S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' From Lemma 6, each u ∈ S ⊆ A3k t is stable black (and all its neighbors are stable white) at the end of round t + log n, with probability at least 1/(6ek).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' It follows that E[|Vt+log n|] ≤ |Vt| − (|Vt|/10) · 1/(6ek) ≤ |Vt| − |Vt|/(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='1 · 105 ln n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' In the next lemma we analyze the remaining case, in which |Vt| = O(ln2(n)/p) and |At| = O(ln(n)/p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' In fact, the lemma does not require a bound on |At|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Unlike the previous lemmas, however, it requires that p = O( � log(n)/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Lemma 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' If |Vt| ≤ 10 ln2(n)/p and p ≤ c � log(n)/n, for some constant c > 0, then there is a constant ǫ = ǫ(c) > 0 such that E[|Vt+2 log n|] ≤ � 1 − ǫ/ ln3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='5 n � |Vt|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='10 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' From property (P1) in Definition 17, the average degree of graph G[Vt] is at most k = max{8p|Vt|, 4 ln n} ≤ 80 ln2 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Let T be a subset of Vt consisting of the min{ln(n)/p, |Vt|/2} vertices u ∈ Vt with the largest degree in G[Vt], and let S = Vt \\ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Then |S| ≥ |T|, and for all u ∈ S, |N(u) ∩ Vt| is at most d = k|Vt|/|T| ≤ k · max{p|Vt|/ ln n, 2} ≤ 800 ln3 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' From property (P4) in Definition 17, the number of edges between S and T is |E(S, T)| ≤ 6|S| ln n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Let R = {u ∈ S : |N(u)∩T| ≤ 12 ln n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Then |R| ≥ |S|/2 ≥ |Vt|/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We will show for some constant ǫ′ = ǫ′(c) that P[u /∈ Vt+2 log n] ≥ ǫ′ ln−α−1 n · (ln ln n)−α, for all u ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' (4) It follows that E[|Vt+2 log n|] ≤ |Vt| − (|Vt|/4) · ǫ′ ln−α−1 n · (ln ln n)−α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Since α = 1/log(4/3) ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='41, the above implies the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' To complete the proof it remains to show (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Let u ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We partition the neighbors of u in G[Vt] into sets N(u) ∩ S and N(u) ∩ T, and let x = P[N(u) ∩ S ∩ A ̸= ∅] and y = P[N(u) ∩ T ∩ B ̸= ∅], where A = At ∪ · · · ∪ At+r−2, B = Bt+r−2 ∪ Bt+r−1 ∪ Bt+r, and r = log(48 ln n) + 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We distinguish the following three cases: x + y ≤ 1/2, x ≥ 1/4, and y ≥ 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Case x + y ≤ 1/2: With probability at least 1 − (x + y) ≥ 1/2, we have N(u) ∩ S ∩ A = ∅ and N(u) ∩ T ∩ B = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' If N(u) ∩ S ∩ A = ∅ then N(u) ∩ S ⊆ Wt+r−2 (it is easy to see that N(u) ∩ S ∩ It+r−2 = ∅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Similarly, if N(u) ∩ T ∩ B = ∅, it is immediate that N(u) ∩ T ⊆ Wt+r−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Thus, with probability at least 1/2, we have that N(u) ⊆ Wt+r−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' If N(u) ⊆ Wt+r−2, then either u ∈ At+r−2 ∩ Wt−r−2 or u ∈ It+r−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Therefore, with probability at least 1/2, either u /∈ Vt+r−2 or u ∈ At+r−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' If u ∈ At+r−2, then u ∈ Ad t+r−2 since |N(u) ∩ Vt| ≤ d, and from Lemma 6, the probability that u ∈ It+r−2+log n is at least (2ed)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Combining the last two statements yields that the probability of u /∈ Vt+r−2+log n is at least (1/2) · (2ed)−1 ≥ (8700 ln3 n)−1, which implies (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Case x ≥ 1/4: With probability at least 1/4, there is a pair v, j such that v ∈ N(u) ∩ S, 0 ≤ j ≤ r − 2, and v ∈ At+j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' And if v ∈ At+j then v ∈ Ad t+j since |N(v) ∩ Vt| ≤ d, and from Lemma 6, the probability that v ∈ It+j+log n is at least (2ed)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We conclude that the probability that u ∈ N +(It+r−2+log n) is at least (1/4) · (2ed)−1 ≥ (17400 ln3 n)−1, which implies (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Case y ≥ 1/4: There exists some v∗ ∈ N(u) ∩ T such that P[v∗ ∈ B] ≥ y/|N(u) ∩ T| ≥ (4 · 12 ln n)−1 = (48 ln n)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' 10If c is super constant, then the proof gives E[|Vt+2 log n|] ≤ � 1 − ǫ/(c2 ln3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='5 n) � |Vt|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' 14 If v∗ ∈ At then we can apply Lemma 15, for ℓ = r − 2 ≥ log(48 ln n) + 2 and br ≥ (48 ln n)−1, to obtain that v∗ ∈ Aλ t ∪ · · · ∪ Aλ t+r−1 with probability at least q = r1−α · (16 · 48 ln n)−α, where λ = θv∗� α log (32r · 48 ln n) � + log (32r · 48 ln n) + log(48 ln n) + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Suppose now that v∗ ∈ Vt \\ At, and let p∗ = P[v∗ ∈ At+1 ∪ · · · ∪ At+r−1] − 2−|N(v∗)∩At|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' If p∗ ≥ P[v∗ ∈ B]/2 ≥ (96 ln n)−1, then we can apply Lemma 14 (using r − 1 in place of r), to obtain that v∗ ∈ Aλ′ t ∪· · ·∪Aλ′ t+r−2 with probability at least q′ = r1−α ·(p∗/2)α ≥ r1−α·(192 ln n)−α, where λ′ = θv∗� α log (4r · 96 ln n) � + log (4r · 96 ln n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' If p∗ < P[v∗ ∈ B]/2, then we can apply Lemma 16, for ℓ = r − 2 ≥ log(48 ln n) + 4 and br = P[v∗ ∈ B] ≥ (48 ln n)−1, to obtain that that v∗ ∈ Aλ′′ t ∪ · · · ∪ Aλ′′ t+r−1 with probability at least q′′ = r1−α · (64 · 48 ln n)−α, where λ′′ = θv∗� α log (128r · 48 ln n) � + log (128r · 48 ln n) + log(4 · 48 ln n) + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' In all the settings above, we have q, q′, q′′ ≥ ε ln−α n · (ln ln n)1−α and λ, λ′, λ′′ ≤ β ln n · ln ln n, for some constants ε, β > 0, where the bound on λ, λ′, λ′′ holds because property (P5) and assumption p ≤ c � log(n)/n imply that for any v ∈ V , θv(i) ≤ i · (6c2 + 4) log n (recall the definition of θv from (3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Therefore, the probability that v∗ is (β · ln n · ln ln n)-active at the end of some round in {t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' , t + r − 1} is at least ε · ln−α n · (ln ln n)1−α, and from Lemma 6, the probability that v∗ ∈ It+r−1+log n is at least ε · ln−α n · (ln ln n)1−α · (2eβ · ln n · ln ln n)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Thus with at least that probability we have u /∈ Vt+r−1+log n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' This completes the proof of (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Putting the Pieces Together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' First, suppose that p ≤ c � log(n)/n for some constant c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' From Lemmas 21 to 23, E[|Vt+2 log n|] ≤ � 1 − ǫ/ ln3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='5 n � E[|Vt|], for any t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Iteratively applying this inequality, we obtain that for any i ≥ 0, E[|V2i log n|] ≤ � 1 − ǫ/ ln3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='5 n �i · n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Substituting i = 3 ln4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='5 n/ǫ yields E � |V(6/ǫ) log n·ln4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='5 n| � ≤ n−2, and by Markov’s inequality, it follows P[|V(6/ǫ) log n·ln4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='5 n| ≥ 1] ≤ n−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' If p ≥ ε/ ln2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='5 n for some constant ε > 0, then we use Lemmas 21 and 22 as above to obtain that P[|Vt| ≥ 10 ln2(n)/p] ≤ n−2 for some t = O(log3 n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We also observe that if |Vt| < 10 ln2(n)/p then the maximum degree of graph (V, E(Vt)) is ∆ < |Vt| ≤ 10 ln2(n)/p ≤ 10 ln4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='5(n)/ε, and Theorem 12 yields a bound of O(∆ log n) = O(log5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='5 n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Combining the two completes the proof of Lemma 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Remark 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Some of the logarithmic factors can be shaved off with a more careful analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' For example, using a “pipelining” argument, one could improve the bound on halving |Vt| obtained from Lemma 23, from O(log n · ln3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='5 n) to O(log n + ln3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='5 n), thus saving one logarithmic factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' 5 Logarithmic Switch and the 3-Color MIS Process We present an extension of the 2-state MIS process, called 3-color MIS process, which uses one additional color, grey, and includes also a sub-process, called logarithmic switch, which runs in parallel to the main process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Then we analyze the 3-color MIS process on Gn,p random graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' 15 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='1 The Logarithmic Switch Process We first introduce an abstract logarithmic switch process, by specifying its properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Then we describe an actual randomized graph process that satisfies these properties with high probability and in a self-stabilizing manner, using 6 states per vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Definition 25 (Logarithmic Switch Process).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' An (a, b)-logarithmic switch process on G = (V, E) generates for each vertex u ∈ V a binary sequence σ0(u), σ1(u), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' , where σt(u) ∈ {on, off} for each t ≥ 0, such that the following properties hold for all u ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' (S1) Every run of consecutive off values in sequence σ0(u), σ1(u), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' has length at most a ln n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' (S2) If diam(G) ≤ 2 then every run of consecutive off values in sequence σt(u), σt+1(u), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' has length at least a 6 ln n, where t = min{i ≥ a 6 ln n: σi(u) = on}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' (S3) If diam(G) ≤ 2 then every run of consecutive on values in sequence σt(u), σt+1(u), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' has length at most b, where t is some constant independent of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Definition 26 (Randomized Logarithmic Switch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' In the randomized logarithmic switch process on G = (V, E), each vertex u ∈ V has a state, called level, that takes on values in the set {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' , 5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The initial value level0(u) of u can be arbitrary, and in each round t ≥ 1 the level of u is updated according to the following rule, which uses a global parameter 0 < ζ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' if level t−1(u) = 5 then choose a random bit bt(u) such that P[bt(u) = 0] = ζ end if (level t−1(u) = 5 and bt(u) = 1) or level t−1(u) = 0 then set level t(u) = 5 else set level t(u) = max{level t−1(v): v ∈ N +(u)} − 1 Finally, we define the following mapping of the levels to the binary on/off values of Definition 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' For each u ∈ V and t ≥ 0, σt(u) = � on if levelt(u) ≤ 2 off if levelt(u) ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Lemma 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' For any graph G = (V, E), the randomized logarithmic switch process with parameter 0 < ζ ≤ 1/2 satisfies properties (S1) to (S3) for a = 4/ζ and b = 3, with probability 1 − O(n−2), during the first n rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Let u ∈ V , and let Sv ⊆ V be the set of vertices at distance at most 2 from u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' If u has level at least 3 in all rounds t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' , t+a ln n, then no vertex v ∈ Su has level 0 in rounds t+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' , t+a ln n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' and at least one vertex v ∈ Su must be at level 5 in all rounds t + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' , t + a ln n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' It follows that the probability there is some u ∈ V and t ≤ n such that u has level at least 3 in all rounds t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' , t + a ln n is at most n2(1 − ζ)a ln n−4 ≤ n2−aζ/(1 − ζ)4 ≤ 16 · n−2, when aζ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Thus, property (S1) holds with probability at least 1 − O(n−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Next we assume diam(G) ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The rest of the proof is similar to that in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Observe that there must be a vertex v and a round t∗ ≤ 5 such that levelt∗(u) = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' And from the end of round t∗ + 2, all vertices “synchronize” in the sense that once a vertex reaches level 2 in a round, all vertices reach level 2 in that round, then the they all reach level 1 in the next round, then level 0, and then 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' It follows that property (S3) holds for b = 3, starting from round t∗ + 2 ≤ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The property holds with probability 1, and for all rounds after round t∗ + 2, not just for the first n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' 16 As mentioned above, after vertices have synchronized, all n vertices move from level 0 to level 5 simultaneously, each time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' When that happens, the number of rounds until there are no vertices left at level 5 is greater than a ln n − 6 with probability at most n(1 − ζ)a ln n−6 ≤ 64 · n−3, as before;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' and is smaller than r = a 6 ln n with probability at most (1 − (1 − ζ)r)n ≤ e−n(1−ζ)r ≤ e−n4−ζr = e−n4−(aζ/6) ln n ≤ e−n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='07 = O(n−3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Combining the above, using a union bound, we obtain that property (S2) holds with probability 1 − O(n−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='2 The 3-Color MIS Process We now define the 3-color MIS process, which is an extensions of the 2-state MIS process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Definition 28 (3-Color MIS Process).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The process consists of two (sub-)processes that run in parallel on G = (V, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The first is an (a, 3)-logarithmic switch process, where a = 512, which gen- erates a value σt(u) ∈ {on, off} for each vertex u ∈ V in each round t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The second is a variant of the 2-state MIS process, where each vertex u ∈ V has a state ct(u) ∈ {black, white, gray}, c0(u) can be arbitrary, and in each round t ≥ 1, u’s state is updated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' let NC t(u) = {ct−1(v): v ∈ N(u)} if ct−1(u) = black and NC t(u) ∋ black then let ct(u) be a uniformly random state from {black,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' gray} else if ct−1(u) = white and NC t(u) ̸∋ black then let ct(u) be a uniformly random state from {black,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' white} else if ct−1(u) = gray and σt−1(u) = on then set ct(u) = white else set ct(u) = ct−1(u) There are precisely two differences in the update rule above compared to that for the 2-state MIS process: a black vertex with a black neighbor changes to gray with probability 1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' rather than to white;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' and a gray vertex changes to white if its switch value is on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Note that a gray vertex is treated similarly to a non-active white vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' A vertex is stable, if it is black and has no black neighbors, or it is not black and has a neighbor that is stable black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Other than that, the remaining definitions and notations are the same as in the 2-state MIS process, namely, of active vertices, stabilization times, Bt, Wt, At, Ak t , It, and Vt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We also let Γt = V \\ (Bt ∪ Wt) denote the set of gray vertices at the end of round t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The definition of the 3-color MIS process above assumes an arbitrary logarithmic switch process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We can use the randomized logarithmic switch from Definition 26, which uses 6 states per vertex, to obtain a 3-color MIS process that uses 6 · 3 = 18 states in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The probability parameter of the randomized switch is ζ = 4/a = 27, thus at most 7 random bits are required per round for each vertex (plus one more for each active vertex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We note that Lemmas 6 and 7 and their proofs carry over to the 3-color MIS process, without changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We will use also the two simple lemmas below that are specific to the 3-color MIS process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Recall that a = 512 is a parameter of the logarithmic switch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Lemma 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' If t ≥ a ln n and u ∈ Γt then u ∈ At−a ln n ∪ · · · ∪ At−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' 17 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' By property (S1) in Definition 25 of the logarithmic switch, a vertex is gray for at most a ln n consecutive rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Also if a vertex becomes gray in round j > 0, it must be active black at the end of round j − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Combining these two facts implies the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Lemma 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' If diam(G) ≤ 2, u ∈ V , t ≥ a 6 ln n, and t′ = t + a 6 ln n, then the expected number of times u is active black between rounds t and t′ is E [|{j : u ∈ Bj ∩ Vj} ∩ {t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' , t′}| | Bt, Wt] ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' From properties (S2) and (S3) in Definition 25, it is easy to see that sequence ct(u), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' , ct′(u) contains at most two runs of consecutive black states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Moreover, the expected length of the prefix of each black run until u becomes stable black or the run finishes (and u becomes gray) is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' It follows that u is non-stable black in at most 4 rounds in expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Lemmas 13 and 14 hold also for the 3-color MIS process, when u ∈ Vt \\ (At ∪ Γt) and thus u ∈ Wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='11 The next simple lemma will be used together with Lemma 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Lemma 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Let t ≥ 0, u ∈ V , and d > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Let t′ ≥ t be the first round when either u is white and has at least d black neighbors, or u is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The expected number of rounds t < j < t′ at which u is black and has at least d black neighbors is at most 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The lemma is obtained using the observations that: each time u’s state changes from white to black, it is equally likely that it remained white;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' and, when u is active black, it becomes gray in the next step with probability 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='3 Analysis of the 3-Color MIS Process on Gn,p We show that the stabilization time of the 3-color MIS process on Gn,p random graphs is poly(log n), for the complete range of values of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Theorem 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The stabilization time of the 3-color MIS process on a random graph drawn from Gn,p is O(log6 n) with probability 1 − O(n−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' As before, it suffices to show that the above bound holds for good graphs, and apply Lemma 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Lemma 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The stabilization time of the 3-color MIS process on any (n, p)-good graph G = (V, E) is O(log6 n) with probability 1 − O(n−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='1 Proof of Lemma 33 The proof strategy is similar to Lemma 20’s: From any vector of vertex states at the end of round t, we show that the process makes sufficient progress in expectation in poly(log n) rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The main difference is that now we show that this is also true even in the case of |Vt| = O(log2(n)/p) when diam(G) ≤ 2, which corresponds to the case of p = Ω( � log(n)/n), by property (P6) in Definition 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' This is precisely the case that we could not handle in the analysis of the 2 state MIS process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The relevant lemma is Lemma 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We first observe that Lemma 21, which considers that case of |At| = Ω(log(n)/p), carries over to the 3-color MIS process, without any changes in the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Next we consider the case where |At| = O(ln(n)/p), |Vt| = Ω(ln2(n)/p), and |Γt| = O(ln2(n)/p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The following lemma is very similar to Lemma 22, except that it requires also a bound on |Γt|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Recall that a = 512 is a parameter of the logarithmic switch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' 11The proofs require just minor modifications, mostly replacing some occurrences of “white” by “not black” or “gray”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' 18 Lemma 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' If |Vt| ≥ 82a ln2(n)/p, |At| ≤ 80 ln(n)/p, and |Γt| ≤ 80a ln2(n)/p, then there is a constant ǫ > 0 such that E[|Vt+log n|] ≤ (1 − ǫ/ ln n) · |Vt|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The proof is very similar to Lemma 22’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' As before, from property (P1) in Definition 17, the average degree of G[At] is at most k = max{8p|At|, 4 ln n} ≤ 640 ln n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We let S be a subset of At consisting of the 2|At|/3 vertices u ∈ At with the smallest degree in G[At], and let T = At \\ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Then for all u ∈ S, |N(u) ∩ At| ≤ 3k, thus S ⊆ A3k t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The set Vt consist of (i) all active vertices, u ∈ At = S ∪T, (ii) all non-active non-stable vertices that have some active neighbor, and (iii) all non-active non-stable vertices have no active neighbors (these vertices are gray).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We can thus partition Vt into the five distinct sets: S, N(S) \\ N(It), T \\ N(S), N(T) \\ N +(S ∪ It), and Vt \\ N +(T ∪ Sf) ⊆ Γt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We have that |Vt \\ N +(T ∪ S)| ≤ |Γt| ≤ 80a ln2(n)/p, |T \\ N(S)| ≤ |T| < |S|, and, by property (P3) in Definition 17, |N(T) \\ N +(S ∪ It)| ≤ |N(S) \\ N(It)| + 8 ln2(n)/p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Using these three inequalities, the fact that the sizes of the five sets above sum to |Vt|, the assump- tion |Vt| ≥ 82a ln2(n)/p, and that a ≥ 8, we obtain |S| + |N(S) \\ N(It)| ≥ (|Vt| − (80a + 8) ln2(n)/p)/2 ≥ (|Vt| − 81a ln2(n)/p)/2 ≥ |Vt|/82a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Therefore, at least |Vt|/82a vertices u ∈ Vt are in S or adjacent to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' And, from Lemma 6, each u ∈ S ⊆ A3k t is in It+log n, with probability at least 1/(6ek).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' It follows E[|Vt+log n|] ≤ |Vt| − (|Vt|/82a) · 1/(6ek) ≤ |Vt| − |Vt|/(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='1 · 82a · 104 ln n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Next we assume |At| = O(ln(n)/p) and |Vt| = Ω(ln2(n)/p), as in the previous lemma, but now |Γt| = Ω(ln2(n)/p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We reduce this case to the previous cases using Lemma 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Lemma 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' If |Vt| ≥ 83a ln2(n)/p, |At| ≤ 80 ln(n)/p, and |Γt| > 80a ln2(n)/p, then there is a constant ǫ > 0 such that E[|Vt+a ln n+log n|] ≤ (1 − ǫ/ ln n) · |Vt|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Let τ = min{j ≥ t: |Vj| ≤ 82a ln2(n)/p or |Aj| ≥ 80 ln(n)/p or |Γj| ≤ 80a ln2(n)/p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We have τ ≤ t + a ln n, because if |Γt+a ln n| > 80a ln2(n)/p, then Lemma 29 implies there is some j ∈ {t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' , t + a ln n − 1} such that |Aj| ≥ |Γt+a ln n|/(a ln n) ≥ 80 ln(n)/p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We distinguish three cases depending on which condition in the definition of τ is satisfied first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' If |Vτ| ≤ 82a ln2(n)/p, then |Vt+a ln n| ≤ |Vτ| ≤ 82a ln2(n)/p ≤ (1 − 1/83) · |Vt|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' If |Aτ| ≥ 80 ln(n)/p, then Lemma 21 yields E[|Vt+a ln n+log n|] ≤ E[|Vt+τ+log n|] ≤ (1 − ǫ) · |Vt|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Last, if |Γτ| ≤ 80a ln2(n)/p and the other two conditions do not hold, then Lemma 34 gives E[|Vt+a ln n+log n|] ≤ (1 − ǫ/ ln n) · |Vt|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The next two lemmas deal with the case of |Vt| = O(ln2(n)/p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The first one assumes diam(G) ≤ 2, and thus covers the case of p = Ω( � log(n)/n), by property (P6) in Definition 17;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' while the second lemma assumes p = O( � log(n)/n) and is similar to Lemma 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Lemma 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' For any t ≥ a 6 ln n, if |Vt| ≤ 83a ln2(n)/p and diam(G) ≤ 2 then there is a constant ǫ > 0 such that E[|Vt+ 7 6 a log n+log n|] ≤ � 1 − ǫ/ ln3 n � |Vt|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' From property (P1) in Definition 17, the average degree of induced subgraph G[Vt] is at most k = max{8p|Vt|, 4 ln n} ≤ 664a ln2 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Let T be a subset of Vt consisting of the min{ln(n)/p, |Vt|/2} 19 vertices u ∈ Vt with the largest degree in G[Vt], and let S = Vt \\ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Then |S| ≥ |T|, and all u ∈ S, |N(u) ∩ Vt| is at most d = k|Vt|/|T| ≤ k · max{p|Vt|/ ln n, 2} ≤ 55112 ln3 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' From property (P4) in Definition 17, |E(S, T)| ≤ 6|S| ln n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Let R = {u ∈ S : |N(u) ∩ T| ≤ 12 ln n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Then |R| ≥ |S|/2 ≥ |Vt|/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We will show that, for some constant ǫ′ > 0, P[u /∈ Vt+ 7 6 a ln n+log n] ≥ ǫ′ ln−3 n, for all u ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' (5) From this, it follows that E[|Vt+ 7 6 a ln n+log n|] ≤ |Vt| − (|Vt|/4) · ǫ′ ln−3 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' To complete the proof of the lemma it remains to prove (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Let u ∈ R, and suppose that u /∈ Γt (we deal with the case u ∈ Γt at the end).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' From Lemma 30, the expected value of � t≤j≤t+ a 6 ln n |(N(u) ∩ T) ∩ (Bj ∩ Vj)|, that is, the total number of times that vertices v ∈ N(u)∩T are active black between rounds t and a 6 ln n, is at most 4·|N(u)∩T| ≤ 4·12 ln n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Then, by Markov’s inequality, that number is at most 5 · 12 ln n with probability at least 1/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' And since a 6 > 5 · 12, it follows that, with probability at least 1/5, there is some j ∈ {t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' , t + a 6 ln n} such that (N(u) ∩ T) ∩ (Bj ∩ Vj) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Next we claim that, if (N(u) ∩ T) ∩ (Bj ∩ Vj) = ∅ for some j ≥ t, then (i) u /∈ Vj, or (ii) u ∈ Aj′ for some t ≤ j′ < j, or (iii) (N +(u) ∩ S) ∩ Aj ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Indeed, suppose that (i) and (ii) do not hold, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=', u ∈ Vj and u /∈ At ∪ · · · ∪ Aj−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' From u ∈ Vj, it follows N +(u) ∩ Ij = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' From u /∈ At ∪ · · · ∪ Aj−1 and the assumption u /∈ Γt, it follows u ∈ Wj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Then, if (N(u) ∩ S) ∩ Bj ̸= ∅, each vertex v ∈ (N(u) ∩ S) ∩ Bj is in Aj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' while if (N(u) ∩ S) ∩ Bj = ∅, then N(u) ∩ Bj = ∅ and u ∈ Aj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Therefore (iii) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' From the above, it follows that with probability at least 1/5, there is some t ≤ j ≤ t + a 6 ln n such that u /∈ Vj or (N +(u) ∩ S) ∩ Aj ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' And if v ∈ (N +(u) ∩ S) ∩ Aj, then v ∈ Ad j, and from Lemma 6, the probability that v ∈ Ij+log n is at least 1/(6ed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We conclude that P[u /∈ Vt+ a 6 ln n+log n] ≥ (1/5) · 1/(6ed) ≥ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='5 · 106 ln3 n)−1, which implies (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Finally, if u /∈ Γt, we consider the first round j > t such that u /∈ Γj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' From property (S1), j ≤ t + a ln n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Then we apply the result for the previous case to complete the proof of (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Lemma 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' If |Vt| ≤ 83a ln2(n)/p and p ≤ c � log(n)/n for some constant c > 0, then there is a constant ǫ = ǫ(c) > 0 such that E[|Vt+log1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='1 n|] ≤ � 1 − ǫ/ ln3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='9 n � |Vt|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Proof Sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We define the set S, T, R and the degree thresholds k, d as in the proof of Lemma 36, and we show P[u /∈ Vt+log1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='1 n] ≥ ǫ′ ln−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='9 n, for all u ∈ R, (6) which implies the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Next we prove (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Let u ∈ R, and suppose that u /∈ Γt (we deal with case u ∈ Γt at the end).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' For each v ∈ N(u)∩T, let tv ≥ t be the first round when either v is white and has at least ℓ = ln n black neighbors, or is stable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' and let xv be the number of rounds t ≤ j ≤ min{tv, t+r} at which v is black and has at least ℓ black neighbors, where r = 12 ln n · ln2 ln n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' From Lemma 31, the probability that xv ≤ ln2 ln n for all v is at least 1 − |N(u) ∩ T| · e−Ω(ln2 ln n) = 1 − e−ω(ln ln n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' For each v ∈ N(u) ∩ T let pv be the conditional probability that v ∈ Btv+1 ∪ · · · ∪ Bt+r, given Bt, Wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' If � v∈N(u)∩T pv ≤ 1/2, then with probability at least 1/2 − e−ω(ln ln n) > 1/3, the total number of rounds in which at least one v ∈ N(u) ∩ T is black and has at least ℓ black neighbors is at 20 most |N(u) ∩ T| · ln2 ln n ≤ 12 ln n · ln2 ln n ≤ r, thus there is some j ∈ {t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' , t + r} such that no v ∈ N(u) ∩ T is black and has at least ℓ black neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Then we can infer that with probability at least 1/3 some vertex in N +(u) is stable black or is d-active at some round in {t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' , t + r}, in the same way as in the proof of Lemma 36, and then obtain (6) using Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' If � v∈N(u)∩T pv > 1/2, then there is some v∗ ∈ N(u) ∩ T such that pv∗ ≥ (2|N(u) ∩ T|)−1 ≥ (24 ln n)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We can then apply Lemma 14 to v∗ at round tv∗ to show that the probability vertex v∗ is z-active at some round in {t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' , t + r}, where z = θu � α log 4r pv∗−2−ℓ � + log 4r pv∗−2−ℓ = O(log n · log log n), is at least r1−α· � pv∗−2−ℓ 2 �α = Ω(ln3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='9 n), as α ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='41 Again we obtain (6) using Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Finally, as before, if u /∈ Γt, we consider the first round j > t such that u /∈ Γj, and apply the result for the previous case to complete the proof of (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We can now conclude the proof of Lemma 33, as we did for Lemma 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' From Lemmas 21 and 34 to 37, we have that for any t ≥ a 6 ln n, E[|Vt+log1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='1 n|] ≤ � 1 − ǫ/ ln3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='9 n � E[|Vt|].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Iteratively applying this inequality, and using by Markov’s inequality, we obtain as before P[|Vc′ ln6 n| ≥ 1] ≤ n−2, for a large enough constant c′ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' This completes the proof of Lemma 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' APPENDIX A Omitted Proofs A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='1 Proof of Lemma 7 We assume k1 ≤ k2 ≤ · · · ≤ kℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' For 1 ≤ i ≤ ℓ, let ri = ⌈log(ki + 1)⌉, let Ei be the event that φt+1(ui) = · · · = φt+ri(ui) = black, and let E = � i Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Then P[E] = 1 − � i � 1 − 1 2ri � ≥ 1 − � i � 1 − 1 2ki � ≥ � 1 − e− � i 1 2ki � ≥ � 1 − e−1� min � 1, � i 1 2ki � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Suppose that E occurs and let j be the smallest index such that Ej occurs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=', ¯E1 ∩ · · · ∩ ¯Ej−1 ∩ Ej occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' If gj = |N(uj) ∩ {u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' , uj−1}|, then the probability that none of the kj vertices v ∈ N(uj) ∩ At satisfies φt+1(v) = · · · = φt+rj(v) = black is (1 − 2−rj)kj−gj ≥ (1 − 2−rj)kj ≥ e−1, similarly to (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Combining this with the previous inequality we obtain that the probability that ui ∈ It+ri for at least one vertex ui ∈ {u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' , uℓ} is at least e−1 · � 1 − e−1� min � 1, � i 1 2ki � ≥ 1 5 · min � 1, � i 1 2ki � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='2 Proof of Lemma 15 Let B be the event u ∈ Bt+ℓ ∪ · · · ∪ Bt+r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' then P[B] = br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Let τ = min{j > t: u ∈ Wj or |N(u) ∩ Bj| ≤ k} be the first round j > t at the end of which u is white or has at most k black neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We have P[τ > t + ℓ] ≤ 2ℓ ≤ br/4, since τ > t + ℓ implies φt+1(u) = · · · = φt+ℓ(u) = black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Thus P[τ ≤ t + ℓ] ≥ 1 − br/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Let x = P[|N(u) ∩ Bτ| ≤ k | τ ≤ t + ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' 21 We distinguish two cases, x ≥ br/4 and x ≤ br/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' First suppose that x ≥ br/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' For any given j > t, P[u ∈ Aj | τ = j, |N(u) ∩ Bτ| ≤ k] = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The reason is that u ∈ Bj−1 and |N(u) ∩ Bj−1| > k > 0 if τ = j > t + 1, and u ∈ At = Aj−1 if τ = j = t+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' In either case u ∈ Aj−1, thus the state of u at the end of round j is chosen uniformly at random, independently of the remaining choices in round j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' In particular, u is black with probability 1/2 when 0 < |N(u)∩Bj| ≤ k, and is white with probability 1/2 when |N(u)∩Bj| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' It follows that P[{u ∈ Aτ} ∩ {|N(u) ∩ Bτ| ≤ k} ∩ {τ ≤ t + ℓ}] ≥ (1/2) · x · (1 − br/4) ≥ 3br/32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Since the event on the left side implies that u is k-active at the end of round τ ≤ t + ℓ < t + r, and 3br/32 is greater than the desired lower bound for qr, the lemma holds in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Suppose now that x ≤ br/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Then P[B ∩ {|N(u) ∩ Bτ| > k} ∩ {τ ≤ t + ℓ}] ≥ P[B] − P[τ > t + ℓ] − x ≥ br − br/4 − br/4 = br/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' If τ ≤ t+ℓ and |N(u)∩Bτ| > k (and thus u ∈ Wτ by τ’s definition), we define the following events: Ak is the event that u ∈ Ak τ+1 ∪ · · · ∪ Ak t+r−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' A is the event that u ∈ Aτ+1 ∪ · · · ∪ At+r−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' and X is the event that the states of vertices at the end of round τ are such that the conditional probability of A, given these states and τ, is at least br/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' If τ ≤ t + ℓ and |N(u) ∩ Bτ| > k, then event B implies A, because vertex u, which is non-active white at the end of round τ, cannot become black before becoming active first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Thus, from the last inequality above, it follows P[A ∩ {|N(u) ∩ Bτ| > k} ∩ {τ ≤ t + ℓ}] ≥ br/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Also P[A ∩ X ∩ {|N(u) ∩ Bτ| > k} ∩ {τ ≤ t + ℓ}] ≥ br/2 − br/4 = br/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We can now apply Lemma 14, starting from round τ ≤ t + ℓ, using d > k ≥ log(1/br) + 3 and pr ≥ br/4, to obtain P[Ak ∩ X ∩ {|N(u) ∩ Bτ| > k} ∩ {τ ≤ t + ℓ}] ≥ r1−α · �br/4 − 2k 2 �α ≥ r1−α · �br/4 − br/8 2 �α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' It follows that qr = P[Ak] ≥ r1−α · � br/4−br/8 2 �α , which concludes the proof of this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='3 Proof of Lemma 16 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We have P[u ∈ At+1] = 2−d and P[u ∈ (At+2 ∪ · · · ∪ At+r−1) \\ At+1] = pr − 2−d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We also note that if u /∈ At+1 then u ∈ Wt+1, and u may become black in a subsequent round only after it becomes active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' It follows that P[{u ∈ (Bt+ℓ ∪ · · · ∪ Bt+r) ∩ At+1] = br − P[u ∈ (Bt+ℓ ∪ · · · ∪ Bt+r) \\ At+1] ≥ br − P[u ∈ (At+2 ∪ · · · ∪ At+r−1) \\ At+1] ≥ br − (pr − 2−d) ≥ br/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' 22 Let X be the event that the states of vertices at the end of round t+1 are such that the conditional probability of u ∈ Bt+ℓ ∪ · · · ∪ Bt+r is at least br/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Then P[{u ∈ (Bt+ℓ ∪ · · · ∪ Bt+r) ∩ At+1} ∩ X] ≥ br/2 − P[{u ∈ (Bt+ℓ ∪ · · · ∪ Bt+r) ∩ At+1} ∩ ¯ X ] ≥ br/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We can now apply Lemma 15, starting from round t + 1 and using br/4 in place of br, to obtain P[{u ∈ (Ak t+1 ∪ · · · ∪ Ak t+r−1) ∩ At+1} ∩ X] ≥ r1−α · (br/64)α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' This implies the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='4 Proof of Lemma 18 The proof of consists of a series of lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' In all these lemmas, the graph G = (V, E) considered is a random graph drawn from Gn,p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Property (P1) holds trivially for sets S of size k ≤ 4 ln n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The next lemma (applied for all k > 4 ln n, and then combining the results using a union bound) shows that G satisfies the property for all larger sets, with probability at least 1 − n−Ω(log n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Lemma 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Let G = (V, E) be a random graph drawn from Gn,p, and let k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' With probability at least 1 − n−k, all subgraphs of G on k vertices have at most max{4pk2, 2k ln n} edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The probability there is a subgraph with k vertices and at least r = max{2k ln n, 4pk2} edges is at most �n k � �k2/2 r � pr ≤ nk · �ek2 2r �r pr = ek ln n−r ln 2r epk2 ≤ ek ln n−2k ln n·ln 8pk2 epk2 ≤ n−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The next lemma shows that G satisfies property (P2) with probability 1 − n−Ω(log n/p) Lemma 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Let G = (V, E) be a random graph drawn from Gn,p, and let k ≥ 40 ln(n)/p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' With probability at least 1 − n−k, every set S ⊆ V of size |S| = k satisfies |{u ∈ V : |N(u) ∩ S)| < pk/2}| ≤ k/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' For any set S of size k, and any vertex u ∈ V \\ S, the expected number of neighbors of u in S is pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' By a Chernoff bound, the probability that u has fewer than pk/2 neighbors in S is at most e−pk/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Then the probability there is some set S of size k such that at least k/2 vertices u ∈ V \\ S have fewer than pk/2 neighbors in S, is at most �n k � �n − k k/2 � e−(k/2)·pk/8 ≤ nk · nk/2 · e−pk2/16 = e(3/2)k ln n−pk2/16 ≤ n−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Lemma 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Let G = (V, E) be a random graph drawn from Gn,p, and let k = 3 ln(n)/p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' With probability at least 1 − n−k, every set S ⊆ V of size |S| ≥ k satisfies |V \\ N +(S)| ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The probability there is a set S of size k with |V \\ N +(S)| ≥ k is at most �n k � �n − k k � (1 − p)k2 ≤ nk · nk · e−pk2 = e2k ln n−pk2 = n−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The next lemma shows that G satisfies property (P3) with probability 1 − n−Ω(log n/p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' 23 Lemma 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Let G = (V, E) be a random graph drawn from Gn,p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' With probability at least 1 − n− ln(n)/p, every triplet of disjoint sets S, T, I ⊆ V , such that |S| ≥ 2|T| and (S ∪ T) ∩ N(I) = ∅, satisfies |N(T) \\ N +(S ∪ I)| − |N(S) \\ N +(I))| ≤ 8 ln2(n)/p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' (7) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' From Lemma 40, with probability at least 1−n−3 ln(n)/p, all sets S, I ⊆ V such that |S∪I| ≥ 3 ln(n)/p satisfy |V \\ N +(S ∪ I)| ≤ 3 ln(n)/p, and thus |N(T) \\ N +(S ∪ I)| ≤ |V \\ N +(S ∪ I)| ≤ 3 ln(n)/p, which implies (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Next we assume that |S ∪ I| ≤ 3 ln(n)/p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Since |S| ≥ 2|S|, we have |S ∪ T ∪ I| ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='5 ln(n)/p, thus there are at most n4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='5 ln(n)/p different triplets S, T, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Choose one such triplet S, T, I, before revealing the edges of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Then reveal the edges incident to vertices u ∈ I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' this determines N(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Let U = V \\ (S ∪ T ∪ N +(I)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The two sets on the left side of (7) can then be expressed as N(T) \\ N +(S ∪ I) = U ∩ N(T) \\ N(S), and N(S) \\ N +(I) = U ∩ N(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' For every u ∈ U, the probability that u ∈ N(T) \\ N(S) is p1 = P[u ∈ N(T) \\ N(S)] = � 1 − (1 − p)|T|� (1 − p)|S|, and the probability that u ∈ N(S) is p2 = P[u ∈ N(S)] = 1 − (1 − p)|S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Letting ε = (1 − p)|T| and using that |S| ≥ 2|T|, we obtain p1 ≤ (1 − ε) · ε2 and p2 ≥ 1 − ε2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Thus p2 p1 ≥ 1 − ε2 (1 − ε) · ε2 = 1 + ε ε2 ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' It follows that, by considering all vertices u ∈ U one after the other, and revealing all edges incident to each u at the moment u is considered, we can analyze the difference D = |U ∩ N(T) \\ N(S)| − |U ∩ N(S)| = |N(T) \\ N +(S ∪ I)| − |N(S) \\ N +(I)| as a biased random walk on the integers starting at 0, and moving to the right with probability p1 and to the left with probability p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The probability that the (infinite) random walk every reaches value i ≥ 1 is know to be (p1/p2)i ≤ 2−i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Thus, P[D ≥ i] ≤ 2−i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' And the probability that D ≥ 8 ln2(n)/p for at least one possible triplet S, T, I is then at most n4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='5 ln(n)/p · 2−8 ln2(n)/p ≤ n−ln n/p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Property (P4) holds trivially for sets S of size k ≤ 6 ln n, since |S| ≥ |T|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The next lemma (applied for all k > 6 ln n) shows that G satisfies the property with probability at least 1−n−Ω(log n) for all larger sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Lemma 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Let G = (V, E) be a random graph drawn from Gn,p, and let k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' With probability at least 1 − n−2k, every pair of disjoint sets S, T ⊆ V , such that |S| = k ≥ |T| and |T| ≤ ln(n)/p, satisfies |E(S, T)| ≤ 6k ln n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' For any given pair S, T, the expected value of |E(S, T)| is p · |S| · |T| ≤ k ln n, and by a Chernoff bound, the probability that |E(S, T)| ≥ 6k ln n is at most 2−6k ln n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Then the probability there is at least one pair S, T such that |E(S, T)| ≥ 6k ln n is at most n|S| · n|T| · 2−6k ln n ≤ n2k · 2−6k ln n ≤ n−2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' 24 Our last lemma implies that properties (P5) and (P6) hold with probability 1 − O(n−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Lemma 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' In a random graph G drawn from Gn,p, the probability that no two vertices have k common neighbors is at least 1 − n2 · (ep2n/k)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' And the probability that diam(G) ≤ 2 is at least 1 − n2 · e−p2(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The probability there is a pair of vertices that have at least k common neighbors is at most �n 2 � �n−2 k � p2k ≤ n2 · � ep2n k �k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' And the probability there is a pair of vertices with no common neighbors and no adjacent to each other is �n 2 � (1 − p) · (1 − p2)n−2 ≤ n2 · e−p2(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' B Other Related Work In 1985, Luby [24] proposed a simple distributed randomized algorithm that finds an MIS in time O(log n) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Simultaneously, Alon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' [3] proposed a similar algorithm with the same performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Both algorithms work with O(log n)-bit messages and need access to O(log n) random bits at each round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Due to various applications in radio sensor networks, restricted distributed models of communi- cation were introduced, in which the MIS problem has been widely studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' In the beeping model, introduced by Cornejo and Kuhn [9], nodes have no knowledge of the local or global structure of the network, do not have access to synchronized clocks and the communication among nodes relies completely on carrier sensing (as described in the introduction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Afek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' [1] show that in the version of the beeping model where nodes are initially asleep and are woken up by an adversary, it is not possible to locally converge to an MIS in sub-polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Therefore, they consider various relaxations on the model, providing algorithms converging to an MIS in a polylogarithmic number of rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' In detail, if the nodes know an upper bound on the size of the network, or if the beeping nodes are awakened by the neighbor’s beep, the MIS can be found in time O(log3 n) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' If the nodes have synchronous clocks, an MIS can be found in time O(log2 n) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We remark that the authors provide a self-stabilizing algorithm just in the first setting, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' when an upper bound on the size of the network is known by the nodes and that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' In all algorithms, the nodes have super-constnt state and have access to a super-constant number of random bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' In the version of the beeping model with synchronized clocks, collision detection, and simul- taneous wakeup, Afek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' [2] had earlier shown that the MIS problem is solved by a biological process in time O(log2 n) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' [1] showed that this bound is also achievable without knowledge of an upper bound on the size of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Jeavons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' [23] improved these results, showing that an MIS can be found in time O(log n) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' An improved analyisis of the local complexity of this algorithm was provided by Ghaffari [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' In the same version of the beeping model without collision detection, Holzer and Lynch in [21], proposed a variant of the algorithm of [15], and showed that it converges locally in time O((log ∆ + log 1/ε) · log 1/ε) with probability at least 1 − ε on a network with maximum degree ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' All these algorithms require super constant space and random bits per round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Emek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' [13] introduced the stone age model, inspired by biological cellular networks or networks of microprocessor devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' In this model, the nodes can communicate by transmitting messages belonging to a finite communication alphabet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The nodes communicate in an asyn- chronous environment, where the pattern is decided by an adversary, and they have no knowledge about the size of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' In the stone age model, the MIS problem was considered by [13, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' In [13], is provided an algorithm that compute a MIS in O(log2 n) rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' However, it assumes that all the nodes have the same initial state, and therefore is not self-stabilizing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' In [12], they 25 provided a self-stabilizing algorithm that stabilizes in time O((D + log n) log n) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=', and the possible number of states of each node is O(D), where D is the diameter of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' In [25], the authors introduced a randomized distributed algorithm that finds an MIS in time O(log n) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' In particular, the algorithm is an adaptation of Luby’s algorithm so that messages of just 1 bit are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' They consider an anonymous network, but in their setting, the vertices can distinguish between their neighbors, and each vertex needs a number of states that depends on n and the node degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' MIS algorithm has also received a lot of attention from the Self-Stabilization community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' For a survey of those algorithms see [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' We first cite here the self-stabilizing algorithm for non-anonymous networks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' where vertices have IDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' In [18], the authors provide a simple deterministic distributed algorithm that stabilizes on an MIS in O(n) time and O(n2) moves (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' total number of state changed), in a synchronous model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' In [22], the authors proposed a deterministic two-state algorithm that works under distributed scheduler (an adversary that, at each time, selects arbitrarily a set of processes to execute).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Both algorithms stabilize in time O(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' In [29], Turau introduces a 3-state self-stabilizing algorithm that stabilizes in O(n) moves, under a distributed scheduler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' A breakthrough was achieved by Barenboim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' [5], who proposed a self-stabilizing algorithm for the MIS and other related problems, in the synchronous model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' They prove that the algorithm stabilizes after O(∆ + log∗ n) rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Assuming anonymous networks Shukla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' [28] proposed two deterministic two-state self- stabilizing algorithms, that work under a centralized scheduler (an adversary that selects one process to execute at each round) and stabilizes in O(n) rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' In [30], Turau introduced a synchronous randomized self-stabilizing algorithm for MIS that stabilizes w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' in O(log n) rounds w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The possible states of the nodes are O(log n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Next, we briefly summarize the best known upper bounds to compute an MIS in the distributed LOCAL model on arbitrary graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Barenboim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' [6] proved that an MIS can be computed with a distributed deterministic algorithm in O(∆ + log∗ n) rounds and Ghaffari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' [17] provide an upper bound of O(log5 n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Regarding distributed randomized algorithms, Ghaffari [15] provides an upper bound of O(log ∆) + 2O(√log log n) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=', which, thanks to [27, 17], was improved to O(log ∆ + log5 log n) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' See also [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' The current best-known lower bound for finding an MIS is proved by Balliu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' [4], who show that computing an MIS in the LOCAL model requires Ω(min{∆, log n/ log log n}) rounds deterministically, and Ω(min{∆, log log n/ log log log n}) rounds with a randomized algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' References [1] Yehuda Afek, Noga Alon, Ziv Bar-Joseph, Alejandro Cornejo, Bernhard Haeupler, and Fabian Kuhn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Beeping a maximal independent set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' Distributed Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=', 26(4):195–208, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' [2] Yehuda Afek, Noga Alon, Omer Barad, Eran Hornstein, Naama Barkai, and Ziv Bar-Joseph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' A biological solution to a fundamental distributed computing 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+page_content=' 7th IBM Symposium on Mathematical Foundations of Computer Science, 1982.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} +page_content=' 28' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E4T4oBgHgl3EQfZwye/content/2301.05059v1.pdf'} diff --git a/0dFJT4oBgHgl3EQfjix9/content/tmp_files/2301.11575v1.pdf.txt b/0dFJT4oBgHgl3EQfjix9/content/tmp_files/2301.11575v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..da0a167286e79f489232cf8b50f57eca6e03ccd3 --- /dev/null +++ b/0dFJT4oBgHgl3EQfjix9/content/tmp_files/2301.11575v1.pdf.txt @@ -0,0 +1,776 @@ +ARiADNE: A Reinforcement learning approach using Attention-based +Deep Networks for Exploration +Yuhong Cao1, Tianxiang Hou1, Yizhuo Wang1, Xian Yi1, Guillaume Sartoretti1† +Abstract— In autonomous robot exploration tasks, a mobile +robot needs to actively explore and map an unknown envi- +ronment as fast as possible. Since the environment is being +revealed during exploration, the robot needs to frequently +re-plan its path online, as new information is acquired by +onboard sensors and used to update its partial map. While +state-of-the-art exploration planners are frontier- and sampling- +based, encouraged by the recent development in deep reinforce- +ment learning (DRL), we propose ARiADNE, an attention- +based neural approach to obtain real-time, non-myopic path +planning for autonomous exploration. ARiADNE is able to +learn dependencies at multiple spatial scales between areas of +the agent’s partial map, and implicitly predict potential gains +associated with exploring those areas. This allows the agent +to sequence movement actions that balance the natural trade- +off between exploitation/refinement of the map in known areas +and exploration of new areas. We experimentally demonstrate +that our method outperforms both learning and non-learning +state-of-the-art baselines in terms of average trajectory length +to complete exploration in hundreds of simplified 2D indoor +scenarios. We further validate our approach in high-fidelity +Robot Operating System (ROS) simulations, where we consider +a real sensor model and a realistic low-level motion controller, +toward deployment on real robots. +I. INTRODUCTION +Autonomous robot exploration (ARE) refers to the task, +in which a robot needs to autonomously explore and map an +unknown environment as efficiently and quickly as possible. +The robot is usually equipped with a sensor (e.g., LiDAR +or camera) to obtain measurements of its surroundings and +build/update a partial map of the environment. In practice, +the (high-dimensional) collected sensor data (e.g., point +cloud) is usually converted into a (simplified) occupancy grid +map or Octomap [1] that can be used for further planning [2], +[3], [4]. Such a task is also known as active SLAM [2]. +Although a robot can always slowly but accurately construct +a map by carefully methodically covering the entire environ- +ment, the objective of ARE is to plan the shortest path to +complete exploration, where small noises/errors in the final +map are tolerable. In consequence, the main challenge of +ARE is to plan a non-myopic path that balances the trade- +off between exploiting surroundings (i.e., refining the map in +already-explored areas) and exploring new (usually, further +away) areas, most importantly with only partial knowledge of +the environment. Such an exploration path is usually planned +† Corresponding author, to whom correspondence should be addressed. +1 +Authors are with the Department of Mechanical Engineering, +College of Design and Engineering, National University of Singapore. +caoyuhong@u.nus.edu, htx24@foxmail.com, +{wy98,yxian11}@u.nus.edu, mpegas@nus.edu.sg +Fig. 1: Illustration of autonomous robot exploration. A +wheeled robot is building a 3D Octomap using an onboard +LiDAR. The purple ball indicates the next viewpoint output +by our planner, which is tracked by the robot using a low- +level motion controller (yellow primitives). +incrementally online, as the partial map is updated using new +measurements along the way. +For example, conventional frontier-based methods [3], [5], +[6], [7] generate multiple candidate paths, each covering +a frontier (i.e., the boundary between explored free area +and unexplored area), and greedily select the path with +maximum gain, usually defined as a combination of utility +(i.e., number of observable frontiers along the path) and +cost (i.e., path length). However, an essential problem of +these methods is that such myopic frontier selection does +not guarantee optimality in the long term. A more recent +approach [4] reasons about the whole path to cover all +current frontiers, thus guaranteeing (near-)optimal paths in +the current partial map. However, since the environment is +only partially known, a previous optimal path often quickly +becomes sub-optimal as more of the environment is revealed, +or even worse, results in redundant movements (e.g., missing +an unexplored shortcut between two rooms which were +previously not known to be connected). Based on experi- +ences from conventional exploration planners, we note that +non-myopicity comes at two different levels in autonomous +exploration. The first level, spatial non-myopicity, requires +the planner to reason about the current partial map to balance +the exploration-exploitation trade-off, while temporal non- +myopicity requires the planner to estimate the future influence +of current decisions (e.g., predict the changes in the partial +map that may stem from given path planning decisions). +To achieve non-myopicity at these two levels, we propose +a deep reinforcement learning (DRL) based approach for +ARE, named ARiADNE, which relies on two attention-based +neural networks. These networks allow the agent to reason +arXiv:2301.11575v1 [cs.RO] 27 Jan 2023 + +about dependencies of different areas in the partial map at +different spatial scales, thus allowing the agent to sequence +spatially non-myopic decisions efficiently without the need +to optimize a long path. Furthermore, our critic network +implicitly provides the robot with the ability to estimate +potential areas that might be found by learning the state- +action value, which furthermore helps make decisions bene- +ficial to the long-term efficiency, thus addressing temporally +non-myopicity. Specifically, we first formulate autonomous +exploration as a sequential decision-making problem on a +collision-free graph that covers the known traversable area, +where one of the nodes is the robot’s current position. +We then use our attention-based neural network to select +one neighboring node of the current robot position as the +next viewpoint for the robot. In this work, we focus on +training and testing our approach in indoor environments +based on 2D occupancy grid maps, ranging from simple +scenarios (single room) to relatively complex ones (multiple +rooms with complicated corridors). There, we experimentally +demonstrate that our approach outperforms state-of-the-art +conventional methods on average. We also validate our +approach in high-fidelity ROS (Robot Operating System) +simulations, where we consider a real sensor model and a +motion controller, showing the generalizability of our model +to realistic environments. +II. RELATED WORK +Frontier-based vs sampling-based approaches The +first +frontier-based method was proposed by Yamauchi [5], in +which the robot is constantly driven towards the nearest +frontier. In more advanced frontier-based methods, selections +of which frontier to visit are evaluated by a gain function, +which considers the effect of both utility and cost [6], +[8], [9]. On the other hand, the past few years have +seen a number of sampling-based methods be developed, +based on Rapidly-exploring Random Trees (RRT) [3], +Rapidly-exploring Random Graphs [10], and Probabilistic +Random Maps (PRM) [11]. Sampling-based methods only +need to compute the gain of sampled paths, which avoids +the complexity of identifying and evaluating all frontiers. +Recent works demonstrated that frontier-based methods are +more suitable when frontiers are sparse (e.g., 2D indoor +scenarios), while sampling-based methods perform better +with dense frontiers (e.g., 3D outdoor scenarios) [7], [11]. +Intuitively, Frontier-based methods become inefficient when +there are too many frontiers to evaluate, while sampling- +based methods underperform when informative paths are +hard to sample. +Planning for long-term objectives Both frontier-based and +sampling-based methods mostly rely on greedy strategies +to plan short-term paths. Since the robot only has access +to a partial map of the environment, such paths inevitably +lead to myopic performance in the long term. Recently, Cao +et al. [4] proposed TARE to optimize the full exploration +path and mitigate myopicity. Utilizing the full knowledge +of the current partial map, TARE was shown to significantly +outperform state-of-the-art sampling-based planners in large- +scale 3D scenarios. Similar methods were also considered +to approach the informative path planning [12] problem, +which considers information gathering usually in obstacle- +free environments (in fact, works on autonomous exploration +and informative path planning mutually promote each other, +e.g., [4] and [12], [13] and [3]). +Learning-based exploration Niroui et al. [14] first com- +bined frontier-based method with deep reinforcement learn- +ing to adaptively tune the parameter of the gain function +for frontier selections and improve performance. Schmid et +al. [15] proposed to learn the underlying gain distribution +based on the partial map by supervised learning, thus help- +ing sampling-based methods more efficiently find next-best- +views and reduce computation. While the above works focus +on improving conventional methods using machine learning, +some other works [16], [17], [18], [19] directly applied deep +reinforcement learning to select a viewpoint to visit, often +relying on convolutional neural networks (CNNs). However, +although [16], [18] argue that DRL-based methods naturally +optimize long-term objectives, it seems that [16], [17], [18], +[19] are only able to reach performance slightly better than +the nearest-frontier method so far. +III. PROBLEM FORMULATION +We consider a bounded and unknown environment E +represented by a x × y 2D occupancy grid map, whose +partial (occupancy grid) map is denoted as P. The partial +map consists of unknown area Pu (i.e., unexplored area) and +known area Pk (i.e., explored area), such that Pu ∪ Pk = P. +The known area Pk is further classified into free area Pf +(i.e., traversable area for the robot) and occupied area Po +(i.e., obstacles) such that Pf ∪ Po = Pk. At the beginning +of exploration, the environment is fully unknown so the +partial map P = Pu. Then, during exploration, the unknown +area in the sensor range ds (the sensor we use is a 360- +degree LiDAR) is classified into either free area or occupied +area according to sensor measurements. The objective of +autonomous exploration is to find the shortest collision-free +robot trajectory ψ∗ to complete exploration: +ψ∗ = argmin +ψ∈Ψ +C(ψ), s.t. Pk = Pg, +(1) +where C : ψ −→ R+ maps a trajectory to its length and Pg +denotes the ground truth of the environment. Although the +ground truth is not accessible in real-world deployments, it +is known and can be utilized to evaluate the performance +of planners in testing. In practice, most works consider the +closure of occupied areas as Pk = Pg [5], [4], [18], [19]. +IV. METHOD +In this section, we cast ARE as an RL problem, and in- +troduce our attention-based policy and critic neural networks +as well as details of our training. +A. Exploration as an RL Problem +Sequential Decision-making Problem Since the free area +is updated based on the robot’s movements, online planning +for ARE is a sequential decision-making problem in nature. + +Following our previous work [20] for informative path plan- +ning, we consider the robot trajectory ψ as a sequence of +viewpoints ψ = (ψ0, ψ1, ...), ψi ∈ Pf. At each decision +step t, we first uniformly distribute candidate viewpoints +Vt = {v0, v1, ...}, ∀ vi = (xi, yi) ∈ Pf in the current free +area Pf, similar to [4]. Then, to find collision-free paths +between viewpoints, we connect each viewpoint with its k +nearest neighbors through a straight line and remove edges +that collide with occupied or unknown areas. In doing so, +we build a collision-free graph Gt = (Vt, Et), with Vt a +set of uniformly distributed nodes (i.e., viewpoints) over the +free area, and Et a set of traversable edges. We finally let +the robot select one neighboring node of its current position +ψt as the next viewpoint. Since the decision will be taken +upon arriving at the last selected viewpoint, the trajectory is +a sequence of waypoints such that ψi ∈ V . +Observation The observation of the agent is ot = (G′ +t, ψt), +where G′ +t = (V ′ +t , Et) is the augmented graph based on the +current collision-free graph Gt, while ψt is the robot current +position. Note that G′ +t shares the same edge set Et as Gt. +In addition to the node coordinates (i.e., vi = (xi, yi)), +The properties of each node v′ +i in the augmented graph +further include a binary signal bi, which indicates if the node +has been visited by the agent already, and the associated +utility ui, such that v′ +i = (xi, yi, ui, bi). We experimentally +found that the binary signal helps improve the learning by +allowing the robot to be aware of its previous movements. +The utility ui represents the number of observable frontiers +at node vi [4]. We consider observable frontiers as frontiers +within light of sight of the node (i.e., lines between the node +and observable frontiers are collision-free and their length +is smaller than the sensor range). The utility ui at node +vi is computed as ui = |Fo,i|, ∀fj ∈ Fo,i, ||fj − vi|| ≤ +ds, L(vi, fj) ∩ (P − Pf) = ∅, where Fo,i denotes the +observable frontiers set at node vi, ds denotes the sensor +range and L(vi, fj) the line between node vi and frontier +fj. In practice, we scale the node coordinates and utility to +[0, 1] before feeding the observation into the neural network. +Action At each decision step t, given the agent’s observation +ot, our attention-based neural network outputs a stochastic +policy to select a node out of all neighboring nodes as the +next viewpoint to visit. The policy is denoted as πθ(at|ot) = +πθ(ψt+1 = vi, (ψt, vi) ∈ Et | ot), where θ represents the set +of weights of the neural network. The robot moves to the +next viewpoint in a straight line, and updates its partial map +based on data collected along the way. +Reward To encourage efficient exploration, after taking +each movement action at, the robot receives a reward +composed of three parts. The first part ro = |Fo,ψt+1| is +the number of observed frontiers at the new viewpoint. +The second part rc = −C(ψt, ψt+1) is a punishment on +the distance between the previous and new viewpoints. A +fixed finishing reward rf = +� 20, +Pk = Pg +0, +otherwise, +is given +at the end of the episode, if and only if the exploration +task was completed. The total reward reads: rt(ot, at) = +a · ro + b · rc + rf, where a and b are scaling parameters (in +Fig. 2: Example decision step in the middle of an +exploration task in our approach, showing the unknown +area (grey cells), free area (white cells), occupied area (black +cells), frontiers (red cells), executed trajectory (blue line), +graph edges (tan lines), candidate viewpoints (small dots, +whose color represents their utility), robot current position +(purple disk), and robot starting position (light blue disk). +practice a = 1/50, b = 1/64). +B. Policy Network +The policy ψθ is output by our attention-based neural +network, which is composed of an encoder and a decoder +(shown in Fig. 3). We first rely on the encoder to extract +salient features from the current partial map, specifically +by learning dependencies between nodes in the associated +augmented graph G′. Based on these features as well as the +current robot position, the decoder then outputs the policy +over neighboring nodes, which can be used to decide which +one to visit next. Note that, while policy-based RL agents +often have a fixed action space, our decoder is inspired by the +Pointer Network [22] to allow the action space to depend on +the number of neighboring nodes input in the network. This +allows our network to naturally adapt to our collision-free +graph, where nodes have arbitrary numbers of neighbors. +Attention Layer We use the attention layer [21] as the +fundamental building block in our model. The input of such +an attention layer is composed of a query vector hq and a +key-and-value vector hk,v. The output of this layer, h′ +i, is the +weighted sum of the value vector, where weights depend on +the similarity between key and query: +qi = W Qhq +i , ki = W Khk,v +i +, vi = W V hk,v +i +, +uij = qT +i · kj +√ +d +, wij = +euij +�n +j=1 euij , h′ +i = +n +� +j=1 +wijvj, +(2) +where W Q, W K, W V ∈ Rd×d are all learnable matrices. +Updated features are then passed through a feed-forward +sublayer, following [21]. +Encoder In the encoder, we first linearly embed the node +inputs V ′ into d-dimensional node features hn, where hn +i = +W lv′ +i +bl. We then calculate an edge mask M where mij = +� 0, (vi, vj) ∈ Et +1, (vi, vj) /∈ Et . The node features are then passed to + +Node Features +Enhanced Node Features +Enhanced Current Node Features +Neighboring Features +Encoder +Partial Map +Policy +Action +Decoder +Filter Neighboring Feature +Augmented Graph +Construct Enhanced +Current Node Feature +Fig. 3: Attention-based policy network. Note that neighboring relationships in the augmented graph (tan) are also used as +the mask [21] in attention layers in the encoder. +multiple (6 in practice) stacked attention layers, where hq = +hk,v = hn, each attention layer taking the output of the +previous one as input. An edge mask is applied to allow each +node access to its neighboring node features only, by setting +wij = 0, ∀(i, j), mij = 1. Despite attention being restricted +to neighboring nodes in each layer, nodes can still obtain +non-neighboring node features by aggregating node features +multiple times through this stacked attention structure. We +empirically found that such structure is more suitable than +graph transformers [23] (like in our previous work [20]) to +learn path finding in maps with cluttered obstacles. We term +the output of the encoder, ˆhe, the enhanced node features, +since each of these updated node features ˆhn +i contains the +dependencies of v′ +i with other nodes. +Decoder We use the decoder to output a policy based on +enhanced node features ˆhe and the current robot position ψt. +Denoting the current robot position as node vc = ψt, we first +select the current node features hc and neighboring features +hnb, ∀ˆhnb +i , (vc, vi) ∈ Et from the corresponding enhanced +node features. We then pass the current node features and +enhanced node features to an attention layer, where hq = +hc, hk,v = ˆhn, concatenate its output with hc, and project it +back to a d-dimensional feature vector. We term this vector +the enhanced current node features ˆhc. After that, we pass +the enhanced current node features and neighboring features +to a pointer layer [22], an attention layer directly outputting +the attention weights w as the output with hq = ˆhc, hk,v = +hnb. We finally take the output of this pointer layer as the +robot’s policy, i.e., πθ(at | ot) = wi. +C. Critic Network +We train the policy network using the soft actor critic +(SAC) algorithm [24], [25] (see details below), where a critic +network is trained to predict state-action values. Since state- +action values approximate long-term returns (the accumu- +lated sum of rewards), we believe that they also implicitly +predict potential gains (i.e., potential areas that might be +found), which further helps the robot sequence non-myopic +decisions. In practice, we train a critic network to approx- +imate soft state-action values Qφ(ot, at), where φ denotes +the set of weights of the critic network. The structure of +the critic network is nearly the same as the policy network, +except that there is no pointer layer at the end of the decoder. +Instead, we directly concatenate the enhanced current node +features and neighboring features, then project them to soft +state-action values. +D. Training +Soft Actor-critic SAC aims to learn a policy that maximizes +return while keeping its entropy as high as possible: +π∗ = argmax E(ot,at)[ +T +� +t=0 +γt(rt + αH(π(.|ot)))], +(3) +where π∗ is the optimal policy, T the number of decision +steps, γ the discount factor, and α the temperature parameter +that tunes the importance of the entropy term versus the +return. In SAC, the soft state value is calculated as: V (ot) = +Eat[Q(ot, at)] − αlog(π(at|ot)). +The critic loss is calculated as: JQ(φ) = Eot[ 1 +2(Qφ(ot, at)− +(rt + γEot+1[V (ot+1)]))2]. +The +policy +loss +loss +is +calculated +as: +Jπ(θ) += +E(ot,at)[αlog(πθ(at|ot)) − Qφ(ot, at)]. +The temperature parameter is auto-tuned during the train- +ing and the temperature loss is calculated as: J(α) = +Eat[−α(logπt(at|ot) + H)], +where H denotes the target entropy. In practice, we use +double target networks for the critic network training, as +in [24], [25]. +Training Details We utilize the same environments pro- +vided in [18] for training, which are generated by a random +dungeon generator. Each environment is a 640 × 480 grid +map, while the sensor range ds = 80. To build the collision- +free graph, 900 points are uniformly distributed to cover +the whole environment, with all points in the known free +area considered as candidate viewpoints V . We check the +k = 20 nearest neighbor of each viewpoint, and connect +them if such an edge is collision-free, to form the edge +set E. We consider the exploration task to be completed +once more than 99% of the ground truth has been explored +(|Pk|/|Pg| > 0.99). During training, we set the max episode +length to 128 decision steps, the discount factor to γ = 1, +the batch size to 256, and the episode buffer size to 10, 000. +Training starts after the episode buffer collects more than +2000 steps data. The target entropy is set to 0.01 · log(k). +Each training step contains 1 iteration and happens after 1 +episode finishes. We use the Adam optimizer with a learning + +TABLE I: Comparison with baseline ARE planners (100 scenarios for each test set). We report the average and standard +deviation of the trajectory length to complete exploration (lower is better). For utility-based methods [6], the numbers 1, 10, +25 represent the value of λ, which is used to tune exploitation and exploration. +Nearest +Utility 1 +Utility 10 +Utility 25 +NBVP +TARE Local +CNN +ARiADNE +easy +772(±253) +736(±266) +732(±256) +764(±258) +745(±268) +692(±228) +779(±281) +663(±257) +medium +1248(±295) +1266(±311) +1179(±300) +1227(±307) +1217(±271) +1170(±275) +1169(±319) +1130(±334) +complex +1669(±332) +1873(±457) +1662(±347) +1711(±352) +1744(±366) +1646(±312) +1647(±422) +1599(±363) +random +1354(±410) +1423(±466) +1268(±396) +1315(±413) +1323(±371) +1266(±388) +1323(±428) +1204(±378) +(a) simple +(b) medium +(c) complex +Fig. 4: Examples scenarios from each different test set. +rate of 10−5 for both policy and critic networks. The target +critic network updates every 256 training steps. Our model is +trained on a workstation equipped with a i9-10980XE CPU +and an NVIDIA GeForce RTX 3090 GPU. We train our +model utilizing Ray, a distributed framework for machine +learning [26], to parallelize and accelerate data collection +(32 instances in practice). The training needs around 24h to +converge. We will release our full code upon acceptance. +V. EXPERIMENT +A. Comparison Analysis +Most previous works often only conduct experiments in +a few scenarios (often less than 10). However, we note +that the performance of exploration planners exhibits high +variance in different scenarios. Therefore, we believe a +convincing comparison should be based on evaluation in +a large number of testing environments. Although building +so many testing environments is tricky and time-consuming +even in ROS, hundreds of simplified scenarios, like the ones +we used for training, can be generated easily. Therefore, we +conduct comparison analyses on a fixed set of simplified +environments, which were never seen by our trained model. +Testing environments are divided in four sets (100 scenarios +each), named random, easy, medium, and complex. Easy +scenarios only contain one room, and complex scenarios +contain multiple rooms with complicated corridors, while the +complexity of medium scenarios lies in-between. Random +scenarios contain a mix of easy, medium, and complex +scenarios (but no repeated scenario from these test sets). +We compare ARiADNE with state-of-the-art conventional +planners, including Nearest Frontier [5], Utility-based Fron- +tier [6], NBVP [3], and TARE Local [4]. Nearest Frontier +always drives the robot towards the nearest frontier, while +Utility-based Frontier evaluates the gain of each frontier +gi = ui · e−λ·C(ψi) and drives the robot to the frontier with +the highest gain, where ui is the utility of frontier i, ψi the +shortest path from the robot’s current position to frontier i, +and λ a tunable parameter used to balance exploration and +exploitation. The same function is also used in NBVP to +evaluate sampled trajectories. We tried a series of values of λ +for Utility-based Frontier and NBVP, and found that λ = 10 +(a) Trajectory Analysis +(b) ARiADNE (1618) +(c) TARE Local (1703) +(d) NBVP (1922) +(e) Utility 10 (1793) +Fig. 5: Visual comparison of our method and baselines +in an example scenario. +generally performs best (see Table I). Finally, TARE Local +refers to the local planner of TARE [4], which explicitly +plans a full trajectory to cover all frontiers (we do not +use TARE’s global planner, since its local planning horizon +already fits our testing environments). NBVP and TARE run +300 and 10 iterations for each decision step respectively +(15 and 1 in default [3], [4]), to make their decisions as +optimal as possible. In our tests, we adopt our collision- +free graph as the trajectory space for all baselines except +NBVP (we found RRTs mostly generate poor zig-zag paths +due to symmetries in our uniform graph), to alleviate the +randomness of sampling and ensure a fair comparison. We +further compare against a CNN-based DRL planner [18]. +Since this CNN-based planner has a fixed observation range, +it only has a partial observation of the (partial) map, and +relies on a frontier-based method for exploration when there +is no nearby frontier in its field-of-view. +We report the average and variance of the total trajectory +length to complete exploration in Table I. Our results indicate +that ARiADNE outperforms all baselines on average, in all +test sets. We do not report the planning time of baseline +methods in Table I, since we focused on implementing fun- +damental inner workings of the baselines, without perfectly +optimizing their computing time. In addition, we observed +that the utility/gain computation generally takes 90% of the +planning time for conventional methods in practice, while +its computing time is determined by the resolution of the +map. Therefore, computing times vary greatly in different +exploration scenarios. Despite this, we note that our method +can be used in real-time. Under our exploration setting, our + +Fig. 6: Attention weights visualization of the critic net- +work decoder. The query vector is the node at the current +robot’s position (purple) and the keys vector are nodes in the +augmented graph (blue). Note how the different heads of the +decoder learn to focus on either local or global dependencies +of areas in the partial map. +method takes 0.7s for the observation generation on average +(utility computation and graph building) and less than 0.02s +for the neural network inference on a i9-10980XE CPU. +As discussed in the related work section, the best-tuned +frontier-based method (Utility 10) performs well in 2D ex- +ploration tasks (better than NBVP). Despite this, since these +frontier-based methods are myopic, they are outperformed by +TARE Local, which plans near-optimal long-term (full) tra- +jectories on the current partial map. While it only constructs +paths one viewpoint at a time, our learning-based method +can not only reason about the whole partial map to construct +efficient, non-myopic exploration trajectories, while learning +to predict the potential long-term gain of decisions. We +believe such an advantage results in the improvement of our +method over conventional baselines (5% better than TARE +Local in our random scenarios). Fig. 5 shows an example +where ARiADNE plans a more efficient trajectory, while +conventional methods suffer from redundant movements. +However, it should be noted that considering long-term paths +and predicting potential gains do not strictly guarantee better +performance in every scenario (e.g., predictions could be +wrong). In fact, ARiADNE plans the shortest path for 33 +scenarios in our random tests, while TARE Local, NBVP, +Utility 10 perform best in 23, 21, 23 scenarios respectively. +Finally, ARiADNE also outperforms the CNN-based plan- +ner. We believe that our main advantage stems from the +attention-based neural network, which efficiently learns fea- +tures at different scales (as shown in Fig. 6, different heads +of the decoder learn to focus on either local or global +dependencies), while CNNs naturally only focus on local +dependencies. Therefore, our model can better learn depen- +dencies between different areas to reason about the entire +partial map and avoid myopic decisions. +B. Experimental validation +We validate ARiADNE in a simulation environment for +exploration provided by [27]. It contains fundamental mod- +ules (e.g., state estimation and motion control), which allow +us to consider a real sensor model and a low-level motion +(a) Ground truth +(b) Constructed Octomap +(c) Constructed occupancy grid map +Fig. 7: Validation of our method in simulation. Note that +ignoring the small left-down corner is actually a wise deci- +sion since the objective is to explore 99% of the environment. +controller. The validation is conducted in a realistic indoor +environment (approximately 70m × 40m) with long and +narrow corridors connected with tables, colums, and lobby +areas (see Fig. 7(a)). We use a wheeled robot equipped with +a 3D Velodyne Lidar with a 130m sensor range. We convert +collected data into an Octomap (see Fig. 7(b)) and then +project it to a occupancy grid map for exploration planning +(see Fig. 7(c)). The resolution of the grid map is 0.2m. +We re-plan the path every 0.2s. Although the sensor model +(i.e., sensor range and sensing frequency) of the robot is +drastically different from the one used in training, our trained +model still makes efficient decisions to avoid redundant +movements for exploration (see the colored trajectory in +Fig. 7(c), highlighting the generalizability of our approach. +VI. CONCLUSION +In this work, we propose ARiADNE, a reinforcement +learning approach that relies on attention-based deep neural +network for autonomous exploration. Our approach allows +the robot to efficiently learn dependencies between different +areas in its partial map and implicitly predict potential gains, +thus allowing it to sequence non-myopic movement decisions +in partially-known environments. In our tests, ARiADNE +exhibits improvement over state-of-the-art frontier-based, +sampling-based, and CNN-based exploration planners, in +terms of average trajectory length to complete exploration. +We also validate our approach in a high-fidelity ROS simula- +tion, where we consider a real sensor model and a low-level +motion controller, towards deployments on real robots. +Future work will focus on extending our approach to +autonomous exploration of 3D environments, where frontiers +are much denser than in 2D. Second, although in this work +we uniformly distribute nodes to construct a graph, we be- +lieve a sparser graph containing more informative viewpoints +may improve performance. Finally, we are also interested in +explicitly predicting the potential gain during exploration to +further boost planning performance. +ACKNOWLEDGMENTS +This work was supported by Temasek Laboratories +(TL@NUS) under grant TL/FS/2022/01. + +L +LREFERENCES +[1] A. Hornung, K. M. Wurm, M. Bennewitz, C. Stachniss, and +W. Burgard, “OctoMap: An efficient probabilistic 3D mapping +framework based on octrees,” Autonomous Robots, 2013, software +available at https://octomap.github.io. [Online]. Available: https: +//octomap.github.io +[2] J. A. Placed, J. Strader, H. Carrillo, N. Atanasov, V. Indelman, +L. Carlone, and J. A. Castellanos, “A survey on active simultaneous +localization and mapping: State of the art and new frontiers,” arXiv +preprint arXiv:2207.00254, 2022. +[3] A. Bircher, M. Kamel, K. Alexis, H. Oleynikova, and R. 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Zhang, +“Autonomous exploration development environment and the planning +algorithms,” in 2022 International Conference on Robotics and Au- +tomation (ICRA). +IEEE, 2022, pp. 8921–8928. + diff --git a/0dFJT4oBgHgl3EQfjix9/content/tmp_files/load_file.txt b/0dFJT4oBgHgl3EQfjix9/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2f688675ce6c2b82a9c47378481f023376b001e1 --- /dev/null +++ b/0dFJT4oBgHgl3EQfjix9/content/tmp_files/load_file.txt @@ -0,0 +1,555 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf,len=554 +page_content='ARiADNE: A Reinforcement learning approach using Attention-based Deep Networks for Exploration Yuhong Cao1, Tianxiang Hou1, Yizhuo Wang1, Xian Yi1, Guillaume Sartoretti1† Abstract— In autonomous robot exploration tasks, a mobile robot needs to actively explore and map an unknown envi- ronment as fast as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Since the environment is being revealed during exploration, the robot needs to frequently re-plan its path online, as new information is acquired by onboard sensors and used to update its partial map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' While state-of-the-art exploration planners are frontier- and sampling- based, encouraged by the recent development in deep reinforce- ment learning (DRL), we propose ARiADNE, an attention- based neural approach to obtain real-time, non-myopic path planning for autonomous exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' ARiADNE is able to learn dependencies at multiple spatial scales between areas of the agent’s partial map, and implicitly predict potential gains associated with exploring those areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' This allows the agent to sequence movement actions that balance the natural trade- off between exploitation/refinement of the map in known areas and exploration of new areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' We experimentally demonstrate that our method outperforms both learning and non-learning state-of-the-art baselines in terms of average trajectory length to complete exploration in hundreds of simplified 2D indoor scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' We further validate our approach in high-fidelity Robot Operating System (ROS) simulations, where we consider a real sensor model and a realistic low-level motion controller, toward deployment on real robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' INTRODUCTION Autonomous robot exploration (ARE) refers to the task, in which a robot needs to autonomously explore and map an unknown environment as efficiently and quickly as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' The robot is usually equipped with a sensor (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=', LiDAR or camera) to obtain measurements of its surroundings and build/update a partial map of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' In practice, the (high-dimensional) collected sensor data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=', point cloud) is usually converted into a (simplified) occupancy grid map or Octomap [1] that can be used for further planning [2], [3], [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Such a task is also known as active SLAM [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Although a robot can always slowly but accurately construct a map by carefully methodically covering the entire environ- ment, the objective of ARE is to plan the shortest path to complete exploration, where small noises/errors in the final map are tolerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' In consequence, the main challenge of ARE is to plan a non-myopic path that balances the trade- off between exploiting surroundings (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=', refining the map in already-explored areas) and exploring new (usually, further away) areas, most importantly with only partial knowledge of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Such an exploration path is usually planned † Corresponding author, to whom correspondence should be addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' 1 Authors are with the Department of Mechanical Engineering, College of Design and Engineering, National University of Singapore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' caoyuhong@u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='nus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='edu, htx24@foxmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='com, {wy98,yxian11}@u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='nus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='edu, mpegas@nus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='sg Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' 1: Illustration of autonomous robot exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' A wheeled robot is building a 3D Octomap using an onboard LiDAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' The purple ball indicates the next viewpoint output by our planner, which is tracked by the robot using a low- level motion controller (yellow primitives).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' incrementally online, as the partial map is updated using new measurements along the way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' For example, conventional frontier-based methods [3], [5], [6], [7] generate multiple candidate paths, each covering a frontier (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=', the boundary between explored free area and unexplored area), and greedily select the path with maximum gain, usually defined as a combination of utility (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=', number of observable frontiers along the path) and cost (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=', path length).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' However, an essential problem of these methods is that such myopic frontier selection does not guarantee optimality in the long term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' A more recent approach [4] reasons about the whole path to cover all current frontiers, thus guaranteeing (near-)optimal paths in the current partial map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' However, since the environment is only partially known, a previous optimal path often quickly becomes sub-optimal as more of the environment is revealed, or even worse, results in redundant movements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=', missing an unexplored shortcut between two rooms which were previously not known to be connected).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Based on experi- ences from conventional exploration planners, we note that non-myopicity comes at two different levels in autonomous exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' The first level, spatial non-myopicity, requires the planner to reason about the current partial map to balance the exploration-exploitation trade-off, while temporal non- myopicity requires the planner to estimate the future influence of current decisions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=', predict the changes in the partial map that may stem from given path planning decisions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' To achieve non-myopicity at these two levels, we propose a deep reinforcement learning (DRL) based approach for ARE, named ARiADNE, which relies on two attention-based neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' These networks allow the agent to reason arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='11575v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='RO] 27 Jan 2023 about dependencies of different areas in the partial map at different spatial scales, thus allowing the agent to sequence spatially non-myopic decisions efficiently without the need to optimize a long path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Furthermore, our critic network implicitly provides the robot with the ability to estimate potential areas that might be found by learning the state- action value, which furthermore helps make decisions bene- ficial to the long-term efficiency, thus addressing temporally non-myopicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Specifically, we first formulate autonomous exploration as a sequential decision-making problem on a collision-free graph that covers the known traversable area, where one of the nodes is the robot’s current position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' We then use our attention-based neural network to select one neighboring node of the current robot position as the next viewpoint for the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' In this work, we focus on training and testing our approach in indoor environments based on 2D occupancy grid maps, ranging from simple scenarios (single room) to relatively complex ones (multiple rooms with complicated corridors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' There, we experimentally demonstrate that our approach outperforms state-of-the-art conventional methods on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' We also validate our approach in high-fidelity ROS (Robot Operating System) simulations, where we consider a real sensor model and a motion controller, showing the generalizability of our model to realistic environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' RELATED WORK Frontier-based vs sampling-based approaches The first frontier-based method was proposed by Yamauchi [5], in which the robot is constantly driven towards the nearest frontier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' In more advanced frontier-based methods, selections of which frontier to visit are evaluated by a gain function, which considers the effect of both utility and cost [6], [8], [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' On the other hand, the past few years have seen a number of sampling-based methods be developed, based on Rapidly-exploring Random Trees (RRT) [3], Rapidly-exploring Random Graphs [10], and Probabilistic Random Maps (PRM) [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Sampling-based methods only need to compute the gain of sampled paths, which avoids the complexity of identifying and evaluating all frontiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Recent works demonstrated that frontier-based methods are more suitable when frontiers are sparse (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=', 2D indoor scenarios), while sampling-based methods perform better with dense frontiers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=', 3D outdoor scenarios) [7], [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Intuitively, Frontier-based methods become inefficient when there are too many frontiers to evaluate, while sampling- based methods underperform when informative paths are hard to sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Planning for long-term objectives Both frontier-based and sampling-based methods mostly rely on greedy strategies to plan short-term paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Since the robot only has access to a partial map of the environment, such paths inevitably lead to myopic performance in the long term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Recently, Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' [4] proposed TARE to optimize the full exploration path and mitigate myopicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Utilizing the full knowledge of the current partial map, TARE was shown to significantly outperform state-of-the-art sampling-based planners in large- scale 3D scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Similar methods were also considered to approach the informative path planning [12] problem, which considers information gathering usually in obstacle- free environments (in fact, works on autonomous exploration and informative path planning mutually promote each other, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=', [4] and [12], [13] and [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Learning-based exploration Niroui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' [14] first com- bined frontier-based method with deep reinforcement learn- ing to adaptively tune the parameter of the gain function for frontier selections and improve performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Schmid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' [15] proposed to learn the underlying gain distribution based on the partial map by supervised learning, thus help- ing sampling-based methods more efficiently find next-best- views and reduce computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' While the above works focus on improving conventional methods using machine learning, some other works [16], [17], [18], [19] directly applied deep reinforcement learning to select a viewpoint to visit, often relying on convolutional neural networks (CNNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' However, although [16], [18] argue that DRL-based methods naturally optimize long-term objectives, it seems that [16], [17], [18], [19] are only able to reach performance slightly better than the nearest-frontier method so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' PROBLEM FORMULATION We consider a bounded and unknown environment E represented by a x × y 2D occupancy grid map, whose partial (occupancy grid) map is denoted as P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' The partial map consists of unknown area Pu (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=', unexplored area) and known area Pk (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=', explored area), such that Pu ∪ Pk = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' The known area Pk is further classified into free area Pf (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=', traversable area for the robot) and occupied area Po (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=', obstacles) such that Pf ∪ Po = Pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' At the beginning of exploration, the environment is fully unknown so the partial map P = Pu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Then, during exploration, the unknown area in the sensor range ds (the sensor we use is a 360- degree LiDAR) is classified into either free area or occupied area according to sensor measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' The objective of autonomous exploration is to find the shortest collision-free robot trajectory ψ∗ to complete exploration: ψ∗ = argmin ψ∈Ψ C(ψ), s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Pk = Pg, (1) where C : ψ −→ R+ maps a trajectory to its length and Pg denotes the ground truth of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Although the ground truth is not accessible in real-world deployments, it is known and can be utilized to evaluate the performance of planners in testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' In practice, most works consider the closure of occupied areas as Pk = Pg [5], [4], [18], [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' METHOD In this section, we cast ARE as an RL problem, and in- troduce our attention-based policy and critic neural networks as well as details of our training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Exploration as an RL Problem Sequential Decision-making Problem Since the free area is updated based on the robot’s movements, online planning for ARE is a sequential decision-making problem in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Following our previous work [20] for informative path plan- ning, we consider the robot trajectory ψ as a sequence of viewpoints ψ = (ψ0, ψ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='), ψi ∈ Pf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' At each decision step t, we first uniformly distribute candidate viewpoints Vt = {v0, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='}, ∀ vi = (xi, yi) ∈ Pf in the current free area Pf, similar to [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Then, to find collision-free paths between viewpoints, we connect each viewpoint with its k nearest neighbors through a straight line and remove edges that collide with occupied or unknown areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' In doing so, we build a collision-free graph Gt = (Vt, Et), with Vt a set of uniformly distributed nodes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=', viewpoints) over the free area, and Et a set of traversable edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' We finally let the robot select one neighboring node of its current position ψt as the next viewpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Since the decision will be taken upon arriving at the last selected viewpoint, the trajectory is a sequence of waypoints such that ψi ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Observation The observation of the agent is ot = (G′ t, ψt), where G′ t = (V ′ t , Et) is the augmented graph based on the current collision-free graph Gt, while ψt is the robot current position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Note that G′ t shares the same edge set Et as Gt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' In addition to the node coordinates (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=', vi = (xi, yi)), The properties of each node v′ i in the augmented graph further include a binary signal bi, which indicates if the node has been visited by the agent already, and the associated utility ui, such that v′ i = (xi, yi, ui, bi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' We experimentally found that the binary signal helps improve the learning by allowing the robot to be aware of its previous movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' The utility ui represents the number of observable frontiers at node vi [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' We consider observable frontiers as frontiers within light of sight of the node (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=', lines between the node and observable frontiers are collision-free and their length is smaller than the sensor range).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' The utility ui at node vi is computed as ui = |Fo,i|, ∀fj ∈ Fo,i, ||fj − vi|| ≤ ds, L(vi, fj) ∩ (P − Pf) = ∅, where Fo,i denotes the observable frontiers set at node vi, ds denotes the sensor range and L(vi, fj) the line between node vi and frontier fj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' In practice, we scale the node coordinates and utility to [0, 1] before feeding the observation into the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Action At each decision step t, given the agent’s observation ot, our attention-based neural network outputs a stochastic policy to select a node out of all neighboring nodes as the next viewpoint to visit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' The policy is denoted as πθ(at|ot) = πθ(ψt+1 = vi, (ψt, vi) ∈ Et | ot), where θ represents the set of weights of the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' The robot moves to the next viewpoint in a straight line, and updates its partial map based on data collected along the way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Reward To encourage efficient exploration, after taking each movement action at, the robot receives a reward composed of three parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' The first part ro = |Fo,ψt+1| is the number of observed frontiers at the new viewpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' The second part rc = −C(ψt, ψt+1) is a punishment on the distance between the previous and new viewpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' A fixed finishing reward rf = � 20, Pk = Pg 0, otherwise, is given at the end of the episode, if and only if the exploration task was completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' The total reward reads: rt(ot, at) = a · ro + b · rc + rf, where a and b are scaling parameters (in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' 2: Example decision step in the middle of an exploration task in our approach, showing the unknown area (grey cells), free area (white cells), occupied area (black cells), frontiers (red cells), executed trajectory (blue line), graph edges (tan lines), candidate viewpoints (small dots, whose color represents their utility), robot current position (purple disk), and robot starting position (light blue disk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' practice a = 1/50, b = 1/64).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Policy Network The policy ψθ is output by our attention-based neural network, which is composed of an encoder and a decoder (shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' We first rely on the encoder to extract salient features from the current partial map, specifically by learning dependencies between nodes in the associated augmented graph G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Based on these features as well as the current robot position, the decoder then outputs the policy over neighboring nodes, which can be used to decide which one to visit next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Note that, while policy-based RL agents often have a fixed action space, our decoder is inspired by the Pointer Network [22] to allow the action space to depend on the number of neighboring nodes input in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' This allows our network to naturally adapt to our collision-free graph, where nodes have arbitrary numbers of neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Attention Layer We use the attention layer [21] as the fundamental building block in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' The input of such an attention layer is composed of a query vector hq and a key-and-value vector hk,v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' The output of this layer, h′ i, is the weighted sum of the value vector, where weights depend on the similarity between key and query: qi = W Qhq i , ki = W Khk,v i , vi = W V hk,v i , uij = qT i · kj √ d , wij = euij �n j=1 euij , h′ i = n � j=1 wijvj, (2) where W Q, W K, W V ∈ Rd×d are all learnable matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Updated features are then passed through a feed-forward sublayer, following [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Encoder In the encoder, we first linearly embed the node inputs V ′ into d-dimensional node features hn, where hn i = W lv′ i +bl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' We then calculate an edge mask M where mij = � 0, (vi, vj) ∈ Et 1, (vi, vj) /∈ Et .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' The node features are then passed to Node Features Enhanced Node Features Enhanced Current Node Features Neighboring Features Encoder Partial Map Policy Action Decoder Filter Neighboring Feature Augmented Graph Construct Enhanced Current Node Feature Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' 3: Attention-based policy network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Note that neighboring relationships in the augmented graph (tan) are also used as the mask [21] in attention layers in the encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' multiple (6 in practice) stacked attention layers, where hq = hk,v = hn, each attention layer taking the output of the previous one as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' An edge mask is applied to allow each node access to its neighboring node features only, by setting wij = 0, ∀(i, j), mij = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Despite attention being restricted to neighboring nodes in each layer, nodes can still obtain non-neighboring node features by aggregating node features multiple times through this stacked attention structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' We empirically found that such structure is more suitable than graph transformers [23] (like in our previous work [20]) to learn path finding in maps with cluttered obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' We term the output of the encoder, ˆhe, the enhanced node features, since each of these updated node features ˆhn i contains the dependencies of v′ i with other nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Decoder We use the decoder to output a policy based on enhanced node features ˆhe and the current robot position ψt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Denoting the current robot position as node vc = ψt, we first select the current node features hc and neighboring features hnb, ∀ˆhnb i , (vc, vi) ∈ Et from the corresponding enhanced node features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' We then pass the current node features and enhanced node features to an attention layer, where hq = hc, hk,v = ˆhn, concatenate its output with hc, and project it back to a d-dimensional feature vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' We term this vector the enhanced current node features ˆhc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' After that, we pass the enhanced current node features and neighboring features to a pointer layer [22], an attention layer directly outputting the attention weights w as the output with hq = ˆhc, hk,v = hnb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' We finally take the output of this pointer layer as the robot’s policy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=', πθ(at | ot) = wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Critic Network We train the policy network using the soft actor critic (SAC) algorithm [24], [25] (see details below), where a critic network is trained to predict state-action values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Since state- action values approximate long-term returns (the accumu- lated sum of rewards), we believe that they also implicitly predict potential gains (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=', potential areas that might be found), which further helps the robot sequence non-myopic decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' In practice, we train a critic network to approx- imate soft state-action values Qφ(ot, at), where φ denotes the set of weights of the critic network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' The structure of the critic network is nearly the same as the policy network, except that there is no pointer layer at the end of the decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Instead, we directly concatenate the enhanced current node features and neighboring features, then project them to soft state-action values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Training Soft Actor-critic SAC aims to learn a policy that maximizes return while keeping its entropy as high as possible: π∗ = argmax E(ot,at)[ T � t=0 γt(rt + αH(π(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='|ot)))], (3) where π∗ is the optimal policy, T the number of decision steps, γ the discount factor, and α the temperature parameter that tunes the importance of the entropy term versus the return.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' In SAC, the soft state value is calculated as: V (ot) = Eat[Q(ot, at)] − αlog(π(at|ot)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' The critic loss is calculated as: JQ(φ) = Eot[ 1 2(Qφ(ot, at)− (rt + γEot+1[V (ot+1)]))2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' The policy loss loss is calculated as: Jπ(θ) = E(ot,at)[αlog(πθ(at|ot)) − Qφ(ot, at)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' The temperature parameter is auto-tuned during the train- ing and the temperature loss is calculated as: J(α) = Eat[−α(logπt(at|ot) + H)], where H denotes the target entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' In practice, we use double target networks for the critic network training, as in [24], [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Training Details We utilize the same environments pro- vided in [18] for training, which are generated by a random dungeon generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Each environment is a 640 × 480 grid map, while the sensor range ds = 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' To build the collision- free graph, 900 points are uniformly distributed to cover the whole environment, with all points in the known free area considered as candidate viewpoints V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' We check the k = 20 nearest neighbor of each viewpoint, and connect them if such an edge is collision-free, to form the edge set E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' We consider the exploration task to be completed once more than 99% of the ground truth has been explored (|Pk|/|Pg| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='99).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' During training, we set the max episode length to 128 decision steps, the discount factor to γ = 1, the batch size to 256, and the episode buffer size to 10, 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Training starts after the episode buffer collects more than 2000 steps data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' The target entropy is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='01 · log(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Each training step contains 1 iteration and happens after 1 episode finishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' We use the Adam optimizer with a learning TABLE I: Comparison with baseline ARE planners (100 scenarios for each test set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' We report the average and standard deviation of the trajectory length to complete exploration (lower is better).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' For utility-based methods [6], the numbers 1, 10, 25 represent the value of λ, which is used to tune exploitation and exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='Nearest ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='Utility 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='Utility 10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='Utility 25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='NBVP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='TARE Local ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='CNN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='ARiADNE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='easy ' metadata={'source': 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+page_content='1323(±428) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='1204(±378) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='(a) simple ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='(b) medium ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='(c) complex ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' 4: Examples scenarios from each different test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' rate of 10−5 for both policy and critic networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' The target critic network updates every 256 training steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Our model is trained on a workstation equipped with a i9-10980XE CPU and an NVIDIA GeForce RTX 3090 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' We train our model utilizing Ray, a distributed framework for machine learning [26], to parallelize and accelerate data collection (32 instances in practice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' The training needs around 24h to converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' We will release our full code upon acceptance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' EXPERIMENT A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Comparison Analysis Most previous works often only conduct experiments in a few scenarios (often less than 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' However, we note that the performance of exploration planners exhibits high variance in different scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Therefore, we believe a convincing comparison should be based on evaluation in a large number of testing environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Although building so many testing environments is tricky and time-consuming even in ROS, hundreds of simplified scenarios, like the ones we used for training, can be generated easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Therefore, we conduct comparison analyses on a fixed set of simplified environments, which were never seen by our trained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Testing environments are divided in four sets (100 scenarios each), named random, easy, medium, and complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Easy scenarios only contain one room, and complex scenarios contain multiple rooms with complicated corridors, while the complexity of medium scenarios lies in-between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Random scenarios contain a mix of easy, medium, and complex scenarios (but no repeated scenario from these test sets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' We compare ARiADNE with state-of-the-art conventional planners, including Nearest Frontier [5], Utility-based Fron- tier [6], NBVP [3], and TARE Local [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Nearest Frontier always drives the robot towards the nearest frontier, while Utility-based Frontier evaluates the gain of each frontier gi = ui · e−λ·C(ψi) and drives the robot to the frontier with the highest gain, where ui is the utility of frontier i, ψi the shortest path from the robot’s current position to frontier i, and λ a tunable parameter used to balance exploration and exploitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' The same function is also used in NBVP to evaluate sampled trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' We tried a series of values of λ for Utility-based Frontier and NBVP, and found that λ = 10 (a) Trajectory Analysis (b) ARiADNE (1618) (c) TARE Local (1703) (d) NBVP (1922) (e) Utility 10 (1793) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' 5: Visual comparison of our method and baselines in an example scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' generally performs best (see Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Finally, TARE Local refers to the local planner of TARE [4], which explicitly plans a full trajectory to cover all frontiers (we do not use TARE’s global planner, since its local planning horizon already fits our testing environments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' NBVP and TARE run 300 and 10 iterations for each decision step respectively (15 and 1 in default [3], [4]), to make their decisions as optimal as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' In our tests, we adopt our collision- free graph as the trajectory space for all baselines except NBVP (we found RRTs mostly generate poor zig-zag paths due to symmetries in our uniform graph), to alleviate the randomness of sampling and ensure a fair comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' We further compare against a CNN-based DRL planner [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Since this CNN-based planner has a fixed observation range, it only has a partial observation of the (partial) map, and relies on a frontier-based method for exploration when there is no nearby frontier in its field-of-view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' We report the average and variance of the total trajectory length to complete exploration in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Our results indicate that ARiADNE outperforms all baselines on average, in all test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' We do not report the planning time of baseline methods in Table I, since we focused on implementing fun- damental inner workings of the baselines, without perfectly optimizing their computing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' In addition, we observed that the utility/gain computation generally takes 90% of the planning time for conventional methods in practice, while its computing time is determined by the resolution of the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Therefore, computing times vary greatly in different exploration scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Despite this, we note that our method can be used in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Under our exploration setting, our Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' 6: Attention weights visualization of the critic net- work decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' The query vector is the node at the current robot’s position (purple) and the keys vector are nodes in the augmented graph (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Note how the different heads of the decoder learn to focus on either local or global dependencies of areas in the partial map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' method takes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='7s for the observation generation on average (utility computation and graph building) and less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='02s for the neural network inference on a i9-10980XE CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' As discussed in the related work section, the best-tuned frontier-based method (Utility 10) performs well in 2D ex- ploration tasks (better than NBVP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Despite this, since these frontier-based methods are myopic, they are outperformed by TARE Local, which plans near-optimal long-term (full) tra- jectories on the current partial map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' While it only constructs paths one viewpoint at a time, our learning-based method can not only reason about the whole partial map to construct efficient, non-myopic exploration trajectories, while learning to predict the potential long-term gain of decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' We believe such an advantage results in the improvement of our method over conventional baselines (5% better than TARE Local in our random scenarios).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' 5 shows an example where ARiADNE plans a more efficient trajectory, while conventional methods suffer from redundant movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' However, it should be noted that considering long-term paths and predicting potential gains do not strictly guarantee better performance in every scenario (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=', predictions could be wrong).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' In fact, ARiADNE plans the shortest path for 33 scenarios in our random tests, while TARE Local, NBVP, Utility 10 perform best in 23, 21, 23 scenarios respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Finally, ARiADNE also outperforms the CNN-based plan- ner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' We believe that our main advantage stems from the attention-based neural network, which efficiently learns fea- tures at different scales (as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' 6, different heads of the decoder learn to focus on either local or global dependencies), while CNNs naturally only focus on local dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Therefore, our model can better learn depen- dencies between different areas to reason about the entire partial map and avoid myopic decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Experimental validation We validate ARiADNE in a simulation environment for exploration provided by [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' It contains fundamental mod- ules (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=', state estimation and motion control), which allow us to consider a real sensor model and a low-level motion (a) Ground truth (b) Constructed Octomap (c) Constructed occupancy grid map Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' 7: Validation of our method in simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Note that ignoring the small left-down corner is actually a wise deci- sion since the objective is to explore 99% of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' The validation is conducted in a realistic indoor environment (approximately 70m × 40m) with long and narrow corridors connected with tables, colums, and lobby areas (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' 7(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' We use a wheeled robot equipped with a 3D Velodyne Lidar with a 130m sensor range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' We convert collected data into an Octomap (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' 7(b)) and then project it to a occupancy grid map for exploration planning (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' 7(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' The resolution of the grid map is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' We re-plan the path every 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='2s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Although the sensor model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=', sensor range and sensing frequency) of the robot is drastically different from the one used in training, our trained model still makes efficient decisions to avoid redundant movements for exploration (see the colored trajectory in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' 7(c), highlighting the generalizability of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' CONCLUSION In this work, we propose ARiADNE, a reinforcement learning approach that relies on attention-based deep neural network for autonomous exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Our approach allows the robot to efficiently learn dependencies between different areas in its partial map and implicitly predict potential gains, thus allowing it to sequence non-myopic movement decisions in partially-known environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' In our tests, ARiADNE exhibits improvement over state-of-the-art frontier-based, sampling-based, and CNN-based exploration planners, in terms of average trajectory length to complete exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' We also validate our approach in a high-fidelity ROS simula- tion, where we consider a real sensor model and a low-level motion controller, towards deployments on real robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Future work will focus on extending our approach to autonomous exploration of 3D environments, where frontiers are much denser than in 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Second, although in this work we uniformly distribute nodes to construct a graph, we be- lieve a sparser graph containing more informative viewpoints may improve performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' Finally, we are also interested in explicitly predicting the potential gain during exploration to further boost planning performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFJT4oBgHgl3EQfjix9/content/2301.11575v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work was supported by Temasek Laboratories (TL@NUS) under grant TL/FS/2022/01.' metadata={'source': 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a/39FKT4oBgHgl3EQfRS0O/content/tmp_files/2301.11770v1.pdf.txt b/39FKT4oBgHgl3EQfRS0O/content/tmp_files/2301.11770v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..44544f5ae9cc5df675392f6083de483973ba70fe --- /dev/null +++ b/39FKT4oBgHgl3EQfRS0O/content/tmp_files/2301.11770v1.pdf.txt @@ -0,0 +1,1644 @@ +arXiv:2301.11770v1 [math.RA] 27 Jan 2023 +CONSTRUCTION OF SOME NON-ASSOCIATIVE ALGEBRAS +FROM ASSOCIATIVE ALGEBRAS WITH A ENDOMORPHISM +OPERATOR, DIFFERENTIAL OPERATOR OR LEFT +AVERAGING OPERATOR. +WILSON ARLEY MARTINEZ, SAMIN INGRITH CERON +Abstract. In this paper, we introduce the concepts of a Endomorphism Op- +erator, Left Averaging Operator, Differential Operator and Rota-Baxter Op- +erator and we construct examples of these linear maps on associative algebras +with a left identity, skew-idempotent or idempotent element. Its maps on asso- +ciative algebra, induces a non-associative algebra structure such as Lie algebra, +Pre-Lie algebra, Jordan algebra, Flexible Algebra or (left) Leibniz algebra. We +consider that the construction of non-associative algebras from associative al- +gebras with Linear Operators as the main results of this work. In this paper +we give a example of non-associative algebras on subspaces of square matrices +M(3 × 3, R). +Introduction +Linear operators can be defined on different algebraic structures, the weell-known +operators are the endomorphism operator and differential operator [15, 24, 16]. By +the 1970’s, new identities for operators have emerged from studies in combinatorics, +probability and analysis. Gian-Carlo Rota was most interested in the following +operators: +Endomorphism operator +R(x · y) = R(x) · R(y), +Differential operator +R(x · y) = R(x) · y + x · R(y), +Rota-Baxter operator of weight λ +where λ is a fixed constant, +R(x) · R(y) = R(x · R(y) + R(x) · y + λx · y), +Average operator +R(x) · R(y) = R(x · R(y)), +Inverse average operator +R(x) · R(y) = R(R(x) · y), +Reynolds operator +R(x) · R(y) = R(x · R(y) + R(x) · y − R(x) · R(y)). +Received by the editors January 27, 2023 and, in revised form, January, 2023. +2020 Mathematics Subject Classification. 17A15;17A32;17A20;17B40;47C05. +Key words and phrases. Associative Algebras, Lie algebra, Pre-Lie algebra, Jordan algebra, +Flexible Algebra, (left) Leibniz algebra, Rota-Baxter Operator, Endomorphism operator, Differ- +ential operator. +This work was completed with the support of the Universidad del Cauca. +The author was also supported by the research group “Estructuras Algebraicas, Divulgaci´on +Matem´atica y Teor´ıas Asociadas. @DiTa”. +1 + +2 +WILSON ARLEY MARTINEZ, SAMIN INGRITH CERON +An endomorphism is a homomorphism from an algebraic structure into itself. +Let A be a non-unital, associative algebra, α an algebra endomorphism on A and +define ∗ : A × A −→ A by a ∗ b = α(ab) for all a, b ∈ A. +Then (A, ∗, α) is a +hom-associative algebra, see [30]. +The study of averaging operators from an algebraic point of view was started +by Kamp´e de F´eriet [11] and continued and elaborated by Birkhoff [5]. Averaging +operators has connection with developments in the theory of turbulence [6, 5], and +was closely related to the probability theory. +In his Ph. D. thesis in 2000 [7], Weili Cao studied averaging operators in the +general context and the algebraic definition. He studied the naturally induced Lie +algebra structures from averaging operators: Let R : A → A be an Averaging +Operator on an algebra A, its map permit us to define a Lie bracket operation on +A, by [x, y] = x · R(y) − y · R(x), ∀x, y ∈ A, see [7, 21]. +Let R : A → A be a Differential Operator on a commutative associative algebra +A, its map induces a new Lie algebra structure called Witt type Lie algebras [29], +defined by the bracket [x, y] = R(x)·y−x·R(y), ∀x, y ∈ A. Commutative associative +algebras with this type of linear maps, permit us to present examples of Lie algebras. +Rota-Baxter operators were introduced by the mathematician Glenn E. Bax- +ter [3], in the study of differential equations applied to probability theory, and +mainly its importance by the works of G.-C. Rota in combinatorics [4, 22, 23]. +A Rota-Baxter algebra, is an associative algebra equipped with a Rota-Baxter +operator. Recently, noncommutative Rota-Baxter algebras have appeared in a wide +range of areas in pure mathematics, for example the works of Loday and Ronco on +dendriform dialgebras and trialgebras, see [18, 17] and too in applied mathematics, +see [8]. +The following result provides a way to construct a pre-Lie algebra structure from +Rota-Baxter operator relation on Lie Algebras or pre-Lie algebras. We find that if +R : A → A es a Rota Baxter-Operator on an Lie Algebra (A, [, ]), its map induces +a pre-Lie algebra structure, defined by x ∗ y = [R(x), y], ∀x, y ∈ A, see [2]. +In the case of the Rota-Baxter relation on pre-Lie algebras, it is known that +if (A, ·) is a pre-Lie algebra and R be a Rota-Baxter operator on A. Then R is +still a Rota-Baxter operator on (A, ∗), and the product given by x ∗ y = [R(x), y] += R(x) · y − y · R(x), x, y ∈ A, defines a new pre-Lie algebra (A, ∗) see [27] . +An element u is said to be skew-idempotent with respect to a product · in +the algebra if: u · u = −u, and an element u is a right identity if: x · u = x +for all element x in the algebra. Associative algebras with a left identity, skew- +idempotent or idempotent element, permit us to build examples of linear maps +as Endomorphism Operator, Left Averaging Operator, Differential Operator and +Rota-Baxter Operator. Associative algebras with this type of linear maps, permit +us to present constructions of non-associative algebras. +1. Construction of Lie algebra from Associative Algebras with a +Endomorphism Operator +In this section, we present in the Proposition 1.1.3 a Lie algebra structure given +by the following bracket [x, y] = x · R(y) − y · R(x) where R is a endomorphism + +CONSTRUCTION OF SOME NON-ASSOCIATIVE ALGEBRAS +3 +operator, and R is defined from a right identity on a subalgebra A: Let (A, ·) be an +algebra, then an element u of H, A ⊆ H, is called a right identity on A if x · u = x +for all x in A. +1.1. Introduction. In the Definition 1.2.1 we define a left and a right identity for +an associative algebra A and in the Proposition 1.2.3 we give a proof of the build +of an endomorphism operator with certain properties on A useding a right identity, +inspired by the well known result of X. Xu, [29], where establish the existence of +Lie Algebras from an associative algebra A with an Differential Operator on the +space A, we establish a new connection between an endomorphism operator with +a construction of Lie algebra structures on A . We start by briefly introducing the +definition of a Lie Algebra, a left and a right identity for A. +Definition 1.1.1. A Lie algebra over a field F is a vector space g over F equipped +with bilinear operation [, ] : g × g → g, called the commutator or (Lie) bracket +which satisfies the following identities: +[x, y] += +−[y, x] +(Antisymmetry) +(1.1.1) +[x, [y, z]] + [z, [x, y]] + [y, [z, x]] += +0 +(Jacobi identity) +(1.1.2) +Remark 1.1.2. It is well known that any associative algebra becomes a Lie algebra +with the Lie bracket given by the commutator: [x, y] = x · y − y · x. Also that the +dimension of a Lie algebra g is its dimension as a vector space over F and Ado’s +theorem states that every finite dimensional Lie algebra g over a field F can be +viewed as a Lie algebra of square matrices with the commutator as bracket. +Proposition 1.1.3. Let A be an associative algebra and let R : A → A a linear +map such that R2(x) = R(x) and R(x) · R(y) = R(x · y) for all x, y ∈ A. Then we +can define a Lie algebra structures on A given by +(1.1.3) +[x, y] = x · R(y) − y · R(x) (respectively [x, y] = R(x) · y − R(y) · x) +Proof. Let x, y, z ∈ A; then, [x, y] = −[y, x] and +[x, [y, z]] = x · R([y, z]) − [y, z] · R(x) += x · R(y · R(z) − z · R(y)) − (y · R(z) − z · R(y)) · R(x) += x · (R(y) · R(z) − R(z) · R(y)) − (y · R(z)) · R(x) + (z · R(y)) · R(x) += x · (R(y) · R(z)) − x · (R(z) · R(y)) − (y · R(z)) · R(x) + (z · R(y)) · R(x) +[y, [z, x]] = y · R([z, x]) − [z, x] · R(y) += y · R(z · R(x) − x · R(z)) − (z · R(x) − x · R(z)) · R(y) += y · (R(z) · R(x) − R(x) · R(z)) − (z · R(x)) · R(y) + (x · R(z)) · R(y) += y · (R(z) · R(x)) − y · (R(x) · R(z)) − (z · R(x)) · R(y) + (x · R(z)) · R(y) +[z, [x, y]] = z · R([x, y]) − [x, y] · R(z) += z · R(x · R(y) − y · R(x)) − (x · R(y) − y · R(x)) · R(z) += z · (R(x) · R(y) − R(y) · R(x)) − (x · R(y)) · R(z) + (y · R(x)) · R(z) += z · (R(x) · R(y)) − z · (R(y) · R(x)) − (x · R(y)) · R(z) + (y · R(x)) · R(z) +Thus, [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0. Therefore, (A, [ , ]) is a Lie algebra. +□ + +4 +WILSON ARLEY MARTINEZ, SAMIN INGRITH CERON +1.2. Examples of Lie algebras from the endomorphism operator. In this +subsection we present the relationship between an endomorphism operator and a +right identity of an associative algebra. In one direction, we show that an associa- +tive algebra A with a right identity gives an endomorphism operator with certain +properties on the associative algebra A that allow us to give examples of lie algebras +from the endomorphism. +Definition 1.2.1. Let A be an algebra and H a set containing A. An element +u ∈ H is called a left identity for A if u · x = x for all x ∈ A. Similarly, u ∈ H is +a right identity for A if x · u = x for all x ∈ A. An element u ∈ A which is both a +left and a right identity for A is an identity element. +Remark 1.2.2. Given an operation (function) · : H × A → A, an element u of H is +called a right identity for · if x · u = x for every element x of A. That is, the map +A → A given by x · u is the identity function on A. +The following proposition and its examples introduce the idea of all the main +results of this paper. +Proposition 1.2.3. Let ( A, · ) be an associative algebra and H a set containing +A, and suppose that there exists u ∈ H such that u · x ∈ A and x · u = x for all +x ∈ A. Then the linear map R : A −→ A defined by R(x) = u · x satisfies +(1.2.1) +R(x) · R(y) = R(x · y) for all x, y ∈ A. +Furthemore, R2(x) = R(x) if u2 = u, or R2(x) = x if u2 = 1. +Proof. Let x, y ∈ A, then R(x · y) = u · (x · y). On the other hand we have, +R(x)·R(y) = (u·x)·(u·y) = u·((x·u)·y) = u·(x·y). Therefore R(x)·R(y) = R(x·y) +for all x, y ∈ A. Now, if u2 = u, then R2(x) = u · (u · x) = (u2 · x) = u · x = R(x). +Therefore R2(x) = R(x) for all x ∈ A. +□ +Example 1.2.4. We consider the subalgebra +A = + + + + + +y +y +0 +n +n +0 +r +r +0 + + : r, n, y ∈ R + + + +under the usual matrix multiplication. +The element u = + + +a +b +c +1 − a +1 − b +−c +e +f +g + + satisfies x · u = x and u · x ∈ A for all +x ∈ A. +Then the linear map R : A −→ A defined by +R + + + + +y +y +0 +n +n +0 +r +r +0 + + + + = + + +a +b +c +1 − a +1 − b +−c +e +f +g + + · + + +y +y +0 +n +n +0 +r +r +0 + + += + + +ay + bn + cr +ay + bn + cr +0 +(1 − a)y + (1 − b)n − cr +(1 − a)y + (1 − b)n − cr +0 +ey + fn + gr +ey + fn + gr +0 + + +satisfies R(x) · R(y) = R(x · y) for all x, y ∈ A. + +CONSTRUCTION OF SOME NON-ASSOCIATIVE ALGEBRAS +5 +Example 1.2.5. We consider the subalgebra +A = + + + + + +x +y +y +w +k +k +m +n +n + + : x, w, m, y, k, n ∈ R + + + +under the usual matrix multiplication. +The element u = + + +1 +0 +0 +0 +0 +1 +0 +1 +0 + + satisfies u2 = I, +x · u = x and u · x ∈ A for all +x ∈ A. +Then the linear map R : A −→ A defined by +R + + + + +x +y +y +w +k +k +m +n +n + + + + = + + +x +y +y +m +n +n +w +k +k + + +satisfies R2(x) = x and R(x) · R(y) = R(x · y) for all x, y ∈ A. +Example 1.2.6. We consider the subalgebra +A = + + + + + +y +y +0 +n +n +0 +r +r +0 + + : y, n, r ∈ R + + + +under the usual matrix multiplication. +The element u = + + +1 +b +b +0 +1 − b +−b +0 +b − 1 +b + + satisfies u2 = u, +x · u = x and u · x ∈ A +for all x ∈ A. +Then the linear map R : A −→ A defined by +R + + + + +y +y +0 +n +n +0 +r +r +0 + + + + = + + +1 +b +b +0 +1 − b +−b +0 +b − 1 +b + + · + + +y +y +0 +n +n +0 +r +r +0 + + += + + +y + bn + br +y + bn + br +0 +n − bn − br +n − bn − br +0 +nb − n + br +nb − n + br +0 + + +satisfies R2(x) = R(x) and R(x) · R(y) = R(x · y) for all x, y ∈ A. Therefore, we +can define a Lie algebra structures on A given by [x, y] = x · R(y) − y · R(x). +2. Construction of Jordan algebra from Commutative Associative +Algebras with a Endomorphism Operator +Jordan algebras were introduced in the early 1930’s by a physicist, P.Jordan, +in an attempt to generalize the formalism of quantum mechanics. Little appears +to have resulted in this direction, but unanticipated relationships between these +algebras and Lie groups and the foundations of geometry have been discovered. +Definition 2.0.1. A (non-commutative) Jordan algebra is a vector space J over +a field F of characteristic ̸= 2 with a binary operation ◦ satisfying for x, y ∈ J the +following identity: +(2.0.1) +(x ◦ y) ◦ x = x ◦ (y ◦ x) +and +((x ◦ x) ◦ y) ◦ x = (x ◦ x) ◦ (y ◦ x). + +6 +WILSON ARLEY MARTINEZ, SAMIN INGRITH CERON +Remark 2.0.2. Given an associative algebra (A, ·) we can modify the product to +obtain a commutative algebra A+ as follows: To construct A+ we define x ∗ y = +x · y + y · x. In this new algebra the Jordan identity is satisfied +(2.0.2) +((x ∗ x) ∗ y) ∗ x = (x ∗ x) ∗ (y ∗ x). +Example 2.0.3. We consider the subalgebra +A+ = + + + + + +y +y +0 +n +n +0 +r +r +0 + + : y, n, r ∈ R + + + +under the product x ∗ y = x · y + y · x is a commutative algebra (non-associative), +where · is the usual matrix multiplication. +The element u = + + +1 +b +b +0 +1 − b +−b +0 +b − 1 +b + + satisfies u2 = u, +x · u = x and u · x ∈ A+ +for all x ∈ A+. +Then the linear map R : A+ −→ A+ defined by +R + + + + +y +y +0 +n +n +0 +r +r +0 + + + + = + + +1 +b +b +0 +1 − b +−b +0 +b − 1 +b + + · + + +y +y +0 +n +n +0 +r +r +0 + + += + + +y + bn + br +y + bn + br +0 +n − bn − br +n − bn − br +0 +nb − n + br +nb − n + br +0 + + +satisfies R2(x) = R(x) and R(x)∗R(y) = R(x∗y) for all x, y ∈ A+. Therefore the +Jordan identity is satisfied on A+, with the product given by x ◦ y = R(x) ∗ R(y). +Proposition 2.0.4. Suppose (A, ·) is a Jordan algebra, and R : A → A is a linear +map, such that +(2.0.3) +R2(x) = R(x) y R(x) · R(y) = R(x · y) for all x, y ∈ A. +Then we can define a new (non-commutative) Jordan algebra structures on A given +by +x ◦ y = R(x) · y +(respectively x ◦ y = x · R(y), x ◦ y = R(x) · R(y)). +Proof. Let A be a Jordan algebra and R : A → A is a linear map such that +R2(x) = R(x) y R(x) · R(y) = R(x · y) for all x, y ∈ A. So +(x ◦ x) ◦ (y ◦ x) = R(R(x) · x) ◦ (R(y) · x) += (R(x) · R(x)) · (R(y) · x) += ((R(x) · R(x)) · R(y)) · x += ((R2(x) · R2(x)) · R(y)) · x += R((R(x) · R(x)) · y) · x += R((R2(x) · R(x)) · y) · x += R(R(R(x) · x) · y) · x += ((x ◦ x) ◦ y) ◦ x. + +CONSTRUCTION OF SOME NON-ASSOCIATIVE ALGEBRAS +7 +This means that (x ◦ x) ◦ (y ◦ x) = ((x ◦ x) ◦ y) ◦ x for all x, y ∈ A. Since +(x ◦ y) ◦ x = R(R(x) · y) · x = (R(x) · R(y)) · x = R(x) · (R(y) · x) = x ◦ (y ◦ x). +So (x ◦ y) ◦ x = x ◦ (y ◦ x) for all x, y ∈ A. Therefore (A, ◦) is a Jordan algebra. +□ +Example 2.0.5. The element u = + + +1 +−1 +1 +1 +−1 +1 +1 +−1 +1 + + satisfy: u2 = u and x · u = x for +all x ∈ A. The algebra of matrices +A = + + + + + +x +−x +x +w +−w +w +p +−p +p + + : x, w, p ∈ R + + + , +is a associative algebra under the usual matrix multiplication. A is a commutative +algebra (non-associative) under the product x∗y = x·y +y ·x , where · is the usual +matrix multiplication. Then the linear map R : A −→ A defined by +R + + + + +x +−x +x +w +−w +w +p +−p +p + + + + = + + +1 +−1 +1 +1 +−1 +1 +1 +−1 +1 + + · + + +x +−x +x +w +−w +w +p +−p +p + + +=(x − w + p) + + +1 +−1 +1 +1 +−1 +1 +1 +−1 +1 + + +satisfies R2(x) = R(x) and R(x) ∗ R(y) = R(x ∗ y) for all x, y ∈ A. Therefore, we +can define a Jordan algebra structures on A given by x ◦ y = R(x) ∗ R(y), and from +it we can define a new Jordan algebra structures on A given by x ◦2 y = R(x) ◦ y. +3. Construction of (left) Leibniz algebra from Associative Algebras +with a Endomorphism Operator. +Leibniz algebras were first introduced by J.-L. Loday in [14] as a non-antisymmetric +version of Lie algebras, and many results of Lie algebras have been extended to +Leibniz algebras. Leibniz algebras play a significant role in different areas of math- +ematics and physics. +Definition 3.0.1. A (left) Leibniz algebra L is a vector space equipped with a +bilinear map +[ , ] : L × L → L +satisfying the (left) Leibniz identity +(3.0.1) +[x, [y, z]] = [[x, y], z] + [y, [x, z]]for all x, y, z ∈ L. +Remark 3.0.2. We may pass from the right to the left Leibniz algebra by considering +a new multiplication x ◦ y = [y, x]. For a Leibniz algebra L, we define left multi- +plication la : L → L by an element a on an element b by la(b) = [a, b]. Similarly, +right multiplication by an element a on an element b is defined by ra(b) = [b, a]. +An algebra L over F is a left Leibniz algebra if for every x ∈ L the corresponding +operator lx of left multiplication is a derivation of L, i.e. the mapping lx satisfies + +8 +WILSON ARLEY MARTINEZ, SAMIN INGRITH CERON +lx(a · b) = lx(a) · b + a · lx(b), lx ∈ Der(L). Thus, left multiplication is a derivation +in a left Leibniz algebra while right multiplication is not necessarily a derivation. +Proposition 3.0.3. Let A be an associative algebra over a field F and let +R : A → A +be an endomorphism of A such that R2 = R. Define the binary operation [ , ] on +A by the following rule: [a, b] = R(a) · b − b · R(a) for all elements a, b ∈ A, Then, +with respect to the operations + and [ , ], A becomes a Leibniz algebra. +Proof. See [25]. +□ +Lemma 3.0.4. Let A be a associative algebra and let R : A → A be a linear map. +Suppose that R2(x) = R(x) y R(x) · R(y) = R(x · y) +for all x, y ∈ A. Then there +exists a Leibniz structures on A given by +(3.0.2) +[x, y] = R(x) · y − R(y) · R(x) +for all x, y ∈ A. +Proof. Let x, y, z ∈ A, then we have +[[x, y], z] = R([x, y]) · z − R(z) · R([x, y]) += R(R(x) · y − R(y) · R(x)) · z − R(z) · R(R(x) · y − R(y) · R(x)) += (R(x) · R(y) − R(y) · R(x)) · z − R(z) · (R(x) · R(y) − R(y) · R(x)) +[y, [x, z]] = R(y) · [x, z] − R([x, z]) · R(y) += R(y) · (R(x) · z − R(z) · R(x)) − R(R(x) · z − R(z) · R(x)) · R(y) += R(y) · (R(x) · z − R(z) · R(x)) − (R(x) · R(z) − R(z) · R(x)) · R(y) +Then +[[x, y], z] + [y, [x, z]] = (R(x) · R(y) − R(y) · R(x)) · z − R(z) · (R(x) · R(y) − R(y) · R(x)) ++ R(y) · (R(x) · z − R(z) · R(x)) − (R(x) · R(z) − R(z) · R(x)) · R(y) +so +[[x, y], z] + [y, [x, z]] = (R(x) · R(y)) · z − (R(x) · R(z)) · R(y) − R(y) · (R(z) · R(x)) ++ R(z) · (R(y) · R(x)) +On the other hand, we have +[x, [y, z]] = R(x) · [y, z] − R([y, z]) · R(x) += R(x) · (R(y) · z − R(z) · R(y)) − R(R(y) · z − R(z) · R(y)) · R(x) += R(x) · (R(y) · z − R(z) · R(y)) − (R(y) · R(z) − R(z) · R(y)) · R(x) +Note that R2(x) = R(x) y R(x) · R(y) = R(x · y) for all x, y ∈ A, implies +[x, [y, z]] = [[x, y], z] + [y, [x, z]] for all x, y, z ∈ A. +□ +Example 3.0.5. The element u = + + +−1 +1 +1 +−1 +1 +1 +−1 +1 +1 + + satisfy: u2 = u and x · u = x for +all x ∈ A. The algebra of matrices +A = + + + + + +−y +y +y +−m +m +m +−t +t +t + + : y, m, t ∈ R + + + , + +CONSTRUCTION OF SOME NON-ASSOCIATIVE ALGEBRAS +9 +is a associative algebra under the usual matrix multiplication. Then the linear map +R : A −→ A defined by +R + + + + +−y +y +y +−m +m +m +−t +t +t + + + + = + + +−1 +1 +1 +−1 +1 +1 +−1 +1 +1 + + · + + +−y +y +y +−m +m +m +−t +t +t + + += (−y + m + t) + + +−1 +1 +1 +−1 +1 +1 +−1 +1 +1 + + +satisfies R2(x) = R(x) and R(x) · R(y) = R(x · y) for all x, y ∈ A. Therefore, we +can define a Leibniz structures structures on A given by +[x, y] = R(x) · y − R(y) · R(x) +4. Construction of Pre-Lie algebra from Commutative Associative +Algebras with a Endomorphism Operator +In this section we present in the Propositions 4.0.3 a construction of Pre-Lie +algebra structure given by x◦y = R(x)·R(y)−y ·R(x) where R is a endomorphism +operator: +Definition 4.0.1. An algebra A over F with a bilinear product ◦ which satisfies +the following identity: +(4.0.1) +(x ◦ y) ◦ z − x ◦ (y ◦ z) = (y ◦ x) ◦ z − y ◦ (x ◦ z) for all x, y, z ∈ A +is called a Left Pre-Lie algebra. +Remark 4.0.2. There is a construction of pre-Lie algebras using a commutative +associative algebra (A, ·) with a derivation D on A, the new product a∗b = a·D(b), +∀a, b ∈ A makes (A, ∗) become a Novikov algebra, Novikov algebra is a pre-Lie +algebra satisfying an additional identity: (xy)z = (xz)y, ∀x, y, z ∈ A. There are +some generalizations of the previous result for a commutative associative algebra +(A, ·). If D is a derivation on A , then the new product x ∗a y = x · D(y) + a · x · y, +∀x, y ∈ A makes (A, ∗a) become a Novikov algebra for a fixed element a ∈ F or +a ∈ A ( [10, 28]). In the case of an associative algebra (A, ·) with a Rota-Baxter +relation of weight 1, it is known that if x ∗ y = R(x) · y − y · R(x) − x · y, ∀x, y ∈ A, +then the product ∗ defines a pre-Lie algebra on A. ( [9, 13] ) +Proposition 4.0.3. Let A be a commutative associative algebra and let R : A → A +a linear map such that R2(x) = R(x) and R(x) · R(y) = R(x · y) for all x, y ∈ A. +Then we can define a Pre-Lie algebra structures on A given by +(4.0.2) x ◦ y = R(x) · R(y) − y · R(x) (respectively x ◦ y = R(x) · y − R(y) · R(x)). + +10 +WILSON ARLEY MARTINEZ, SAMIN INGRITH CERON +Proof. Let x, y, z ∈ A, then we have +(x ◦ y) ◦ z = (R(x) · R(y) − y · R(x)) ◦ z += R(R(x) · R(y) − y · R(x)) · R(z) − z · R(R(x) · R(y) − y · R(x)) += (R(x) · R(y) − R(y) · R(x)) · R(z) − z · (R(x) · R(y) − R(y) · R(x)) += 0 +x ◦ (y ◦ z) = x ◦ (R(y) · R(z) − z · R(y)) += R(x) · R(R(y) · R(z) − z · R(y)) − (R(y) · R(z) − z · R(y)) · R(x) += R(x) · (R(y) · R(z) − R(z) · R(y)) − (R(y) · R(z) − z · R(y)) · R(x) += −(R(y) · R(z) − z · R(y)) · R(x) +On the other hand, we have +(y ◦ x) ◦ z = (R(y) · R(x) − x · R(y)) ◦ z += R(R(y) · R(x) − x · R(y)) · R(z) − z · R(R(y) · R(x) − x · R(y)) += (R(y) · R(x) − R(x) · R(y)) · R(z) − z · (R(y) · R(x) − R(x) · R(y)) += 0 +y ◦ (x ◦ z) = y ◦ (R(x) · R(z) − z · R(x)) += R(y) · R(R(x) · R(z) − z · R(x)) − (R(x) · R(z) − z · R(x)) · R(y) += R(y) · (R(x) · R(z) − R(z) · R(x)) − (R(x) · R(z) − z · R(x)) · R(y) += −(R(x) · R(z) − z · R(x)) · R(y) +Therefore, (x ◦ y) ◦ z − x ◦ (y ◦ z) = (y ◦ x) ◦ z − y ◦ (x ◦ z). +□ +4.1. Examples of Pre-Lie algebras from commutative associative algebras +with the endomorphism operator. In this subsection we present in the example +4.1.1 a commutative associative algebra A with an unusual matrix multiplication, +that allow us to give an example of Pre-lie algebras from a endomorphism operator +with certain properties on the algebra A. +Example 4.1.1. Let A be the vector space of all 2 × 2 matrices over R. +A = +�� +m +x +n +y +� +: m, n, x, y ∈ R +� +. +A is a commutative associative algebra with the unusual matrix multiplication +defined by: +(4.1.1) +� +m +x +n +y +� +∗ +� +p +z +r +w +� += +� +mp +xz +nr +yw +� +. +The element I = +� 1 +1 +1 +1 +� +is the multiplicative identity. +Example 4.1.2. The element u = +� 1 +0 +0 +0 +� +with the usual matrix multiplication +satisfy: u2 = u, u · x ∈ A and x · u = x for all x ∈ A. The algebra of matrices +A = +�� m +0 +n +0 +� +: m, n ∈ R +� +, + +CONSTRUCTION OF SOME NON-ASSOCIATIVE ALGEBRAS +11 +is a commutative associative under the unusual matrix multiplication defined above +in 4.1.1. Then the linear map R : A −→ A defined by +R +�� m +0 +n +0 +�� += +� 1 +0 +0 +0 +� +· +� m +0 +n +0 +� += +� +m +0 +0 +0 +� +satisfies R2(x) = R(x) and R(x) ∗ R(y) = R(x ∗ y) for all x, y ∈ A. Therefore, we +can define a Pre-Lie algebra structures on A given by +x ◦ y = R(x) ∗ R(y) − y ∗ R(x) +5. Construction of Pre-Lie algebra from Commutative Associative +Algebras with a Differential Operator +We now present in the Propositions 5.0.1 a construction of Pre-Lie algebra struc- +ture given by x◦y = R(x)·y where R is a differential operator, and we also introduce +the notion of left zero divisor on A: Let ( H, · ) be a set H with a binary operation +· on it, then an element u of H is called a left zero divisor on A ⊆ H if x · u = 0 for +all x in A. +Proposition 5.0.1. Let A be a commutative associative algebra and let R : A → A +a linear map such that R2(x) = α. x and R(x)·y+x·R(y) = R(x·y) for all x, y ∈ A. +Then we can define a Pre-Lie algebra structures on A given by x ◦ y = R(x) · y. +Proof. Let x, y, z ∈ A, then +(x ◦ y) ◦ z − x ◦ (y ◦ z) = (R(x) · y) ◦ z − x ◦ (R(y) · z) += R(R(x) · y) · z − R(x) · (R(y) · z) += (R2(x) · y + R(x) · R(y)) · z − R(x) · (R(y) · z) += (R2(x) · y) · z += ((α. x) · y) · z. +On the other hand, we have +(y ◦ x) ◦ z − y ◦ (x ◦ z) = (R(y) · x) ◦ z − y ◦ (R(x) · z) += R(R(y) · x) · z − R(y) · (R(x) · z) += (R2(y) · x + R(y) · R(x)) · z − R(y) · (R(x) · z) += (R2(y) · x) · z += ((α. y) · x) · z. +Therefore, (x ◦ y) ◦ z − x ◦ (y ◦ z) = (y ◦ x) ◦ z − y ◦ (x ◦ z) +□ +5.1. Examples of Pre-Lie algebras from commutative associative algebras +with a Differential Operator. +Proposition 5.1.1. Let ( A, · ) be an associative subalgebra of an algebra H, and +suppose that there exists u ∈ H such that u · x ∈ A and x · u = 0 for all x ∈ A . +Then the linear map R : A −→ A defined by R(x) = u · x satisfies +(5.1.1) +R(x) · y + x · R(y) = R(x · y) for all x, y ∈ A. + +12 +WILSON ARLEY MARTINEZ, SAMIN INGRITH CERON +Proof. Let x, y ∈ A, then +R(x) · y + x · R(y) = (u · x) · y + x · (u · y) += u · (x · y) + (x · u) · y += u · (x · y) + (0 · y) = u · (x · y). +On the other hand, R(x · y) = u · (x · y). Therefore R(x) · y + x · R(y) = R(x · y) for +all x, y ∈ A. +□ +Example 5.1.2. We consider the subalgebra +A = + + + + + +y +y +0 +n +n +0 +r +r +0 + + : r, n, y ∈ R + + + +under the usual matrix multiplication. The element u = + + +a +b +c +−a +−b +−c +e +f +g + + satis- +fies x · u = 0 and u · x ∈ A for all x ∈ A. +Then the linear map R : A −→ A defined by +R + + + + +y +y +0 +n +n +0 +r +r +0 + + + + = + + +a +b +c +−a +−b +−c +e +f +g + + · + + +y +y +0 +n +n +0 +r +r +0 + + += + + +ay + bn + cr +ay + bn + cr +0 +−ay − bn − cr +−ay − bn − cr +0 +ey + fn + gr +ey + fn + gr +0 + + +satisfies R(x) · y + x · R(y) = R(x · y) for all x, y ∈ A. +Example 5.1.3. The algebra of matrices +A = + + +α + + +an +ap +aq +bn +bp +bq +n +p +q + + : α ∈ R + + + +where a = −βb and b, n, p, q ∈ R is a commutative associative algebra. +The element u = + + +a +βa +λβa +b +βb +λβb +1 +β +λβ + + where λ, β ∈ R, satisfies +u2 = (λβ)u, +x · u = 0 (⇔ an + bp + q = 0) and u · x ∈ A for all x ∈ A. +Then the linear map R : A −→ A defined by +R + + + + +an +ap +aq +bn +bp +bq +n +p +q + + + + = + + +a +βa +λβa +b +βb +λβb +1 +β +λβ + + · + + +an +ap +aq +bn +bp +bq +n +p +q + + += (λβ) + + +an +ap +aq +bn +bp +bq +n +p +q + + +satisfies R2(x) = (λβ)R(x) and R(x) · y + x · R(y) = R(x · y) for all x, y ∈ A. +Therefore, we can define a Pre-Lie algebra structures on A given by x◦y = R(x)·y. + +CONSTRUCTION OF SOME NON-ASSOCIATIVE ALGEBRAS +13 +6. Construction of flexible algebra from Associative Algebras with +a left averaging operator. +Definition 6.0.1. A flexible algebra is a vector space J over a field F of charac- +teristic ̸= 2 with a binary operation ◦ satisfying for x, y ∈ J the following identity: +(6.0.1) +(x ◦ y) ◦ x = x ◦ (y ◦ x). +Remark 6.0.2. The flexible algebras was initiated by Albert ([1]) and investigated +by the authors Myung, Okubo, Laufer, Tomber and Santilli, see for example ([20]). +Proposition 6.0.3. Suppose (A, ·) is a flexible algebra, and R : A → A is a linear +map, such that +(6.0.2) +R(x) · R(y) = R(R(x) · y) = R(x · y) for all x, y ∈ A. +Then we can define a new flexible algebra structures on A given by +x ◦ y = R(x) · y. +Proof. Let A be a flexible algebra and R : A → A is a linear map such that +R(x) · R(y) = R(R(x) · y) = R(x · y) for all x, y ∈ A. So (x ◦ y) ◦ x = (R(x) · y) ◦ x = +R(R(x) · y) · x = (R(x) · R(y)) · x. Since x ◦ (y ◦ x) = x ◦ (R(y) · x) = R(x) · (R(y) · x) +and (A, ·) is a flexible algebra, then we have (x ◦ y) ◦ x = x ◦ (y ◦ x) for all x, y ∈ A. +Therefore (A, ◦) is a flexible algebra. +□ +6.1. Examples of Flexible algebras from associative algebras with a left +averaging operator. +Proposition 6.1.1. Let ( A, · ) be an associative subalgebra of an algebra H. Sup- +pose H has the following property: +There exists u ∈ H, such that u · x ∈ A and x · u = u · x for all +x ∈ A. +Then the linear map R : A −→ A defined by R(a) = u ·x satisfies the left averaging +identity +(6.1.1) +R(a) · R(b) = R(R(a) · b) for all a, b ∈ A. +Furthemore, if u2 = u. Then +(6.1.2) +R(a) · R(b) = R(R(a) · b) = R(a · b) for all a, b ∈ A. +Proof. Let x, y ∈ A, by the hypothesis there exists u ∈ H such that u · x ∈ A and +x · u = u · x for all x ∈ A, then +R(x) · R(y) = (u · x) · (u · y) += u · ((x · u) · y) += u · ((u · x) · y) += R(R(x) · y). +If u2 = u, then R(x) · R(y) = u · ((u · x) · y) = u2 · (x · y) = u · (x · y) = R(x · y). +Therefore R(x) · R(y) = R(R(x) · y) = R(x · y) for all x, y ∈ A. +□ +Example 6.1.2. We consider the subalgebra +A = + + + + + +y +n +0 +0 +y +0 +0 +0 +r + + : r, n, y ∈ R + + + + +14 +WILSON ARLEY MARTINEZ, SAMIN INGRITH CERON +under the usual matrix multiplication. +The element u = + + +1 +0 +0 +0 +1 +0 +0 +0 +0 + + satisfies u2 = u, x · u = u · x and u · x ∈ A for +all x ∈ A. Then the linear map R : A −→ A defined by +R + + + + +y +n +0 +0 +y +0 +0 +0 +r + + + + = + + +1 +0 +0 +0 +1 +0 +0 +0 +0 + + · + + +y +n +0 +0 +y +0 +0 +0 +r + + = + + +y +n +0 +0 +y +0 +0 +0 +0 + + +satisfies R(x) · R(y) = R(R(x) · y) = R(x · y) for all x, y ∈ A. We have A is a +Lie algebra with the product [x, y] = x · R(y) − y · R(x) (see Proposition 1.1.3), +Therefore (A, [, ]) is a flexive algebra. +7. Construction of Rota-Baxter Operator +In this section we present in the Propositions 7.0.5 constructions of Rota-Baxter +Operators of weight λ = 1 and λ = 0 from associative algebra with an element +u skew-idempotent or nilpotent of index 2 respectively, and we also introduce the +notion of Rota-Baxter Operator of weight (λ, β). +We recall from the Introduction: +Definition 7.0.1. Let (A, ·) be an associative algebra. A linear map R : A → A +is called a Rota-Baxter operator of weight λ on A if R satisfies +(7.0.1) +R(x) · R(y) = R (R(x) · y + x · R(y) + λ x · y) , +for all x, y ∈ A. A Rota-Baxter algebra (also known as a Baxter algebra) is an +associative algebra A with a Rota-Baxter operator. +Remark 7.0.2. One importance of the Rota-Baxter Algebra is its close relationship +with other algebraic structures. For example pre-Lie algebras come naturally from +a Rota Baxter-Operator on an Lie Algebras. [12], [19] . +Definition 7.0.3. An elemet u ̸= 0 of an algebra A is called nilpotent if un = 0 +for some integer n . The least such integer is called the index of u. +Definition 7.0.4. An elemet u ̸= 0 of an algebra A is said to be skew-idempotent +with respect to a product · in the algebra A if: u · u = −u. +Proposition 7.0.5. Let ( A, · ) be an associative algebra and suppose that there +exists u ∈ A such that u2 = −u and u · x ∈ A for all x ∈ A. Then the linear map +R : A −→ A defined by R(x) = u · x satisfies +(7.0.2) +R ( R(x) · y + x · R(y) + x · y ) = R(x) · R(y) for all x, y ∈ A. +Furthemore, if u2 = 0, then R is a Rota-Baxter operator of weight zero on A. +Proof. Let x, y ∈ A, then we have R(x) · R(y) = (u · x) · (u · y). On the other hand, +R(R(x) · y + x · R(y) + x · y) = R((u · x) · y + x · (u · y) + x · y) += u · ((u · x) · y + x · (u · y) + x · y) += u2 · (x · y) + (u · x) · (u · y) + u · (x · y) += (u2 + u) · (x · y) + (u · x) · (u · y). + +CONSTRUCTION OF SOME NON-ASSOCIATIVE ALGEBRAS +15 +Therefore R(R(x)·y +x·R(y)+x·y) = R(x)·R(y) for all x, y ∈ A. Now, if u2 = 0, +then +R(R(x) · y + x · R(y)) = (u · x) · (u · y). +Therefore R(R(x) · y + x · R(y)) = R(x) · R(y) for all x, y ∈ A. +□ +Example 7.0.6. The element u = +� xy +−x2 +y2 +−xy +� +satisfies u2 = 0 and u · x ∈ A for +all x in the algebra of matrices +A = +�� +0 +a +0 +b +� +: a, b ∈ R +� +considered as a subalgebra of B = M2×2 under the usual matrix multiplication. If +we define the map R : A −→ A by +R( +� +0 +a +0 +b +� +) = +� +xy +−x2 +y2 +−xy +� � +0 +a +0 +b +� += +� +0 +xya − x2b +0 +y2a − xyb +� +, +then R satisfies R(R(x) · y + y · R(x)) = R(x) · R(y); for all x, y ∈ A. +Example 7.0.7. The element u = +� +x +y +−x2−x +y +−x − 1 +� +, y ̸= 0 is an skew-idempotent +in the algebra of matrices under the usual matrix multiplication, that is u2 = −u. +We observe that u · x ∈ A for all x ∈ A. If we define the map R : A −→ A by +R( +� +0 +a +0 +b +� +) = +� +x +y +−x2−x +y +−x − 1 +� � +0 +a +0 +b +� += +� +0 +xa + yb +0 +( −x2−x +y +)a − (x + 1)b +� +then R satisfies R(x) · R(y) = R(R(x) · y + x · R(y) + x · y); for all x, y ∈ A. +Definition 7.0.8. Let (A, ·) be an associative algebra. A linear map R : A → A +is called a Rota-Baxter operador of weight (λ, β) on A if R satisfies +(7.0.3) +R(x) · R(y) = R +� +R(x) · y + x · R(y) + λ x · y +� ++ β x · y, for all x, y ∈ A. +Remark 7.0.9. A Rota-Baxter operador of weight (λ, β) for associative algebras +allows to build examples of Dyckm-algebras [26], the main result of this section is +the construction of Rota-Baxter operador of weight (λ, β) on associative algebras. +Proposition 7.0.10. Let ( A, · ) be an associative algebra and suppose that there +exists u ∈ A such that u2 = −λu − β1A and u · x ∈ A for all x ∈ A. Then the +linear map R : A −→ A defined by R(x) = u · x satisfies +(7.0.4) +R ( R(x) · y + x · R(y) + λx · y ) + βx · y = R(x) · R(y) for all x, y ∈ A. +Proof. Let x, y ∈ A, then we have +R(R(x) · y + x · R(y) + λx · y) + βx · y = R((u · x) · y + x · (u · y) + λx · y) + βx · y += u · ((u · x) · y + x · (u · y) + λx · y) + βx · y += u2 · (x · y) + (u · x) · (u · y) + λu · (x · y) + βx · y += (u2 + λu + β1A) · (x · y) + (u · x) · (u · y) += (u · x) · (u · y) +On the other hand, R(x) · R(y) = (u · x) · (u · y). Therefore +R(R(x) · y + x · R(y) + λx · y) + βx · y = R(x) · R(y) for all x, y ∈ A + +16 +WILSON ARLEY MARTINEZ, SAMIN INGRITH CERON +□ +Example 7.0.11. The element u = +� +x +y +−x2−λx−β +y +−x − λ +� +, where y ̸= 0 satisfy: +u2 = −λu − β1A and u · x ∈ A for all x ∈ A. If we define the map R : A −→ A by +R( +� 0 +a +0 +b +� +) = +� +x +y +−x2−λx−β +y +−x − λ +� � 0 +a +0 +b +� += +� +0 +xa + yb +0 +( −x2−λx−β +y +)a − (x + λ)b +� +then R satisfies R(x)·R(y) = R(R(x)·y + x·R(y)+ λx·y)+ βx·y; for all x, y ∈ A. +Acknowledgment. We thank to Universidad del Cauca for the support to our +research group “Estructuras Algebraicas, Divulgaci´on Matem´atica y Teor´ıas Aso- +ciadas. @DiTa” under the research project with ID 5773, entitled ”Aplicaciones de +Estructuras Algebraicas”. We also thank to the anonymous referee for their help- +ful comments. This work is dedicated to my daughters especially to Clara Isabel +Martinez Ceron (January 17, 2017). +References +1. A.A. ALBERT, Power associative rings, Trans. Amer. math. Soc. 64 (1948), 552–597. +2. Huihui An and Chengming Bai, From rota–baxter algebras to pre-lie algebras, Journal of +Physics A: Mathematical and Theoretical 41 (2008), no. 01. +3. G. 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Dongping HOU Xiuxian LI and Chengming BAI, Rota-baxter operators on pre-lie algebras, +Journal of Nonlinear Mathematical Physics 14 (2007), no. 2, 269–289. +28. X. Xu, On simple novikov algebras and their irreducible modules, J. Algebra 185 (1996), +905–934. +29. +, New generalized simple lie algebras of cartan type over a field with characteristic +zero, J. Algebra 224 (2000), 23–58. +30. D. Yau, Hom-algebras and homology, J. Lie Theory 19 (2009), no. 02, 409–421. +Martinez, W.A.; Departmento de Matem´aticas, Universidad del Cauca , Popay´an , +Colombia +Email address: wamartinez@unicauca.edu.co +Ceron, S.I.; Departmento de Matem´aticas, Universidad del Cauca , Popay´an , Colom- +bia +Email address: sicbravo@gmail.com + diff --git a/39FKT4oBgHgl3EQfRS0O/content/tmp_files/load_file.txt b/39FKT4oBgHgl3EQfRS0O/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c7bc7b311d2df4c83a8402e920cebde7977a2cf0 --- /dev/null +++ b/39FKT4oBgHgl3EQfRS0O/content/tmp_files/load_file.txt @@ -0,0 +1,695 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf,len=694 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='11770v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='RA] 27 Jan 2023 CONSTRUCTION OF SOME NON-ASSOCIATIVE ALGEBRAS FROM ASSOCIATIVE ALGEBRAS WITH A ENDOMORPHISM OPERATOR, DIFFERENTIAL OPERATOR OR LEFT AVERAGING OPERATOR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' WILSON ARLEY MARTINEZ, SAMIN INGRITH CERON Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' In this paper, we introduce the concepts of a Endomorphism Op- erator, Left Averaging Operator, Differential Operator and Rota-Baxter Op- erator and we construct examples of these linear maps on associative algebras with a left identity, skew-idempotent or idempotent element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Its maps on asso- ciative algebra, induces a non-associative algebra structure such as Lie algebra, Pre-Lie algebra, Jordan algebra, Flexible Algebra or (left) Leibniz algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' We consider that the construction of non-associative algebras from associative al- gebras with Linear Operators as the main results of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' In this paper we give a example of non-associative algebras on subspaces of square matrices M(3 × 3, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Introduction Linear operators can be defined on different algebraic structures, the weell-known operators are the endomorphism operator and differential operator [15, 24, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' By the 1970’s, new identities for operators have emerged from studies in combinatorics, probability and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Gian-Carlo Rota was most interested in the following operators: Endomorphism operator R(x · y) = R(x) · R(y), Differential operator R(x · y) = R(x) · y + x · R(y), Rota-Baxter operator of weight λ where λ is a fixed constant, R(x) · R(y) = R(x · R(y) + R(x) · y + λx · y), Average operator R(x) · R(y) = R(x · R(y)), Inverse average operator R(x) · R(y) = R(R(x) · y), Reynolds operator R(x) · R(y) = R(x · R(y) + R(x) · y − R(x) · R(y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Received by the editors January 27, 2023 and, in revised form, January, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' 17A15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='17A32;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='17A20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='17B40;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='47C05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Associative Algebras, Lie algebra, Pre-Lie algebra, Jordan algebra, Flexible Algebra, (left) Leibniz algebra, Rota-Baxter Operator, Endomorphism operator, Differ- ential operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' This work was completed with the support of the Universidad del Cauca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' The author was also supported by the research group “Estructuras Algebraicas, Divulgaci´on Matem´atica y Teor´ıas Asociadas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' @DiTa”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' 1 2 WILSON ARLEY MARTINEZ, SAMIN INGRITH CERON An endomorphism is a homomorphism from an algebraic structure into itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Let A be a non-unital, associative algebra, α an algebra endomorphism on A and define ∗ : A × A −→ A by a ∗ b = α(ab) for all a, b ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Then (A, ∗, α) is a hom-associative algebra, see [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' The study of averaging operators from an algebraic point of view was started by Kamp´e de F´eriet [11] and continued and elaborated by Birkhoff [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Averaging operators has connection with developments in the theory of turbulence [6, 5], and was closely related to the probability theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' In his Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' thesis in 2000 [7], Weili Cao studied averaging operators in the general context and the algebraic definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' He studied the naturally induced Lie algebra structures from averaging operators: Let R : A → A be an Averaging Operator on an algebra A, its map permit us to define a Lie bracket operation on A, by [x, y] = x · R(y) − y · R(x), ∀x, y ∈ A, see [7, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Let R : A → A be a Differential Operator on a commutative associative algebra A, its map induces a new Lie algebra structure called Witt type Lie algebras [29], defined by the bracket [x, y] = R(x)·y−x·R(y), ∀x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Commutative associative algebras with this type of linear maps, permit us to present examples of Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Rota-Baxter operators were introduced by the mathematician Glenn E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Bax- ter [3], in the study of differential equations applied to probability theory, and mainly its importance by the works of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Rota in combinatorics [4, 22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' A Rota-Baxter algebra, is an associative algebra equipped with a Rota-Baxter operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Recently, noncommutative Rota-Baxter algebras have appeared in a wide range of areas in pure mathematics, for example the works of Loday and Ronco on dendriform dialgebras and trialgebras, see [18, 17] and too in applied mathematics, see [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' The following result provides a way to construct a pre-Lie algebra structure from Rota-Baxter operator relation on Lie Algebras or pre-Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' We find that if R : A → A es a Rota Baxter-Operator on an Lie Algebra (A, [, ]), its map induces a pre-Lie algebra structure, defined by x ∗ y = [R(x), y], ∀x, y ∈ A, see [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' In the case of the Rota-Baxter relation on pre-Lie algebras, it is known that if (A, ·) is a pre-Lie algebra and R be a Rota-Baxter operator on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Then R is still a Rota-Baxter operator on (A, ∗), and the product given by x ∗ y = [R(x), y] = R(x) · y − y · R(x), x, y ∈ A, defines a new pre-Lie algebra (A, ∗) see [27] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' An element u is said to be skew-idempotent with respect to a product · in the algebra if: u · u = −u, and an element u is a right identity if: x · u = x for all element x in the algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Associative algebras with a left identity, skew- idempotent or idempotent element, permit us to build examples of linear maps as Endomorphism Operator, Left Averaging Operator, Differential Operator and Rota-Baxter Operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Associative algebras with this type of linear maps, permit us to present constructions of non-associative algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Construction of Lie algebra from Associative Algebras with a Endomorphism Operator In this section, we present in the Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='3 a Lie algebra structure given by the following bracket [x, y] = x · R(y) − y · R(x) where R is a endomorphism CONSTRUCTION OF SOME NON-ASSOCIATIVE ALGEBRAS 3 operator, and R is defined from a right identity on a subalgebra A: Let (A, ·) be an algebra, then an element u of H, A ⊆ H, is called a right identity on A if x · u = x for all x in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' In the Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='1 we define a left and a right identity for an associative algebra A and in the Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='3 we give a proof of the build of an endomorphism operator with certain properties on A useding a right identity, inspired by the well known result of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Xu, [29], where establish the existence of Lie Algebras from an associative algebra A with an Differential Operator on the space A, we establish a new connection between an endomorphism operator with a construction of Lie algebra structures on A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' We start by briefly introducing the definition of a Lie Algebra, a left and a right identity for A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' A Lie algebra over a field F is a vector space g over F equipped with bilinear operation [, ] : g × g → g, called the commutator or (Lie) bracket which satisfies the following identities: [x, y] = −[y, x] (Antisymmetry) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='1) [x, [y, z]] + [z, [x, y]] + [y, [z, x]] = 0 (Jacobi identity) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='2) Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' It is well known that any associative algebra becomes a Lie algebra with the Lie bracket given by the commutator: [x, y] = x · y − y · x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Also that the dimension of a Lie algebra g is its dimension as a vector space over F and Ado’s theorem states that every finite dimensional Lie algebra g over a field F can be viewed as a Lie algebra of square matrices with the commutator as bracket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Let A be an associative algebra and let R : A → A a linear map such that R2(x) = R(x) and R(x) · R(y) = R(x · y) for all x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Then we can define a Lie algebra structures on A given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='3) [x, y] = x · R(y) − y · R(x) (respectively [x, y] = R(x) · y − R(y) · x) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Let x, y, z ∈ A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' [x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' y] = −[y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' x] and [x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' [y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' z]] = x · R([y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' z]) − [y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' z] · R(x) = x · R(y · R(z) − z · R(y)) − (y · R(z) − z · R(y)) · R(x) = x · (R(y) · R(z) − R(z) · R(y)) − (y · R(z)) · R(x) + (z · R(y)) · R(x) = x · (R(y) · R(z)) − x · (R(z) · R(y)) − (y · R(z)) · R(x) + (z · R(y)) · R(x) [y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' [z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' x]] = y · R([z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' x]) − [z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' x] · R(y) = y · R(z · R(x) − x · R(z)) − (z · R(x) − x · R(z)) · R(y) = y · (R(z) · R(x) − R(x) · R(z)) − (z · R(x)) · R(y) + (x · R(z)) · R(y) = y · (R(z) · R(x)) − y · (R(x) · R(z)) − (z · R(x)) · R(y) + (x · R(z)) · R(y) [z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' [x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' y]] = z · R([x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' y]) − [x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' y] · R(z) = z · R(x · R(y) − y · R(x)) − (x · R(y) − y · R(x)) · R(z) = z · (R(x) · R(y) − R(y) · R(x)) − (x · R(y)) · R(z) + (y · R(x)) · R(z) = z · (R(x) · R(y)) − z · (R(y) · R(x)) − (x · R(y)) · R(z) + (y · R(x)) · R(z) Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' [x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' [y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' z]] + [y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' [z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' x]] + [z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' [x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' y]] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Therefore, (A, [ , ]) is a Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' □ 4 WILSON ARLEY MARTINEZ, SAMIN INGRITH CERON 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Examples of Lie algebras from the endomorphism operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' In this subsection we present the relationship between an endomorphism operator and a right identity of an associative algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' In one direction, we show that an associa- tive algebra A with a right identity gives an endomorphism operator with certain properties on the associative algebra A that allow us to give examples of lie algebras from the endomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Let A be an algebra and H a set containing A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' An element u ∈ H is called a left identity for A if u · x = x for all x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Similarly, u ∈ H is a right identity for A if x · u = x for all x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' An element u ∈ A which is both a left and a right identity for A is an identity element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Given an operation (function) · : H × A → A, an element u of H is called a right identity for · if x · u = x for every element x of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' That is, the map A → A given by x · u is the identity function on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' The following proposition and its examples introduce the idea of all the main results of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Let ( A, · ) be an associative algebra and H a set containing A, and suppose that there exists u ∈ H such that u · x ∈ A and x · u = x for all x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Then the linear map R : A −→ A defined by R(x) = u · x satisfies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='1) R(x) · R(y) = R(x · y) for all x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Furthemore, R2(x) = R(x) if u2 = u, or R2(x) = x if u2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Let x, y ∈ A, then R(x · y) = u · (x · y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' On the other hand we have, R(x)·R(y) = (u·x)·(u·y) = u·((x·u)·y) = u·(x·y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Therefore R(x)·R(y) = R(x·y) for all x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Now, if u2 = u, then R2(x) = u · (u · x) = (u2 · x) = u · x = R(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Therefore R2(x) = R(x) for all x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' □ Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' We consider the subalgebra A = \uf8f1 \uf8f2 \uf8f3 \uf8eb \uf8ed y y 0 n n 0 r r 0 \uf8f6 \uf8f8 : r, n, y ∈ R \uf8fc \uf8fd \uf8fe under the usual matrix multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' The element u = \uf8eb \uf8ed a b c 1 − a 1 − b −c e f g \uf8f6 \uf8f8 satisfies x · u = x and u · x ∈ A for all x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Then the linear map R : A −→ A defined by R \uf8eb \uf8ed \uf8eb \uf8ed y y 0 n n 0 r r 0 \uf8f6 \uf8f8 \uf8f6 \uf8f8 = \uf8eb \uf8ed a b c 1 − a 1 − b −c e f g \uf8f6 \uf8f8 · \uf8eb \uf8ed y y 0 n n 0 r r 0 \uf8f6 \uf8f8 = \uf8eb \uf8ed ay + bn + cr ay + bn + cr 0 (1 − a)y + (1 − b)n − cr (1 − a)y + (1 − b)n − cr 0 ey + fn + gr ey + fn + gr 0 \uf8f6 \uf8f8 satisfies R(x) · R(y) = R(x · y) for all x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' CONSTRUCTION OF SOME NON-ASSOCIATIVE ALGEBRAS 5 Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' We consider the subalgebra A = \uf8f1 \uf8f2 \uf8f3 \uf8eb \uf8ed x y y w k k m n n \uf8f6 \uf8f8 : x, w, m, y, k, n ∈ R \uf8fc \uf8fd \uf8fe under the usual matrix multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' The element u = \uf8eb \uf8ed 1 0 0 0 0 1 0 1 0 \uf8f6 \uf8f8 satisfies u2 = I, x · u = x and u · x ∈ A for all x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Then the linear map R : A −→ A defined by R \uf8eb \uf8ed \uf8eb \uf8ed x y y w k k m n n \uf8f6 \uf8f8 \uf8f6 \uf8f8 = \uf8eb \uf8ed x y y m n n w k k \uf8f6 \uf8f8 satisfies R2(x) = x and R(x) · R(y) = R(x · y) for all x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' We consider the subalgebra A = \uf8f1 \uf8f2 \uf8f3 \uf8eb \uf8ed y y 0 n n 0 r r 0 \uf8f6 \uf8f8 : y, n, r ∈ R \uf8fc \uf8fd \uf8fe under the usual matrix multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' The element u = \uf8eb \uf8ed 1 b b 0 1 − b −b 0 b − 1 b \uf8f6 \uf8f8 satisfies u2 = u, x · u = x and u · x ∈ A for all x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Then the linear map R : A −→ A defined by R \uf8eb \uf8ed \uf8eb \uf8ed y y 0 n n 0 r r 0 \uf8f6 \uf8f8 \uf8f6 \uf8f8 = \uf8eb \uf8ed 1 b b 0 1 − b −b 0 b − 1 b \uf8f6 \uf8f8 · \uf8eb \uf8ed y y 0 n n 0 r r 0 \uf8f6 \uf8f8 = \uf8eb \uf8ed y + bn + br y + bn + br 0 n − bn − br n − bn − br 0 nb − n + br nb − n + br 0 \uf8f6 \uf8f8 satisfies R2(x) = R(x) and R(x) · R(y) = R(x · y) for all x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Therefore, we can define a Lie algebra structures on A given by [x, y] = x · R(y) − y · R(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Construction of Jordan algebra from Commutative Associative Algebras with a Endomorphism Operator Jordan algebras were introduced in the early 1930’s by a physicist, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='Jordan, in an attempt to generalize the formalism of quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Little appears to have resulted in this direction, but unanticipated relationships between these algebras and Lie groups and the foundations of geometry have been discovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' A (non-commutative) Jordan algebra is a vector space J over a field F of characteristic ̸= 2 with a binary operation ◦ satisfying for x, y ∈ J the following identity: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='1) (x ◦ y) ◦ x = x ◦ (y ◦ x) and ((x ◦ x) ◦ y) ◦ x = (x ◦ x) ◦ (y ◦ x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' 6 WILSON ARLEY MARTINEZ, SAMIN INGRITH CERON Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Given an associative algebra (A, ·) we can modify the product to obtain a commutative algebra A+ as follows: To construct A+ we define x ∗ y = x · y + y · x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' In this new algebra the Jordan identity is satisfied (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='2) ((x ∗ x) ∗ y) ∗ x = (x ∗ x) ∗ (y ∗ x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' We consider the subalgebra A+ = \uf8f1 \uf8f2 \uf8f3 \uf8eb \uf8ed y y 0 n n 0 r r 0 \uf8f6 \uf8f8 : y, n, r ∈ R \uf8fc \uf8fd \uf8fe under the product x ∗ y = x · y + y · x is a commutative algebra (non-associative), where · is the usual matrix multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' The element u = \uf8eb \uf8ed 1 b b 0 1 − b −b 0 b − 1 b \uf8f6 \uf8f8 satisfies u2 = u, x · u = x and u · x ∈ A+ for all x ∈ A+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Then the linear map R : A+ −→ A+ defined by R \uf8eb \uf8ed \uf8eb \uf8ed y y 0 n n 0 r r 0 \uf8f6 \uf8f8 \uf8f6 \uf8f8 = \uf8eb \uf8ed 1 b b 0 1 − b −b 0 b − 1 b \uf8f6 \uf8f8 · \uf8eb \uf8ed y y 0 n n 0 r r 0 \uf8f6 \uf8f8 = \uf8eb \uf8ed y + bn + br y + bn + br 0 n − bn − br n − bn − br 0 nb − n + br nb − n + br 0 \uf8f6 \uf8f8 satisfies R2(x) = R(x) and R(x)∗R(y) = R(x∗y) for all x, y ∈ A+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Therefore the Jordan identity is satisfied on A+, with the product given by x ◦ y = R(x) ∗ R(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Suppose (A, ·) is a Jordan algebra, and R : A → A is a linear map, such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='3) R2(x) = R(x) y R(x) · R(y) = R(x · y) for all x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Then we can define a new (non-commutative) Jordan algebra structures on A given by x ◦ y = R(x) · y (respectively x ◦ y = x · R(y), x ◦ y = R(x) · R(y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Let A be a Jordan algebra and R : A → A is a linear map such that R2(x) = R(x) y R(x) · R(y) = R(x · y) for all x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' So (x ◦ x) ◦ (y ◦ x) = R(R(x) · x) ◦ (R(y) · x) = (R(x) · R(x)) · (R(y) · x) = ((R(x) · R(x)) · R(y)) · x = ((R2(x) · R2(x)) · R(y)) · x = R((R(x) · R(x)) · y) · x = R((R2(x) · R(x)) · y) · x = R(R(R(x) · x) · y) · x = ((x ◦ x) ◦ y) ◦ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' CONSTRUCTION OF SOME NON-ASSOCIATIVE ALGEBRAS 7 This means that (x ◦ x) ◦ (y ◦ x) = ((x ◦ x) ◦ y) ◦ x for all x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Since (x ◦ y) ◦ x = R(R(x) · y) · x = (R(x) · R(y)) · x = R(x) · (R(y) · x) = x ◦ (y ◦ x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' So (x ◦ y) ◦ x = x ◦ (y ◦ x) for all x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Therefore (A, ◦) is a Jordan algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' □ Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' The element u = \uf8eb \uf8ed 1 −1 1 1 −1 1 1 −1 1 \uf8f6 \uf8f8 satisfy: u2 = u and x · u = x for all x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' The algebra of matrices A = \uf8f1 \uf8f2 \uf8f3 \uf8eb \uf8ed x −x x w −w w p −p p \uf8f6 \uf8f8 : x, w, p ∈ R \uf8fc \uf8fd \uf8fe , is a associative algebra under the usual matrix multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' A is a commutative algebra (non-associative) under the product x∗y = x·y +y ·x , where · is the usual matrix multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Then the linear map R : A −→ A defined by R \uf8eb \uf8ed \uf8eb \uf8ed x −x x w −w w p −p p \uf8f6 \uf8f8 \uf8f6 \uf8f8 = \uf8eb \uf8ed 1 −1 1 1 −1 1 1 −1 1 \uf8f6 \uf8f8 · \uf8eb \uf8ed x −x x w −w w p −p p \uf8f6 \uf8f8 =(x − w + p) \uf8eb \uf8ed 1 −1 1 1 −1 1 1 −1 1 \uf8f6 \uf8f8 satisfies R2(x) = R(x) and R(x) ∗ R(y) = R(x ∗ y) for all x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Therefore, we can define a Jordan algebra structures on A given by x ◦ y = R(x) ∗ R(y), and from it we can define a new Jordan algebra structures on A given by x ◦2 y = R(x) ◦ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Construction of (left) Leibniz algebra from Associative Algebras with a Endomorphism Operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Leibniz algebras were first introduced by J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Loday in [14] as a non-antisymmetric version of Lie algebras, and many results of Lie algebras have been extended to Leibniz algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Leibniz algebras play a significant role in different areas of math- ematics and physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' A (left) Leibniz algebra L is a vector space equipped with a bilinear map [ , ] : L × L → L satisfying the (left) Leibniz identity (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='1) [x, [y, z]] = [[x, y], z] + [y, [x, z]]for all x, y, z ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' We may pass from the right to the left Leibniz algebra by considering a new multiplication x ◦ y = [y, x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' For a Leibniz algebra L, we define left multi- plication la : L → L by an element a on an element b by la(b) = [a, b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Similarly, right multiplication by an element a on an element b is defined by ra(b) = [b, a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' An algebra L over F is a left Leibniz algebra if for every x ∈ L the corresponding operator lx of left multiplication is a derivation of L, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' the mapping lx satisfies 8 WILSON ARLEY MARTINEZ, SAMIN INGRITH CERON lx(a · b) = lx(a) · b + a · lx(b), lx ∈ Der(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Thus, left multiplication is a derivation in a left Leibniz algebra while right multiplication is not necessarily a derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Let A be an associative algebra over a field F and let R : A → A be an endomorphism of A such that R2 = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Define the binary operation [ , ] on A by the following rule: [a, b] = R(a) · b − b · R(a) for all elements a, b ∈ A, Then, with respect to the operations + and [ , ], A becomes a Leibniz algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' See [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Let A be a associative algebra and let R : A → A be a linear map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Suppose that R2(x) = R(x) y R(x) · R(y) = R(x · y) for all x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Then there exists a Leibniz structures on A given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='2) [x, y] = R(x) · y − R(y) · R(x) for all x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Let x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' z ∈ A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' then we have [[x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' y],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' z] = R([x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' y]) · z − R(z) · R([x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' y]) = R(R(x) · y − R(y) · R(x)) · z − R(z) · R(R(x) · y − R(y) · R(x)) = (R(x) · R(y) − R(y) · R(x)) · z − R(z) · (R(x) · R(y) − R(y) · R(x)) [y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' [x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' z]] = R(y) · [x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' z] − R([x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' z]) · R(y) = R(y) · (R(x) · z − R(z) · R(x)) − R(R(x) · z − R(z) · R(x)) · R(y) = R(y) · (R(x) · z − R(z) · R(x)) − (R(x) · R(z) − R(z) · R(x)) · R(y) Then [[x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' y],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' z] + [y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' [x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' z]] = (R(x) · R(y) − R(y) · R(x)) · z − R(z) · (R(x) · R(y) − R(y) · R(x)) + R(y) · (R(x) · z − R(z) · R(x)) − (R(x) · R(z) − R(z) · R(x)) · R(y) so [[x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' y],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' z] + [y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' [x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' z]] = (R(x) · R(y)) · z − (R(x) · R(z)) · R(y) − R(y) · (R(z) · R(x)) + R(z) · (R(y) · R(x)) On the other hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' we have [x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' [y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' z]] = R(x) · [y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' z] − R([y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' z]) · R(x) = R(x) · (R(y) · z − R(z) · R(y)) − R(R(y) · z − R(z) · R(y)) · R(x) = R(x) · (R(y) · z − R(z) · R(y)) − (R(y) · R(z) − R(z) · R(y)) · R(x) Note that R2(x) = R(x) y R(x) · R(y) = R(x · y) for all x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' y ∈ A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' implies [x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' [y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' z]] = [[x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' y],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' z] + [y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' [x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' z]] for all x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' z ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' □ Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' The element u = \uf8eb \uf8ed −1 1 1 −1 1 1 −1 1 1 \uf8f6 \uf8f8 satisfy: u2 = u and x · u = x for all x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' The algebra of matrices A = \uf8f1 \uf8f2 \uf8f3 \uf8eb \uf8ed −y y y −m m m −t t t \uf8f6 \uf8f8 : y, m, t ∈ R \uf8fc \uf8fd \uf8fe , CONSTRUCTION OF SOME NON-ASSOCIATIVE ALGEBRAS 9 is a associative algebra under the usual matrix multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Then the linear map R : A −→ A defined by R \uf8eb \uf8ed \uf8eb \uf8ed −y y y −m m m −t t t \uf8f6 \uf8f8 \uf8f6 \uf8f8 = \uf8eb \uf8ed −1 1 1 −1 1 1 −1 1 1 \uf8f6 \uf8f8 · \uf8eb \uf8ed −y y y −m m m −t t t \uf8f6 \uf8f8 = (−y + m + t) \uf8eb \uf8ed −1 1 1 −1 1 1 −1 1 1 \uf8f6 \uf8f8 satisfies R2(x) = R(x) and R(x) · R(y) = R(x · y) for all x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Therefore, we can define a Leibniz structures structures on A given by [x, y] = R(x) · y − R(y) · R(x) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Construction of Pre-Lie algebra from Commutative Associative Algebras with a Endomorphism Operator In this section we present in the Propositions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='3 a construction of Pre-Lie algebra structure given by x◦y = R(x)·R(y)−y ·R(x) where R is a endomorphism operator: Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' An algebra A over F with a bilinear product ◦ which satisfies the following identity: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='1) (x ◦ y) ◦ z − x ◦ (y ◦ z) = (y ◦ x) ◦ z − y ◦ (x ◦ z) for all x, y, z ∈ A is called a Left Pre-Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' There is a construction of pre-Lie algebras using a commutative associative algebra (A, ·) with a derivation D on A, the new product a∗b = a·D(b), ∀a, b ∈ A makes (A, ∗) become a Novikov algebra, Novikov algebra is a pre-Lie algebra satisfying an additional identity: (xy)z = (xz)y, ∀x, y, z ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' There are some generalizations of the previous result for a commutative associative algebra (A, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' If D is a derivation on A , then the new product x ∗a y = x · D(y) + a · x · y, ∀x, y ∈ A makes (A, ∗a) become a Novikov algebra for a fixed element a ∈ F or a ∈ A ( [10, 28]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' In the case of an associative algebra (A, ·) with a Rota-Baxter relation of weight 1, it is known that if x ∗ y = R(x) · y − y · R(x) − x · y, ∀x, y ∈ A, then the product ∗ defines a pre-Lie algebra on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' ( [9, 13] ) Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Let A be a commutative associative algebra and let R : A → A a linear map such that R2(x) = R(x) and R(x) · R(y) = R(x · y) for all x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Then we can define a Pre-Lie algebra structures on A given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='2) x ◦ y = R(x) · R(y) − y · R(x) (respectively x ◦ y = R(x) · y − R(y) · R(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' 10 WILSON ARLEY MARTINEZ, SAMIN INGRITH CERON Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Let x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' z ∈ A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' then we have (x ◦ y) ◦ z = (R(x) · R(y) − y · R(x)) ◦ z = R(R(x) · R(y) − y · R(x)) · R(z) − z · R(R(x) · R(y) − y · R(x)) = (R(x) · R(y) − R(y) · R(x)) · R(z) − z · (R(x) · R(y) − R(y) · R(x)) = 0 x ◦ (y ◦ z) = x ◦ (R(y) · R(z) − z · R(y)) = R(x) · R(R(y) · R(z) − z · R(y)) − (R(y) · R(z) − z · R(y)) · R(x) = R(x) · (R(y) · R(z) − R(z) · R(y)) − (R(y) · R(z) − z · R(y)) · R(x) = −(R(y) · R(z) − z · R(y)) · R(x) On the other hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' we have (y ◦ x) ◦ z = (R(y) · R(x) − x · R(y)) ◦ z = R(R(y) · R(x) − x · R(y)) · R(z) − z · R(R(y) · R(x) − x · R(y)) = (R(y) · R(x) − R(x) · R(y)) · R(z) − z · (R(y) · R(x) − R(x) · R(y)) = 0 y ◦ (x ◦ z) = y ◦ (R(x) · R(z) − z · R(x)) = R(y) · R(R(x) · R(z) − z · R(x)) − (R(x) · R(z) − z · R(x)) · R(y) = R(y) · (R(x) · R(z) − R(z) · R(x)) − (R(x) · R(z) − z · R(x)) · R(y) = −(R(x) · R(z) − z · R(x)) · R(y) Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' (x ◦ y) ◦ z − x ◦ (y ◦ z) = (y ◦ x) ◦ z − y ◦ (x ◦ z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Examples of Pre-Lie algebras from commutative associative algebras with the endomorphism operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' In this subsection we present in the example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='1 a commutative associative algebra A with an unusual matrix multiplication, that allow us to give an example of Pre-lie algebras from a endomorphism operator with certain properties on the algebra A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Let A be the vector space of all 2 × 2 matrices over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' A = �� m x n y � : m, n, x, y ∈ R � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' A is a commutative associative algebra with the unusual matrix multiplication defined by: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='1) � m x n y � ∗ � p z r w � = � mp xz nr yw � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' The element I = � 1 1 1 1 � is the multiplicative identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' The element u = � 1 0 0 0 � with the usual matrix multiplication satisfy: u2 = u, u · x ∈ A and x · u = x for all x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' The algebra of matrices A = �� m 0 n 0 � : m, n ∈ R � , CONSTRUCTION OF SOME NON-ASSOCIATIVE ALGEBRAS 11 is a commutative associative under the unusual matrix multiplication defined above in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Then the linear map R : A −→ A defined by R �� m 0 n 0 �� = � 1 0 0 0 � � m 0 n 0 � = � m 0 0 0 � satisfies R2(x) = R(x) and R(x) ∗ R(y) = R(x ∗ y) for all x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Therefore, we can define a Pre-Lie algebra structures on A given by x ◦ y = R(x) ∗ R(y) − y ∗ R(x) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Construction of Pre-Lie algebra from Commutative Associative Algebras with a Differential Operator We now present in the Propositions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='1 a construction of Pre-Lie algebra struc- ture given by x◦y = R(x)·y where R is a differential operator, and we also introduce the notion of left zero divisor on A: Let ( H, · ) be a set H with a binary operation on it, then an element u of H is called a left zero divisor on A ⊆ H if x · u = 0 for all x in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Let A be a commutative associative algebra and let R : A → A a linear map such that R2(x) = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' x and R(x)·y+x·R(y) = R(x·y) for all x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Then we can define a Pre-Lie algebra structures on A given by x ◦ y = R(x) · y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Let x, y, z ∈ A, then (x ◦ y) ◦ z − x ◦ (y ◦ z) = (R(x) · y) ◦ z − x ◦ (R(y) · z) = R(R(x) · y) · z − R(x) · (R(y) · z) = (R2(x) · y + R(x) · R(y)) · z − R(x) · (R(y) · z) = (R2(x) · y) · z = ((α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' x) · y) · z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' On the other hand, we have (y ◦ x) ◦ z − y ◦ (x ◦ z) = (R(y) · x) ◦ z − y ◦ (R(x) · z) = R(R(y) · x) · z − R(y) · (R(x) · z) = (R2(y) · x + R(y) · R(x)) · z − R(y) · (R(x) · z) = (R2(y) · x) · z = ((α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' y) · x) · z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Therefore, (x ◦ y) ◦ z − x ◦ (y ◦ z) = (y ◦ x) ◦ z − y ◦ (x ◦ z) □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Examples of Pre-Lie algebras from commutative associative algebras with a Differential Operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Let ( A, · ) be an associative subalgebra of an algebra H, and suppose that there exists u ∈ H such that u · x ∈ A and x · u = 0 for all x ∈ A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Then the linear map R : A −→ A defined by R(x) = u · x satisfies (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='1) R(x) · y + x · R(y) = R(x · y) for all x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' 12 WILSON ARLEY MARTINEZ, SAMIN INGRITH CERON Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Let x, y ∈ A, then R(x) · y + x · R(y) = (u · x) · y + x · (u · y) = u · (x · y) + (x · u) · y = u · (x · y) + (0 · y) = u · (x · y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' On the other hand, R(x · y) = u · (x · y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Therefore R(x) · y + x · R(y) = R(x · y) for all x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' □ Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' We consider the subalgebra A = \uf8f1 \uf8f2 \uf8f3 \uf8eb \uf8ed y y 0 n n 0 r r 0 \uf8f6 \uf8f8 : r, n, y ∈ R \uf8fc \uf8fd \uf8fe under the usual matrix multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' The element u = \uf8eb \uf8ed a b c −a −b −c e f g \uf8f6 \uf8f8 satis- fies x · u = 0 and u · x ∈ A for all x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Then the linear map R : A −→ A defined by R \uf8eb \uf8ed \uf8eb \uf8ed y y 0 n n 0 r r 0 \uf8f6 \uf8f8 \uf8f6 \uf8f8 = \uf8eb \uf8ed a b c −a −b −c e f g \uf8f6 \uf8f8 · \uf8eb \uf8ed y y 0 n n 0 r r 0 \uf8f6 \uf8f8 = \uf8eb \uf8ed ay + bn + cr ay + bn + cr 0 −ay − bn − cr −ay − bn − cr 0 ey + fn + gr ey + fn + gr 0 \uf8f6 \uf8f8 satisfies R(x) · y + x · R(y) = R(x · y) for all x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' The algebra of matrices A = \uf8f1 \uf8f2 \uf8f3α \uf8eb \uf8ed an ap aq bn bp bq n p q \uf8f6 \uf8f8 : α ∈ R \uf8fc \uf8fd \uf8fe where a = −βb and b, n, p, q ∈ R is a commutative associative algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' The element u = \uf8eb \uf8ed a βa λβa b βb λβb 1 β λβ \uf8f6 \uf8f8 where λ, β ∈ R, satisfies u2 = (λβ)u, x · u = 0 (⇔ an + bp + q = 0) and u · x ∈ A for all x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Then the linear map R : A −→ A defined by R \uf8eb \uf8ed \uf8eb \uf8ed an ap aq bn bp bq n p q \uf8f6 \uf8f8 \uf8f6 \uf8f8 = \uf8eb \uf8ed a βa λβa b βb λβb 1 β λβ \uf8f6 \uf8f8 · \uf8eb \uf8ed an ap aq bn bp bq n p q \uf8f6 \uf8f8 = (λβ) \uf8eb \uf8ed an ap aq bn bp bq n p q \uf8f6 \uf8f8 satisfies R2(x) = (λβ)R(x) and R(x) · y + x · R(y) = R(x · y) for all x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Therefore, we can define a Pre-Lie algebra structures on A given by x◦y = R(x)·y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' CONSTRUCTION OF SOME NON-ASSOCIATIVE ALGEBRAS 13 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Construction of flexible algebra from Associative Algebras with a left averaging operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' A flexible algebra is a vector space J over a field F of charac- teristic ̸= 2 with a binary operation ◦ satisfying for x, y ∈ J the following identity: (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='1) (x ◦ y) ◦ x = x ◦ (y ◦ x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' The flexible algebras was initiated by Albert ([1]) and investigated by the authors Myung, Okubo, Laufer, Tomber and Santilli, see for example ([20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Suppose (A, ·) is a flexible algebra, and R : A → A is a linear map, such that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='2) R(x) · R(y) = R(R(x) · y) = R(x · y) for all x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Then we can define a new flexible algebra structures on A given by x ◦ y = R(x) · y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Let A be a flexible algebra and R : A → A is a linear map such that R(x) · R(y) = R(R(x) · y) = R(x · y) for all x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' So (x ◦ y) ◦ x = (R(x) · y) ◦ x = R(R(x) · y) · x = (R(x) · R(y)) · x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Since x ◦ (y ◦ x) = x ◦ (R(y) · x) = R(x) · (R(y) · x) and (A, ·) is a flexible algebra, then we have (x ◦ y) ◦ x = x ◦ (y ◦ x) for all x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Therefore (A, ◦) is a flexible algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Examples of Flexible algebras from associative algebras with a left averaging operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Let ( A, · ) be an associative subalgebra of an algebra H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Sup- pose H has the following property: There exists u ∈ H, such that u · x ∈ A and x · u = u · x for all x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Then the linear map R : A −→ A defined by R(a) = u ·x satisfies the left averaging identity (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='1) R(a) · R(b) = R(R(a) · b) for all a, b ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Furthemore, if u2 = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Then (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='2) R(a) · R(b) = R(R(a) · b) = R(a · b) for all a, b ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Let x, y ∈ A, by the hypothesis there exists u ∈ H such that u · x ∈ A and x · u = u · x for all x ∈ A, then R(x) · R(y) = (u · x) · (u · y) = u · ((x · u) · y) = u · ((u · x) · y) = R(R(x) · y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' If u2 = u, then R(x) · R(y) = u · ((u · x) · y) = u2 · (x · y) = u · (x · y) = R(x · y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Therefore R(x) · R(y) = R(R(x) · y) = R(x · y) for all x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' □ Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' We consider the subalgebra A = \uf8f1 \uf8f2 \uf8f3 \uf8eb \uf8ed y n 0 0 y 0 0 0 r \uf8f6 \uf8f8 : r, n, y ∈ R \uf8fc \uf8fd \uf8fe 14 WILSON ARLEY MARTINEZ, SAMIN INGRITH CERON under the usual matrix multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' The element u = \uf8eb \uf8ed 1 0 0 0 1 0 0 0 0 \uf8f6 \uf8f8 satisfies u2 = u, x · u = u · x and u · x ∈ A for all x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Then the linear map R : A −→ A defined by R \uf8eb \uf8ed \uf8eb \uf8ed y n 0 0 y 0 0 0 r \uf8f6 \uf8f8 \uf8f6 \uf8f8 = \uf8eb \uf8ed 1 0 0 0 1 0 0 0 0 \uf8f6 \uf8f8 · \uf8eb \uf8ed y n 0 0 y 0 0 0 r \uf8f6 \uf8f8 = \uf8eb \uf8ed y n 0 0 y 0 0 0 0 \uf8f6 \uf8f8 satisfies R(x) · R(y) = R(R(x) · y) = R(x · y) for all x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' We have A is a Lie algebra with the product [x, y] = x · R(y) − y · R(x) (see Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='3), Therefore (A, [, ]) is a flexive algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Construction of Rota-Baxter Operator In this section we present in the Propositions 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='5 constructions of Rota-Baxter Operators of weight λ = 1 and λ = 0 from associative algebra with an element u skew-idempotent or nilpotent of index 2 respectively, and we also introduce the notion of Rota-Baxter Operator of weight (λ, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' We recall from the Introduction: Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Let (A, ·) be an associative algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' A linear map R : A → A is called a Rota-Baxter operator of weight λ on A if R satisfies (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='1) R(x) · R(y) = R (R(x) · y + x · R(y) + λ x · y) , for all x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' A Rota-Baxter algebra (also known as a Baxter algebra) is an associative algebra A with a Rota-Baxter operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' One importance of the Rota-Baxter Algebra is its close relationship with other algebraic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' For example pre-Lie algebras come naturally from a Rota Baxter-Operator on an Lie Algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' [12], [19] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' An elemet u ̸= 0 of an algebra A is called nilpotent if un = 0 for some integer n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' The least such integer is called the index of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' An elemet u ̸= 0 of an algebra A is said to be skew-idempotent with respect to a product · in the algebra A if: u · u = −u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Let ( A, · ) be an associative algebra and suppose that there exists u ∈ A such that u2 = −u and u · x ∈ A for all x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Then the linear map R : A −→ A defined by R(x) = u · x satisfies (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='2) R ( R(x) · y + x · R(y) + x · y ) = R(x) · R(y) for all x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Furthemore, if u2 = 0, then R is a Rota-Baxter operator of weight zero on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Let x, y ∈ A, then we have R(x) · R(y) = (u · x) · (u · y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' On the other hand, R(R(x) · y + x · R(y) + x · y) = R((u · x) · y + x · (u · y) + x · y) = u · ((u · x) · y + x · (u · y) + x · y) = u2 · (x · y) + (u · x) · (u · y) + u · (x · y) = (u2 + u) · (x · y) + (u · x) · (u · y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' CONSTRUCTION OF SOME NON-ASSOCIATIVE ALGEBRAS 15 Therefore R(R(x)·y +x·R(y)+x·y) = R(x)·R(y) for all x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Now, if u2 = 0, then R(R(x) · y + x · R(y)) = (u · x) · (u · y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Therefore R(R(x) · y + x · R(y)) = R(x) · R(y) for all x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' □ Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' The element u = � xy −x2 y2 −xy � satisfies u2 = 0 and u · x ∈ A for all x in the algebra of matrices A = �� 0 a 0 b � : a, b ∈ R � considered as a subalgebra of B = M2×2 under the usual matrix multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' If we define the map R : A −→ A by R( � 0 a 0 b � ) = � xy −x2 y2 −xy � � 0 a 0 b � = � 0 xya − x2b 0 y2a − xyb � , then R satisfies R(R(x) · y + y · R(x)) = R(x) · R(y);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' for all x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' The element u = � x y −x2−x y −x − 1 � , y ̸= 0 is an skew-idempotent in the algebra of matrices under the usual matrix multiplication, that is u2 = −u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' We observe that u · x ∈ A for all x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' If we define the map R : A −→ A by R( � 0 a 0 b � ) = � x y −x2−x y −x − 1 � � 0 a 0 b � = � 0 xa + yb 0 ( −x2−x y )a − (x + 1)b � then R satisfies R(x) · R(y) = R(R(x) · y + x · R(y) + x · y);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' for all x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Let (A, ·) be an associative algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' A linear map R : A → A is called a Rota-Baxter operador of weight (λ, β) on A if R satisfies (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='3) R(x) · R(y) = R � R(x) · y + x · R(y) + λ x · y � + β x · y, for all x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' A Rota-Baxter operador of weight (λ, β) for associative algebras allows to build examples of Dyckm-algebras [26], the main result of this section is the construction of Rota-Baxter operador of weight (λ, β) on associative algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Let ( A, · ) be an associative algebra and suppose that there exists u ∈ A such that u2 = −λu − β1A and u · x ∈ A for all x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Then the linear map R : A −→ A defined by R(x) = u · x satisfies (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='4) R ( R(x) · y + x · R(y) + λx · y ) + βx · y = R(x) · R(y) for all x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Let x, y ∈ A, then we have R(R(x) · y + x · R(y) + λx · y) + βx · y = R((u · x) · y + x · (u · y) + λx · y) + βx · y = u · ((u · x) · y + x · (u · y) + λx · y) + βx · y = u2 · (x · y) + (u · x) · (u · y) + λu · (x · y) + βx · y = (u2 + λu + β1A) · (x · y) + (u · x) · (u · y) = (u · x) · (u · y) On the other hand, R(x) · R(y) = (u · x) · (u · y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Therefore R(R(x) · y + x · R(y) + λx · y) + βx · y = R(x) · R(y) for all x, y ∈ A 16 WILSON ARLEY MARTINEZ, SAMIN INGRITH CERON □ Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' The element u = � x y −x2−λx−β y −x − λ � , where y ̸= 0 satisfy: u2 = −λu − β1A and u · x ∈ A for all x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' If we define the map R : A −→ A by R( � 0 a 0 b � ) = � x y −x2−λx−β y −x − λ � � 0 a 0 b � = � 0 xa + yb 0 ( −x2−λx−β y )a − (x + λ)b � then R satisfies R(x)·R(y) = R(R(x)·y + x·R(y)+ λx·y)+ βx·y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' for all x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Acknowledgment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' We thank to Universidad del Cauca for the support to our research group “Estructuras Algebraicas, Divulgaci´on Matem´atica y Teor´ıas Aso- ciadas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' @DiTa” under the research project with ID 5773, entitled ”Aplicaciones de Estructuras Algebraicas”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' We also thank to the anonymous referee for their help- ful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' This work is dedicated to my daughters especially to Clara Isabel Martinez Ceron (January 17, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' ALBERT, Power associative rings, Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} +page_content=' math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf'} 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b/39FRT4oBgHgl3EQfozcR/content/tmp_files/2301.13610v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..442fc48c0b0ad0172dbbeee1912bca458c01a108 --- /dev/null +++ b/39FRT4oBgHgl3EQfozcR/content/tmp_files/2301.13610v1.pdf.txt @@ -0,0 +1,1019 @@ +arXiv:2301.13610v1 [gr-qc] 30 Jan 2023 +Non-Singular Bouncing Model in Energy +Momentum Squared Gravity +Z. Yousaf1 ∗, M. Z. Bhatti1 †, H. Aman1 ‡, P.K. Sahoo2 § +1Department of Mathematics, University of the Punjab, +Quaid-i-Azam Campus, Lahore-54590, Pakistan +2Department of Mathematics, +Birla Institute of Technology and Science-Pilani, +Hyderabad Campus, Hyderabad-500078, India. +Abstract +This work is concerned to study the bouncing nature of the universe for an isotropic +configuration of fluid Tαβ and Friedmann-Lemaˆıtre-Robertson-Walker metric scheme. +This work is carried out under the novel f(G, TαβT αβ) gravitation by assuming a spe- +cific model i.e, f(G, T 2) = G + αG2 + 2λT 2 with α and λ are constants, serving as free +parameters. The terms G and T 2 served as an Gauss-Bonnet invariant and square of +the energy-momentum trace term as an inclusion in the gravitational action respec- +tively, and is proportional to T 2 = TαβT αβ. A specific functional form of the Hubble +parameter is taken to provide the evolution of cosmographic parameters. A well known +equation of state parameter, ω(t) = − k log(t+ǫ) +t +− 1 is used to represent the dynamical +behavior of energy density, matter pressure and energy conditions. A detailed graph- +ical analysis is also provided to review the bounce. Furthermore, all free parameters +are set in a way, to make the supposed Hubble parameter act as the bouncing solution +and ensure the viability of energy conditions. Conclusively, all necessary conditions for +a bouncing model are checked. +Keywords: cosmography; Hubble Parameter; f(G, TαβT αβ). +PACS: 98.80.-k; 04.20.Cv; 04.50.Kd. +∗zeeshan.math@pu.edu.pk +†mzaeem.math@pu.edu.pk +‡huzaifaaman971@gmail.com +§pksahoo@hyderabad.bits-pilani.ac.in +1 + +1 +Introduction +According to the big bang hypothesis, the whole universe was created by a single explosion, +with all matter in the cosmos as an infinite speck [1, 2]. +This hypothesis works well in +order to study the beginning, but lack to define different cosmological problems. +These +problems include the horizon problem, the flatness problem, the singularity problem, etc. In +order to resolve these big cosmic challenges, different cosmic theories have been developed +in literature [3–5]. The bouncing hypothesis is one of the major independent theories that +came up with the answers related to the starting of the universe and should be enough to +resolve the major cosmic problem of singularity. +The bouncing cosmology works on the +scheme of an oscillatory universe, i.e, a universe that came into being from the pre-existing +universe without undergoing the singularity [6–8]. This whole transition of the universe not +only explains the big-bang cosmology but also reduces one of the major issues. +For the +bouncing, the universe moves into the contraction phase as a matter-dominated the era of +the universe. After the contraction, the universe starts to expand in a nonsingular manner +for which gravity dominates the matter [9,10]. Also, density perturbations can be produced +during the bounce era. This idea of the origination of the universe is highly accepted and +appreciated in literature. +General relativity (GR) was presented by Einstein and it was thought to be one of the +best theories to explain different cosmological issues. It explains the gravity under the fabric +of space-time. However, to understand gravity much more effectively and to provide the +answers to the effect of gravity, dark energy, and accelerated expansion of the universe under +the addition of different scalar fields, different attempts have been made in past to modify +GR. +These modifications change the geometric or matter or both parts of the Einstein +field equations accordingly. These could help to discuss the effects of couplings of matter +and curvature terms on the above-described items. Roshan and Shojai [11] presented the +nonlinear form of matter term i.e, T 2 = TαβT αβ, naming it f(R, T ∈). They further indicated +that the use of nonlinear terms may provide the prevention of early time singularities. Since +the functional form of curvature terms has helped to introduce new gravitational theories, so +it was considered to be effective to modify the generic action integral of GR as corrections. +These modifications give light to the f(G) theory, for which the term G is defined as G = +RξζαβRξζαβ − 4RξζRξζ + R2. Nojiri and Odintsov [12] introduced this f(G) theory for the +first time in their work. They tested solar systems for this formalism and reported the phase +change of acceleration to deceleration for the achievement of phantom line, which cooperated +to study dark energy. +Odintsov and Oikonomou [13] considered R + f(G) form of the +gravitational theory to provide their contribution to the study of gravitational baryogenesis. +Their work included the higher-order derivatives of Gauss-Bonnet terms that work in order to +produce the baryon asymmetry. Sharif and Ikram [14] gives rise to a new theory by following +2 + +the footsteps of Harko. They coupled the matter part T with the geometric part of the f(G) +theory, making it f(G, T ) cosmology. They investigated the validity of their theory with the +help of energy conditions. Later on, Bhatti et al. [15] worked on the f(G, T ) theory to carry +out the investigation of some physically feasible features of compact star formation. They +inferred that the compactness of a star model grows at the core whereas the energy conditions +remain constant. Yousaf and his mates [16] inspired by [17], have recently developed a novel +f(G, T 2) to present the complexity of structural scalars from the use of Herrera’s method +of splitting scalars. They considered the exponential coupling of Gauss-Bonnet terms as a +functional form as f(G, T 2) = αGn(βGm + 1) + ηT 2, to explore the validity of their solutions +for the Darmois and Israel conditions. They also worked on the non-static complex structures +under the same theory to describe the effects of an electromagnetic field. They used specific +model configuration i.e, f(G, T 2) = k1Gm(k2Gn + 1) + λT 2, in their work. +Bouncing cosmology has gained much reputation over the past few years, because of +its independent hypothetical nature from different standard comic problems. +Guth [18] +during 1980′s, had put forward his inflationary theory to tackle early and late time cosmic +evolutionary problems. He remained successful in solving some related problems, but the +answer to the initial singularity is still under concern. One of the best hypotheses to answer +the singularity problem is the bouncing nature of the universe. The nature of the bouncing +universe allows a certain universe model to transit from a pre-big crunch (contracted) phase +into a new big bang (expanded) phase with the exclusion of singularity during the whole event +[19]. Steinhardt and Ijjas [20] are considered to be the pioneers of the bouncing hypothesis. +They devised a wedge diagram for a smooth bouncing method to explore the consequences +of some cosmological problems. Sahoo et al. [21] worked on the non-singular bouncing by +assuming the specific coupling of R and T as f(R, T ) = R + χRT , for 0 < χ < π +4. They +allowed such a parametric approach for the Hubble parameter to provide no singularity +during the bounce. +They used quintom and phantom scalar field configurations for the +bouncing paradigm. Bamba and his collaborators [22] inspected the singularity-free concept +of bounce by considering an exponential form of scale factor a(t) = σ exp(λt) + τ exp(λt) +under the effect of f(G) gravity. They checked the stability of their assumed solution under +the restricted parametric scheme. +Yousaf et al. [23, 24] explored the bouncing universe with a specific functional form of +Hubble parameter by taking exponential f(G, T ) form. Different cosmic models are under +consideration for the scale factor in order to determine the value of expansion and contraction +at the current cosmic phase and also to predict the current phase equation of state. These +models predicted different results in the literature. However, cosmography provided us a +benefit in processing cosmological data for explaining the universal kinematics without the +involvement of the gravity model and hence provided that the cosmography can be employed +with the Taylor expansions as an alternative scheme. Also, the cosmographic analysis for +3 + +the FLRW universe, is helpful in such a way that it can put aside the effect of the dy- +namical field equations [25]. Gruber et al. [26] studied an alternative approach to describe +cosmography by extending the conventional methodology. They resulted from numerical +values of the cosmographic parameters by applying the Pad´e approximations. The testing +of the ΛCDM model had been conveyed by Busti et al. [27] with the use of cosmographical +analysis. Capozziello et al. provided cosmography as a non-predictive phenomenon when +the redshift parameter becomes z ≈ 1. They used the pad´e approximations for the fifth +order and resulted the divergence of data at the higher levels of the approximations. Lobo +et al. [28] evaluated the dynamics of the redshift drift. They used the expanding FLRW +universe to produce a general matter and low redshift model with the use of different vari- +ables. However, the cross-correlation of large-scale quasars can be used and translated with +the CMB and BAO scale data to produce the best for Hubble parameter H(z) and angular +diametric distance SA. Also, the cosmic chronometers approach can be done to predict the +model independent H(z) measurements which have been extensively used for cosmological +applications [29–31]. The low redshift data set with the inclusion of the megamasers and +chronometers had been presented by Krishnan and others [32]. They result that the Hubble +constant H0, showed descending behavior with the redshift and having non-zero slop when +fitted on the line by statistical means. +Font et al. [33] studied correlation technique for +quasars by using Lya absorption and produced the best line of fit for Planck’s data. They +generated different results on the measurements of the Hubble parameter and the angular +distance. One important thing is to develop such a cosmic Hubble parameter that comes +from early to late span in such a way that it changes from a low to a high value. The Gaus- +sian method helped to predict but provided a non-transitional behavior for both Λ and ω +epochs. The null energy condition also proved to be an important restriction for the cut-off +model, when compared with Hubble parameter data [34]. King et al. [35] studied the future +approximations of the redshift by the inclusion of dark energy. They tested the equation of +state by the linear parametrization technique. Hu et al. [34] reported different values of the +Hubble constant by the Gaussian method. Their research produced an effective reduction +in the Hubble crisis and proposed the non-transitional behavior of the Hubble constant. +Different dark energy models respective to holography and agegraphy had been conducted +by Zhang et al. [36]. They produced different energy conditions for different red shift values +and resulted in an effective role of energy conditions for different cosmic ages. +In this article, we implemented a functional form of the Hubble parameter that evolves +periodically with cosmic time t and investigate the bouncing nature of the universe in +f(G, TαβT αβ) gravity using a flat FLRW peacetime. This analysis of the bouncing uni- +verse involves one of the most important forms of EoS parameter proposed in the litera- +ture [37–39]. The outline is given as: Sect.2 provides a brief introduction to f(G, TαβT αβ) +gravity with the necessary formalism of FLRW metric and modified field equations. Sect.3 +4 + +builds the Hubble parameter as a bouncing solution for the produced field equations. The +cosmographic parameters are also evaluated in this section. We provide the mathematical +expressions of energy density and matter pressure for the assumed EoS parameter form in +Sect.4. The energy conditions are also formulated in the same fashion. Detailed graphical +profiles of energy conditions are represented in the same section to discuss the evolution +of the universe under the influence of restricted free parameters. Finally, the concluding +remarks are made in Sect.5. +2 +f(G, TαβT αβ) Formalism +The modified action for the f(G, TαβT αβ) gravity theory is defined as [16] +Af(G,TαβT αβ) = +√−g +2κ2 +�� +d4x[f(G, TαβT αβ) + R] + +� +d4xLm +� +, +(1) +where R and G symbolize the Ricci scalar and the Gauss-Bonnet scalar terms, respectively +and are provided as +R ≡ gαβRαβ, +G ≡ RξζαβRξζαβ − 4RξζRξζ + R2, +(2) +and κ2 = 8πG (G be the gravitational constant) and Lm = −p. Also, the term g implies the +trace of the metric tensor gαβ with Tαβ, Rξζαβ and Rαβ indicate the stress energy-momentum +tensor, the Riemannian tensor, and the Ricci tensor respectively. The expression for Tαβ is +given as +Tαβ = +−2 +√−g +δ(√−gLm) +δgαβ +. +(3) +Equation (3) yields the following expression, due the dependency of the matter Lagrangian +Lm on gαβ components +Tαβ = gαβLm − 2∂Lm +∂gαβ . +(4) +Now, by taking the variation of Eq.(1) with respect to the term gαβ, we get the following +field equations for the f(G, TαβT αβ) theory as +Rαβ − 1 +2Rgαβ = T eff +αβ , +(5) +where the term T eff +αβ takes the following form +T eff +αβ += +κ2Tαβ − ΘαβfT 2(G, T 2) + 1 +2gαβf(G, T 2) − (2RRαβ − 4Rε +αRεβ − 4RαεβηRεη +5 + ++2Rεηδ +α Rβεηδ)fG(G, T 2) − (2R∇2gαβ − 2R∇α∇β − 4Rαβ∇2 − 4gαβRεη∇ε∇η ++4Rε +α∇β∇ε + 4∇ε∇αRε +β + 4Rαεβη∇ε∇η)fG(G, T 2), +(6) +where, +Θαβ ≡ δ(TµνT µν) +δgαβ += 2T ξ +α Tβξ − T Tαβ − 4T µν ∂2Lm +∂gαβgµν − 2Lm(Tαβ − 1 +2T gαβ) +(7) +T 2 = TαβT αβ, +∇2 = ∇α∇α +(8) +The terms fG(G, T 2) and fT 2(G, T 2) used above are defined as +fG(G, T 2) ≡ df(G, T 2) +dG +, +and fT 2(G, T 2) ≡ df(G, T 2) +dT 2 +. +(9) +The trace of the above-defined field equations is produced as +T − ΘfT 2(G, T 2) + 2GfG(G, T 2) − 2R∇2fG(G, T 2) + 4Rαβ∇α∇βfG(G, T 2) = 0. +(10) +Equation (10) shows the non-conversed situation of the stress energy-momentum tensor. +Also, the properties of GR can be recovered for f(G, T 2) = 0. Similarly if we put f(G, T 2) = +f(G), we get the properties of f(G) gravity. +Now, as we are concerned to study the bouncing nature of the universe, so we consider +the fluid distribution to be perfect throughout the cosmic evolution. For this, we take +Tαβ = (ρ + p)VαVβ − pgαβ, +(11) +here, the four-vector velocity is defined by V β with +V β = (1, 0, 0, 0), +V βVβ = 1 , V β∇ζVζ = 0. +(12) +In addition, ρ defines the energy density part and p defines the pressure part of the stress +energy-momentum tensor. Also the geometric background considered to be in a FLRW +space time [40], so it implies +ds2 = dt2 − a2(t)Σidx2 +i , +i = 1, 2, 3. +(13) +The metric component a(t) symbolizes the scale factor, that contributes to the Hubble +parameter as H = +˙a(t) +a(t). +Using Eq.(13) and Eq.(7) in Eq.(5), we get the following field +equations +6 +� ˙a +a +�2 +− 24 +� ˙a +a +�3 +˙fG + 24 +�¨a +a +� � ˙a +a +�2 +fG − f − 2(ρ2 + 3p2 + 4ρp)fT 2 = 2ρκ2, +(14) +6 + +− 2 +� +2¨a +a + +� ˙a +a +�2� ++ 16 +�¨a˙a +a2 +� +˙fG + 8 +� ˙a +a +�2 +¨fG − 24 +�¨a +a +� � ˙a +a +�2 +fG + f = 2pκ2. +(15) +To draw the conclusions on the field equations, we just need some functional form of f(G, T 2). +As, there are many functional forms regarding the interaction of matter with the curvature +terms, in order to deal with the issues of cosmic evolution. Various coupling models can be +used to evaluate the formations of both energy density and matter pressure, like one can +take f(G, T 2) = G + 2f(T 2) that may help to provide an analysis about ΛCDM epoch. +However, the other choice is f(G, T 2) = f1(G) + f2(T 2) that may be worked as a correction +to f(G) gravity theory because of f2(T 2). +Similar forms have been explored in [41, 42] +and provided some distinct results due to the direct minimal curvature matter coupling. +Also, f(G, T 2) = f1(G) + f2(G)f3(T 2) can be taken because of an explicit non-minimally +coupling nature between geometric parameters and matter variables [43]. So, we considered +the following form to produce the validating results. +f(G, T 2) = f1(G) + f2(T 2). +(16) +To produce a bouncing universe, we need some functional forms of f1 and f2 that not only +describe the accelerating expansion of the universe but also explain inflation to a great +extent. For this, the higher power curvature terms perform well to eliminate such issues. +Elizalde [44] introduced the power forms of the curvature scalar as ηRn (n ≥ 1) and produced +the cosmological dynamics, so we consider the specific form of the f1 as the quadratic power +model, so +f1(G) = G + αG2. +(17) +Also, we take χ2 as +f2(T 2) = 2λT 2. +(18) +So, by using Eq.s (17) and (18) in the field equations, we get +6H2 − 48αH3G ˙G + αG2 = 2κ2ρ + 6λρ2 + 18λp2 + 16λρp, +(19) +and +−2(2 ˙H + 3H2) + 32( ˙H + H2)αG ˙G + 16αH2( ˙G2 + G ¨G) − αG2 = 2κ2p − 2λρ2 − 6λp2. +(20) +In order to reduce the complexity of the Eq.(19) and Eq.(20), we utilize p = ωρ, as the EoS +used in [37–39]. So we get the relations, +(3λ + 9λω2 + 8λω)ρ2 + κ2ρ − (3H2 − 24αH3G ˙G + α +2 G2) = 0 +(21) +7 + +and +(− λ +ω2 − 3λ)p2 + κ2p + ((2 ˙H + 3H2) − 16( ˙H + H2)αG ˙G − 8αH2( ˙G2 + G ¨G) + α +2 G2) = 0. (22) +where, G = 24H2( ˙H + H2). Yousaf and his collaborators checked the stability of cosmic +models in various modified gravity theories [45–48]. +3 +Hubble Parameter and Cosmography +This section mainly focuses on describing the evolutionary behavior of these above-described +dynamical terms. Hence, we consider a trigonometric form of the H(t) which feasibly provides +a bounce solution [44,49], as follows +H(t) = ζ sin(φt)h(t). +(23) +This parameterized form of H(t) includes ζ and φ, which are considered to be constants +here. The choice of h(t) depends on the periodic values of the function sin(φt), so the form +of h(t) can be chosen as periodic, that cooperates with the non-vanishing values of the above +trigonometric function. This artificial approach of choosing such an ansatz can be considered +as a numerical analysis of making the bouncing solution. One interesting feature is possessed +by the term ζ, which can work well as a phase changer for the value of H(t). We consider +h(t) as +h(t) = exp(ϕt), +(24) +where ϕ acts as a constant. Finally, we have +H(t) = ζ sin(φt) exp(ϕt). +(25) +This functional form of the Hubble parameter is helpful to study cosmic evolutionary expan- +sion and contraction. This form of the Hubble parameter gives us the bounce at t = 313, +depending upon the values of ϕ = 0.001 and φ = 0.01 provided in Fig.1. +We have re- +stricted the values of H(t) in the positive era of time. The basic scale factor form for this +parameterized Hubble parameter becomes +a(t) = exp +�ζ exp(ϕt)(ϕ sin(φt) − φ sin(φt)) +ϕ2 + φ2 +� +. +(26) +Similarly, the set of dynamical parameters that are derived from the Taylor series expansion +of the scale factor is termed as cosmographic factors. These factors helped to obtain the +8 + +cosmological concordance with the assumptions of the universal homogeneity and isotropy +on large cosmic scales [27,50]. These include deceleration, jerk and snap parameters. These +factors allow us to check the compatibility of the scale factor and the Hubble parameter. +The negative value of the deceleration parameter q describes the accelerated expansion of +the universe. Similarly, jerk j and snap s determine the expansion rate of the toy universe +model. The mathematical interpretation for these cosmography elements are defined as +q = − 1 +H2 +1 +a +d2a +dt2 = −1 − 1 +ζ (e−ϕt csc(φt)(φ cot(φt) + ϕ)), +(27) +j = 1 +H3 +1 +a +d3a +dt3 += +1 + 1 +ζ2(e−2ϕt csc(φt)(3ζeϕt(φ cot(φt) + ϕ) ++ csc(φt)(2ϕφ cot(φt) + ϕ2 − φ2))), +(28) +and +s = 1 +H4 +1 +a +d4a +dt4 = − +1 +3ζ(3ζeϕt + 2 csc(φt)(φ cot(φt) + ϕ))(2e−ϕt csc(φt)(csc(φt) +(3ζϕeϕt sin(φt) + ϕ2 − φ2) + φ cot(φt)(3ζeϕt + 2ϕ csc(φt)))). +(29) +Fig.1 shows the progression of the Hubble (left panel) and scale parameters (right panel) +along the positive time axis. Similarly, the development of jerk (left panel) and snap factors +(right panel) are provided in the fig.2. The evolution of the deceleration parameter towards +the negative value i.e, q → −1, before the bouncing point, provided in fig.6, shows the +accelerating universe. +4 +Energy Conditions under the EoS Parameter +For a specific cosmology model, energy conditions play an important role to make its val- +idation for the restricted free parameters. These energy conditions help to maintain the +specifications of the certain cosmic model [51–55]. Similarly, these energy conditions also +work for the bouncing cosmology and provide a reasonable approach to validate the proce- +dure for our toy bouncing model. These conditions are described as +• Dominant energy condition (DEC)⇔ ρ ≥ 0 , ρ ± p ≥ 0. +• Strong energy condition (SEC)⇔ ρ + 3p ≥ 0 , ρ + p ≥ 0. +• Weak energy condition (WEC)⇔ ρ ≥ 0 , ρ + p ≥ 0. +9 + +H � 0 +Bouncing +H � +H � 0 +300 +305 +310 +315 +320 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +t +a +a�t� � 0 +0 +50 +100 +150 +200 +250 +300 +350 +�0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +t +a +Figure 1: The illustrations of Hubble parameter and scale factor with fixed values of ϕ = +0.001 and φ = 0.01. +280 +290 +300 +310 +320 +330 +340 +�4 +�2 +0 +2 +4 +t +j +300 +310 +320 +330 +340 +�1.0 +�0.5 +0.0 +0.5 +1.0 +t +s +0 +50 +100 +150 +200 +250 +300 +350 +�1.0 +�0.5 +0.0 +0.5 +1.0 +t +s +Figure 2: The illustration of jerk and snap factors with fixed values of ϕ = 0.001 and +φ = 0.01. +10 + +6F +-=1 +2=0.8 +■=0.6 +M5=0.4 +■2=0.2 +0 +100 +200 +300 +400 +500H +量5=0.8 +拉2=0.6 +2=0.4 +=0.2 +0 +100 +200 +300 +400 +500 +600 +t• Null energy condition (N EC)⇔ ρ + p ≥ 0. +• Trace energy condition (T EC)⇔ ρ − 3p ≥ 0. +The positivity of DEC, SEC and WEC passes on the validity and necessity of the bouncing +concept. However, the violation of N EC has a major role. This violation is different in the +GR context. Universal bouncing scenario is one of those ideas that provides a chance to +discuss the singularity-free universal beginning. Many proposals in the literature suggested +avoiding this singularity through quantum aspects, but these don’t have such reliability to fit +in the gravitational theory. So, at this point gravitational theories allow a specific mechanism +to check the validity of the bounce model and as well its own. Null energy condition is one +such tool to help achieve the task. Also, it has been proved that in the context of GR, the +violation of N EC is extremely difficult to be achieved for local-field models. So, effective +field theories provide a chance to recognize the violation of the N EC and to allow a non- +singular bounce [56–59]. +One such effective field is f(G, TαβT αβ) theory that provides a +chance to study the quadratic nature of the energy terms i.e, energy density and matter +pressure [16, 60]. +However, it also allows getting a non-singular bounce for the assumed +gravity model form. For an excellent bouncing model, the value of H(t) turns out to be +˙H = −4Gρπ(1 + ω) > 0 for the formulation of GR. However, if the N EC gets violated, +we have the surety to get a bouncing scenario. To provide the mathematical formulation +of the energy conditions, we consider Eqs. +(21) and (22). +Also, the EoS parameter in +the negative regime provides the present cosmic evolution [61–63] and becomes favorable in +the bouncing context with ω(t) ≈ −1. However, bouncing cosmology provides the possible +geodesic evolution of the universe by avoiding the singularity along with the resolution of the +horizon problem, flatness problem, entropy problem and many more [5]. For the modified +gravity, EoS parameter enables us to study the universal dynamics. In this study, we used +EoS parameter [44] to obtain the possible chance of obtaining a bounce solution in f(G, T 2) +as +ω(t) = −k log(t + ǫ) +t +− 1, +(30) +here k is assumed to be a constant. This particular form of the EoS parameters allows us to +study the contracting and expanding behavior without involving the Hubble parameter as +well as the scale factor. Elizalde et al. [44] produces cosmological dynamics by considering +R2 gravity and logarithmic trace terms. They checked the effects of the λ parameter in the +gravity model f(R, T ) = R+λR2+2β ln(T ) along with the bouncing solution depending on +the two EOS parameters. Our work first described the choice of Hubble parameter and its +effects on the dynamical field equations and then involves the EOS parameter. We only took +one of the ω(t) value, because this state factor after the bouncing point remains negative and +11 + +becomes ω(t) ≈ −1. Also, the current cosmic expansion and Λ − CDM can be verified by +this state factor. However, the dynamic properties are greatly affected under the influence +of this EoS parameter form. Hence, the general forms of the Eqs.(21) and (22), under the +influence of Eq.30, are presented as +ρ += +− +1 +2λ(9ω2 + 8ω + 3)(κ2 + (κ4 − 12ζ2λ(9ω2 + 8ω + 3)e2ϕt sin2(φt)(2304αζ7e7ϕt sin4(φt) +(sin(φt)(ζeϕt sin(φt) + ϕ) + φ cos(φt))(4ζϕeϕt sin3(φt) + 2ζφeϕt sin(2φt) sin(φt) +− +(φ2 − 3ϕ2) sin2(φt) + 2φ2 cos2(φt) + 3ϕφ sin(2φt)) − 96αζ4e4ϕt sin2(φt) +(sin(φt)(ζeϕt sin(φt) + ϕ) + φ cos(φt))2 − 1)) +1 +2 +(31) +p += +1 +2(3λω2 + λ)(κ2ω2 + (κ4ω4 + 4ζω2(3λω2 + λ)eϕt(18432αζ9e9tϕ(2ϕ(2ϕ + 1) − φ2) +sin10(tφt) + 4608αζ8e8tϕ(ϕ2(25ϕ + 22) − (13ϕ + 2)φ2) sin9(φt) + 9216αζ7φ3e7ϕt +sin5(φt) cos3(φt)(7ζeϕt sin(φt)10ϕ + 8) + 288αζ7e7ϕt(144ϕ4 + 320ϕ3 − 16(9ϕ + 4) +ϕφ2 − 1) sin8(φt) + 576αζ6φ4e6ϕt sin4(2φt)(ζeϕt + 2 csc(φt)) + 576αζ6ϕe6ϕt +(48ϕ3 − 16ϕφ2 − 1) sin7(φt) − 96αζ5ϕ(3ϕ − 8)e5ϕt sin6(φt) + 192αζ4e4ϕt(3ϕ2 − φ2) +sin5(φt) + 2 sin(φt)(5760αζ6ϕφ3e6ϕt sin3(2φt) + 48αζ4φ2e4ϕt sin2(2φt) − ϕ) + 288α +ζ5φ2e5ϕt sin4(φt) cos2(φt)(192ζ4e4ϕt sin4(φt) + 32ζ3(31ϕ + 10)e3ϕt sin3(φt) ++16ζ2e2ϕt(ϕ(45ϕ + 56) − 7φ2) sin2(φt) + 32ζeϕt(17ϕ2 − φ2) sin(φt) − 1) − 2φ cos(φt) +(−18432αζ9(4ϕ + 1)e9ϕt sin9(φt) − 2304αζ8e8ϕt(ϕ(75ϕ + 44) − 11φ2) sin8(φt) − 9216αζ7 +e7ϕt(3ϕ2(3ϕ + 5) − (4ϕ + 1)φ2) sin7(φt) − 288αζ6e6ϕt(192ϕ3 − 32ϕφ2 − 1) +sin6(φt) + 96αζ5(3ϕ − 4)e5ϕt sin5(φt) − 576αζ4ϕe4ϕt sin4(φt) + 1) +−3ζeϕt sin2(φt))) +1 +2 +(32) +Now, the profiles of energy density and pressure under the presence of Eq.(30), are provided +in fig.3. The plots indicate that the energy density suffers a positive behavior for the assumed +values of free parameters. Similarly, the negative behavior for the pressure term indicates +that the universe is in the accelerated expansion phase. However, the positive density proves a +strong validation for the verification of the energy conditions. Also, one can get the positive +and alternate trends of the both terms for different time periods due to the oscillatory +behavior of the assumed Hubble parameter. We restrict our work for the positive density +and negative pressure behavior to ascertain the energy conditions. The evolutionary profiles +of the energy conditions are provided in the figs. 4 and 5. The N EC plot shows the violation +with in the bouncing regime and confirms the major verification for the universe to attain +12 + +Ζ +t +Ρ +Ζ +t +p +Figure 3: The illustration of energy density and matter pressure with fixed values of α = +0.005, k = 0.5, ϕ = 0.001, ǫ = 0.001, φ = 0.01, κ = 1 and λ = −0.005. +a bounce with in the framework of FLRW spacetime. The violated WEC and SEC are +given in the left plots of the figs. 3 and 4. The violated SEC also maintains the recent +observations for the accelerating universe [52]. One important energy condition i.e, T EC has +also been given in this recent study. The positive profiles for the DEC and T EC are given +in the fig.5. The evolution of these energy conditions is strictly dependent on the values +of the free parameters used in this study. However, one can get another configuration of +these physical factors by implementing the different free parameters. The evolution of EoS +parameter is provided in fig.6 to encounter the negative value i.e, ω(t) ≈ −1, for the current +expansion phase of the universe. +5 +Discussions +This work involves the study of bouncing cosmology for an isotropic configuration of fluid +Tαβ and FLRW metric. We comprehend this work under f(G, TαβT αβ) theory of gravitation +by assuming a specific model i.e, f(G, T 2) = G + αG2 + 2λT 2 with α and λ are constants, +serving as free parameters. This is the first-ever attempt to cover bouncing cosmology in +the f(G, TαβT αβ) theory. By the consideration of a specific functional form of the Hubble +parameter, we discuss the evolution of cosmographic parameters. +The assumption of a +well-known equation of state (EoS) parameter, ω(t) = −k log(t+ǫ) +t +− 1, is used as a direct +implementation to represent the dynamical behavior of energy density, matter pressure, and +energy conditions. The free parameters are restricted to the special values provided in each +13 + +-51 +52 +1.0 +53 +54 +54 +100 +200 +300 +20.048 +46 +44 +1.0 +42 +20.5 +40 +100 +38 +200 +300 +0.0Ρ � p +t +Ζ +Ρ � 3 p +t +Ζ +Figure 4: The illustration of N EC and SEC with fixed values of α = 0.005, k = 0.5, +ϕ = 0.001, ǫ = 0.001, φ = 0.01, κ = 1 and λ = −0.005. +Ρ � p +Ζ +t +Ρ � 3 p +t +Ζ +Figure 5: The illustration of DEC and T EC with fixed values of α = 0.005, k = 0.5, +ϕ = 0.001, ǫ = 0.001, φ = 0.01, κ = 1 and λ = −0.005. +14 + +204 +202 +305 +1.0 +200 +2005 +195 +198 +70.5 +100 +196 +200 +300 +0.098 +96- +96 +94 +100 +200 +300 +30.0105 +-110 +110 +1.0 +115 +120 +120 +100 +125 +200 +300 +0.01.0 +10 +0.5 +-15 +100 +200 +00 +0.00 +50 +100 +150 +200 +250 +300 +350 +�2.0 +�1.5 +�1.0 +�0.5 +0.0 +t +Ω +290 +300 +310 +320 +330 +�10 +�5 +0 +5 +10 +t +q +0 +50 +100 +150 +200 +250 +300 +350 +�10 +�5 +0 +5 +10 +t +q +Figure 6: The illustration of EoS and deceleration parameters with fixed values of k = 0.5 +and ǫ = 0.001. +graph plot and are used for H(t) to act as the bouncing solution. The viability of energy +conditions is studied with the help of a graphical approach. Following are the concluding +remarks for this present work. +• The Hubble parameter H(t) used in this study is considered to have a trigonomet- +ric functional form. The evolutionary behavior of different cosmographic factors is +described under the same form of H(t). This parameterized form of H(t) depends +on the periodic values of the function sin(φt) and h(t). We considered this h(t) as +a nonvanishing function for the periodic values of sin(φt). A perfect bouncing model +allows the Hubble parameter to show the contraction phase i.e, H < 0, and when the +universe expands it becomes H > 0. During this expansion and contraction phase, +there is the point in between, at where H(t) becomes zero. So, in order to produce +such a scenario, we have arranged the constants (φ and ϕ) in the Hubble parameter +(H(t) = ζ sin(φt) exp(ϕt)) to some specific values and notice the bounce at t = 313. +However, t = 313 is significant in such a way that all the energy conditions necessary +for the bounce, get satisfied accordingly till t = 313, depending on the values of φ +and ϕ. One can also produce other values of t for bounce by restricting other values +of φ and ϕ. The plot of H(t) is given in fig.1. The Hubble parameter gives us the +bounce at t = 313 which is the future singularity in the scale factor, see fig.1. The +mathematical forms of deceleration, jerk, and snap are evaluated with the same H(t). +The deceleration parameter tends to have a negative trend i.e, q(t) approaches −1, +which can be seen in fig.6. Similarly, the trends of jerk and snaps are given in fig.2 +with j(t) approaches to 1 and s(t) approaches to 0. All these values show a deflection +15 + +at the bouncing point, that fits in for the bouncing universe. +• We ensure the configuration of the bouncing cosmology by studying energy conditions. +These energy conditions are provided in terms of energy density and matter pressure +derived from the modified field equations. We assumed a specific EoS parameter in the +form ω(t) = −k log(t+ǫ) +t +− 1. This EoS parameter helped to maintain the positive and +negative growth of energy density and matter pressure for the limited bouncing time +period. The profiles of ρ and p are provided in the fig.3. However, the mathematical +expression for these terms is evaluated in Eqs.(31) and (32). +• Under the restricted values of the free parameters, α = 0.005, k = 0.5, ϕ = 0.001, +ǫ = 0.001, φ = 0.01, κ = 1 and λ = −0.005, we get the violation of the N EC and SEC. +The violated N EC derives the bouncing nature of the universe. However, the violated +SEC and WEC provide the phase of cosmic expansion. suitable with the observational +data. The left plots of figs.3 and 4 shows the violated SEC and WEC. Similarly, the +positive behavior of DEC and T EC assure that the assumed model configuration is +valid. Figure 5 represents the illustration of DEC and T EC. Also, the evolution of EoS +can be seen in fig.6, showing that ω(t) → −1. This value of ω(t) favors the current +accelerated expansion phase of the universe [61–63]. +• The above discussion provides that the bouncing evolution of the universe, studied in +the framework of f(G, T 2) = G + αG2 + 2λT 2 and agrees with the recent astronomical +observations [64, 65] i.e, all the energy conditions are fully satisfied, a great negative +pressure behavior had been observed and provided help to study the late time acceler- +ated universe [44]. However, this study can be used in the future for different models +of the scale factors and Hubble parameters. +• We finally conclude that the bouncing evolution of the universe can be studied effec- +tively with the oscillating nature of the scale factor under the flat FLRW regime. +References +[1] C. J. Hogan, The little book of the big bang: A cosmic primer. Springer Science & +Business Media, 1998. +[2] A. H. Guth, “Eternal inflation,” Ann. N. Y. Acad. Sci., vol. 950, no. 1, pp. 66–82, 2001. +16 + +[3] T. Padmanabhan and T. R. Seshadri, “Does inflation solve the horizon problem?,” +Class. Quantum Gravity, vol. 5, no. 1, p. 221, 1988. +[4] J. Earman and J. Mosterin, “A critical look at inflationary cosmology,” Philos. Sci., +vol. 66, no. 1, pp. 1–49, 1999. +[5] A. Ijjas and P. J. Steinhardt, “Bouncing cosmology made simple,” Class. Quantum +Grav., vol. 35, no. 13, p. 135004, 2018. +[6] E. Alesci, G. Botta, F. Cianfrani, and S. 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D, vol. 75, no. 6, +p. 063509, 2007. +21 + diff --git a/39FRT4oBgHgl3EQfozcR/content/tmp_files/load_file.txt b/39FRT4oBgHgl3EQfozcR/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0987ae40f5c06f32fcde3154895127276db7fa41 --- /dev/null +++ b/39FRT4oBgHgl3EQfozcR/content/tmp_files/load_file.txt @@ -0,0 +1,1076 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf,len=1075 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='13610v1 [gr-qc] 30 Jan 2023 Non-Singular Bouncing Model in Energy Momentum Squared Gravity Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Yousaf1 ∗, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Bhatti1 †, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Aman1 ‡, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Sahoo2 § 1Department of Mathematics, University of the Punjab, Quaid-i-Azam Campus, Lahore-54590, Pakistan 2Department of Mathematics, Birla Institute of Technology and Science-Pilani, Hyderabad Campus, Hyderabad-500078, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Abstract This work is concerned to study the bouncing nature of the universe for an isotropic configuration of fluid Tαβ and Friedmann-Lemaˆıtre-Robertson-Walker metric scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' This work is carried out under the novel f(G, TαβT αβ) gravitation by assuming a spe- cific model i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='e, f(G, T 2) = G + αG2 + 2λT 2 with α and λ are constants, serving as free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' The terms G and T 2 served as an Gauss-Bonnet invariant and square of the energy-momentum trace term as an inclusion in the gravitational action respec- tively, and is proportional to T 2 = TαβT αβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' A specific functional form of the Hubble parameter is taken to provide the evolution of cosmographic parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' A well known equation of state parameter, ω(t) = − k log(t+ǫ) t − 1 is used to represent the dynamical behavior of energy density, matter pressure and energy conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' A detailed graph- ical analysis is also provided to review the bounce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Furthermore, all free parameters are set in a way, to make the supposed Hubble parameter act as the bouncing solution and ensure the viability of energy conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Conclusively, all necessary conditions for a bouncing model are checked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Keywords: cosmography;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Hubble Parameter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' f(G, TαβT αβ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' PACS: 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='-k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' 04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='Cv;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' 04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='Kd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' ∗zeeshan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='math@pu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='pk †mzaeem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='math@pu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='pk ‡huzaifaaman971@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='com §pksahoo@hyderabad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='bits-pilani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='in 1 1 Introduction According to the big bang hypothesis, the whole universe was created by a single explosion, with all matter in the cosmos as an infinite speck [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' This hypothesis works well in order to study the beginning, but lack to define different cosmological problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' These problems include the horizon problem, the flatness problem, the singularity problem, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' In order to resolve these big cosmic challenges, different cosmic theories have been developed in literature [3–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' The bouncing hypothesis is one of the major independent theories that came up with the answers related to the starting of the universe and should be enough to resolve the major cosmic problem of singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' The bouncing cosmology works on the scheme of an oscillatory universe, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='e, a universe that came into being from the pre-existing universe without undergoing the singularity [6–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' This whole transition of the universe not only explains the big-bang cosmology but also reduces one of the major issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' For the bouncing, the universe moves into the contraction phase as a matter-dominated the era of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' After the contraction, the universe starts to expand in a nonsingular manner for which gravity dominates the matter [9,10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Also, density perturbations can be produced during the bounce era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' This idea of the origination of the universe is highly accepted and appreciated in literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' General relativity (GR) was presented by Einstein and it was thought to be one of the best theories to explain different cosmological issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' It explains the gravity under the fabric of space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' However, to understand gravity much more effectively and to provide the answers to the effect of gravity, dark energy, and accelerated expansion of the universe under the addition of different scalar fields, different attempts have been made in past to modify GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' These modifications change the geometric or matter or both parts of the Einstein field equations accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' These could help to discuss the effects of couplings of matter and curvature terms on the above-described items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Roshan and Shojai [11] presented the nonlinear form of matter term i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='e, T 2 = TαβT αβ, naming it f(R, T ∈).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' They further indicated that the use of nonlinear terms may provide the prevention of early time singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Since the functional form of curvature terms has helped to introduce new gravitational theories, so it was considered to be effective to modify the generic action integral of GR as corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' These modifications give light to the f(G) theory, for which the term G is defined as G = RξζαβRξζαβ − 4RξζRξζ + R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Nojiri and Odintsov [12] introduced this f(G) theory for the first time in their work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' They tested solar systems for this formalism and reported the phase change of acceleration to deceleration for the achievement of phantom line, which cooperated to study dark energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Odintsov and Oikonomou [13] considered R + f(G) form of the gravitational theory to provide their contribution to the study of gravitational baryogenesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Their work included the higher-order derivatives of Gauss-Bonnet terms that work in order to produce the baryon asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Sharif and Ikram [14] gives rise to a new theory by following 2 the footsteps of Harko.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' They coupled the matter part T with the geometric part of the f(G) theory, making it f(G, T ) cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' They investigated the validity of their theory with the help of energy conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Later on, Bhatti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' [15] worked on the f(G, T ) theory to carry out the investigation of some physically feasible features of compact star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' They inferred that the compactness of a star model grows at the core whereas the energy conditions remain constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Yousaf and his mates [16] inspired by [17], have recently developed a novel f(G, T 2) to present the complexity of structural scalars from the use of Herrera’s method of splitting scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' They considered the exponential coupling of Gauss-Bonnet terms as a functional form as f(G, T 2) = αGn(βGm + 1) + ηT 2, to explore the validity of their solutions for the Darmois and Israel conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' They also worked on the non-static complex structures under the same theory to describe the effects of an electromagnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' They used specific model configuration i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='e, f(G, T 2) = k1Gm(k2Gn + 1) + λT 2, in their work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Bouncing cosmology has gained much reputation over the past few years, because of its independent hypothetical nature from different standard comic problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Guth [18] during 1980′s, had put forward his inflationary theory to tackle early and late time cosmic evolutionary problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' He remained successful in solving some related problems, but the answer to the initial singularity is still under concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' One of the best hypotheses to answer the singularity problem is the bouncing nature of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' The nature of the bouncing universe allows a certain universe model to transit from a pre-big crunch (contracted) phase into a new big bang (expanded) phase with the exclusion of singularity during the whole event [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Steinhardt and Ijjas [20] are considered to be the pioneers of the bouncing hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' They devised a wedge diagram for a smooth bouncing method to explore the consequences of some cosmological problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Sahoo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' [21] worked on the non-singular bouncing by assuming the specific coupling of R and T as f(R, T ) = R + χRT , for 0 < χ < π 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' They allowed such a parametric approach for the Hubble parameter to provide no singularity during the bounce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' They used quintom and phantom scalar field configurations for the bouncing paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Bamba and his collaborators [22] inspected the singularity-free concept of bounce by considering an exponential form of scale factor a(t) = σ exp(λt) + τ exp(λt) under the effect of f(G) gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' They checked the stability of their assumed solution under the restricted parametric scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Yousaf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' [23, 24] explored the bouncing universe with a specific functional form of Hubble parameter by taking exponential f(G, T ) form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Different cosmic models are under consideration for the scale factor in order to determine the value of expansion and contraction at the current cosmic phase and also to predict the current phase equation of state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' These models predicted different results in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' However, cosmography provided us a benefit in processing cosmological data for explaining the universal kinematics without the involvement of the gravity model and hence provided that the cosmography can be employed with the Taylor expansions as an alternative scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Also, the cosmographic analysis for 3 the FLRW universe, is helpful in such a way that it can put aside the effect of the dy- namical field equations [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Gruber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' [26] studied an alternative approach to describe cosmography by extending the conventional methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' They resulted from numerical values of the cosmographic parameters by applying the Pad´e approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' The testing of the ΛCDM model had been conveyed by Busti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' [27] with the use of cosmographical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Capozziello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' provided cosmography as a non-predictive phenomenon when the redshift parameter becomes z ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' They used the pad´e approximations for the fifth order and resulted the divergence of data at the higher levels of the approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Lobo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' [28] evaluated the dynamics of the redshift drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' They used the expanding FLRW universe to produce a general matter and low redshift model with the use of different vari- ables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' However, the cross-correlation of large-scale quasars can be used and translated with the CMB and BAO scale data to produce the best for Hubble parameter H(z) and angular diametric distance SA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Also, the cosmic chronometers approach can be done to predict the model independent H(z) measurements which have been extensively used for cosmological applications [29–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' The low redshift data set with the inclusion of the megamasers and chronometers had been presented by Krishnan and others [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' They result that the Hubble constant H0, showed descending behavior with the redshift and having non-zero slop when fitted on the line by statistical means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Font et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' [33] studied correlation technique for quasars by using Lya absorption and produced the best line of fit for Planck’s data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' They generated different results on the measurements of the Hubble parameter and the angular distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' One important thing is to develop such a cosmic Hubble parameter that comes from early to late span in such a way that it changes from a low to a high value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' The Gaus- sian method helped to predict but provided a non-transitional behavior for both Λ and ω epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' The null energy condition also proved to be an important restriction for the cut-off model, when compared with Hubble parameter data [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' King et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' [35] studied the future approximations of the redshift by the inclusion of dark energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' They tested the equation of state by the linear parametrization technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' [34] reported different values of the Hubble constant by the Gaussian method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Their research produced an effective reduction in the Hubble crisis and proposed the non-transitional behavior of the Hubble constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Different dark energy models respective to holography and agegraphy had been conducted by Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' They produced different energy conditions for different red shift values and resulted in an effective role of energy conditions for different cosmic ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' In this article, we implemented a functional form of the Hubble parameter that evolves periodically with cosmic time t and investigate the bouncing nature of the universe in f(G, TαβT αβ) gravity using a flat FLRW peacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' This analysis of the bouncing uni- verse involves one of the most important forms of EoS parameter proposed in the litera- ture [37–39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' The outline is given as: Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='2 provides a brief introduction to f(G, TαβT αβ) gravity with the necessary formalism of FLRW metric and modified field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='3 4 builds the Hubble parameter as a bouncing solution for the produced field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' The cosmographic parameters are also evaluated in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' We provide the mathematical expressions of energy density and matter pressure for the assumed EoS parameter form in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' The energy conditions are also formulated in the same fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Detailed graphical profiles of energy conditions are represented in the same section to discuss the evolution of the universe under the influence of restricted free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Finally, the concluding remarks are made in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' 2 f(G, TαβT αβ) Formalism The modified action for the f(G, TαβT αβ) gravity theory is defined as [16] Af(G,TαβT αβ) = √−g 2κ2 �� d4x[f(G, TαβT αβ) + R] + � d4xLm � , (1) where R and G symbolize the Ricci scalar and the Gauss-Bonnet scalar terms, respectively and are provided as R ≡ gαβRαβ, G ≡ RξζαβRξζαβ − 4RξζRξζ + R2, (2) and κ2 = 8πG (G be the gravitational constant) and Lm = −p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Also, the term g implies the trace of the metric tensor gαβ with Tαβ, Rξζαβ and Rαβ indicate the stress energy-momentum tensor, the Riemannian tensor, and the Ricci tensor respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' The expression for Tαβ is given as Tαβ = −2 √−g δ(√−gLm) δgαβ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' (3) Equation (3) yields the following expression, due the dependency of the matter Lagrangian Lm on gαβ components Tαβ = gαβLm − 2∂Lm ∂gαβ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' (4) Now, by taking the variation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' (1) with respect to the term gαβ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' we get the following field equations for the f(G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' TαβT αβ) theory as Rαβ − 1 2Rgαβ = T eff αβ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' (5) where the term T eff αβ takes the following form T eff αβ = κ2Tαβ − ΘαβfT 2(G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' T 2) + 1 2gαβf(G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' T 2) − (2RRαβ − 4Rε αRεβ − 4RαεβηRεη 5 +2Rεηδ α Rβεηδ)fG(G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' T 2) − (2R∇2gαβ − 2R∇α∇β − 4Rαβ∇2 − 4gαβRεη∇ε∇η +4Rε α∇β∇ε + 4∇ε∇αRε β + 4Rαεβη∇ε∇η)fG(G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' T 2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' (6) where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Θαβ ≡ δ(TµνT µν) δgαβ = 2T ξ α Tβξ − T Tαβ − 4T µν ∂2Lm ∂gαβgµν − 2Lm(Tαβ − 1 2T gαβ) (7) T 2 = TαβT αβ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' ∇2 = ∇α∇α (8) The terms fG(G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' T 2) and fT 2(G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' T 2) used above are defined as fG(G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' T 2) ≡ df(G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' T 2) dG ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' and fT 2(G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' T 2) ≡ df(G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' T 2) dT 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' (9) The trace of the above-defined field equations is produced as T − ΘfT 2(G, T 2) + 2GfG(G, T 2) − 2R∇2fG(G, T 2) + 4Rαβ∇α∇βfG(G, T 2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' (10) Equation (10) shows the non-conversed situation of the stress energy-momentum tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Also, the properties of GR can be recovered for f(G, T 2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Similarly if we put f(G, T 2) = f(G), we get the properties of f(G) gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Now, as we are concerned to study the bouncing nature of the universe, so we consider the fluid distribution to be perfect throughout the cosmic evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' For this, we take Tαβ = (ρ + p)VαVβ − pgαβ, (11) here, the four-vector velocity is defined by V β with V β = (1, 0, 0, 0), V βVβ = 1 , V β∇ζVζ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' (12) In addition, ρ defines the energy density part and p defines the pressure part of the stress energy-momentum tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Also the geometric background considered to be in a FLRW space time [40], so it implies ds2 = dt2 − a2(t)Σidx2 i , i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' (13) The metric component a(t) symbolizes the scale factor, that contributes to the Hubble parameter as H = ˙a(t) a(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' (13) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' (7) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' (5), we get the following field equations 6 � ˙a a �2 − 24 � ˙a a �3 ˙fG + 24 �¨a a � � ˙a a �2 fG − f − 2(ρ2 + 3p2 + 4ρp)fT 2 = 2ρκ2, (14) 6 − 2 � 2¨a a + � ˙a a �2� + 16 �¨a˙a a2 � ˙fG + 8 � ˙a a �2 ¨fG − 24 �¨a a � � ˙a a �2 fG + f = 2pκ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' (15) To draw the conclusions on the field equations, we just need some functional form of f(G, T 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' As, there are many functional forms regarding the interaction of matter with the curvature terms, in order to deal with the issues of cosmic evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Various coupling models can be used to evaluate the formations of both energy density and matter pressure, like one can take f(G, T 2) = G + 2f(T 2) that may help to provide an analysis about ΛCDM epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' However, the other choice is f(G, T 2) = f1(G) + f2(T 2) that may be worked as a correction to f(G) gravity theory because of f2(T 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Similar forms have been explored in [41, 42] and provided some distinct results due to the direct minimal curvature matter coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Also, f(G, T 2) = f1(G) + f2(G)f3(T 2) can be taken because of an explicit non-minimally coupling nature between geometric parameters and matter variables [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' So, we considered the following form to produce the validating results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' f(G, T 2) = f1(G) + f2(T 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' (16) To produce a bouncing universe, we need some functional forms of f1 and f2 that not only describe the accelerating expansion of the universe but also explain inflation to a great extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' For this, the higher power curvature terms perform well to eliminate such issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Elizalde [44] introduced the power forms of the curvature scalar as ηRn (n ≥ 1) and produced the cosmological dynamics, so we consider the specific form of the f1 as the quadratic power model, so f1(G) = G + αG2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' (17) Also, we take χ2 as f2(T 2) = 2λT 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' (18) So, by using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='s (17) and (18) in the field equations, we get 6H2 − 48αH3G ˙G + αG2 = 2κ2ρ + 6λρ2 + 18λp2 + 16λρp, (19) and −2(2 ˙H + 3H2) + 32( ˙H + H2)αG ˙G + 16αH2( ˙G2 + G ¨G) − αG2 = 2κ2p − 2λρ2 − 6λp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' (20) In order to reduce the complexity of the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' (19) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' (20), we utilize p = ωρ, as the EoS used in [37–39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' So we get the relations, (3λ + 9λω2 + 8λω)ρ2 + κ2ρ − (3H2 − 24αH3G ˙G + α 2 G2) = 0 (21) 7 and (− λ ω2 − 3λ)p2 + κ2p + ((2 ˙H + 3H2) − 16( ˙H + H2)αG ˙G − 8αH2( ˙G2 + G ¨G) + α 2 G2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' (22) where, G = 24H2( ˙H + H2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Yousaf and his collaborators checked the stability of cosmic models in various modified gravity theories [45–48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' 3 Hubble Parameter and Cosmography This section mainly focuses on describing the evolutionary behavior of these above-described dynamical terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Hence, we consider a trigonometric form of the H(t) which feasibly provides a bounce solution [44,49], as follows H(t) = ζ sin(φt)h(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' (23) This parameterized form of H(t) includes ζ and φ, which are considered to be constants here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' The choice of h(t) depends on the periodic values of the function sin(φt), so the form of h(t) can be chosen as periodic, that cooperates with the non-vanishing values of the above trigonometric function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' This artificial approach of choosing such an ansatz can be considered as a numerical analysis of making the bouncing solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' One interesting feature is possessed by the term ζ, which can work well as a phase changer for the value of H(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' We consider h(t) as h(t) = exp(ϕt), (24) where ϕ acts as a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Finally, we have H(t) = ζ sin(φt) exp(ϕt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' (25) This functional form of the Hubble parameter is helpful to study cosmic evolutionary expan- sion and contraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' This form of the Hubble parameter gives us the bounce at t = 313, depending upon the values of ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='001 and φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='01 provided in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' We have re- stricted the values of H(t) in the positive era of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' The basic scale factor form for this parameterized Hubble parameter becomes a(t) = exp �ζ exp(ϕt)(ϕ sin(φt) − φ sin(φt)) ϕ2 + φ2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' (26) Similarly, the set of dynamical parameters that are derived from the Taylor series expansion of the scale factor is termed as cosmographic factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' These factors helped to obtain the 8 cosmological concordance with the assumptions of the universal homogeneity and isotropy on large cosmic scales [27,50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' These include deceleration, jerk and snap parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' These factors allow us to check the compatibility of the scale factor and the Hubble parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' The negative value of the deceleration parameter q describes the accelerated expansion of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Similarly, jerk j and snap s determine the expansion rate of the toy universe model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' The mathematical interpretation for these cosmography elements are defined as q = − 1 H2 1 a d2a dt2 = −1 − 1 ζ (e−ϕt csc(φt)(φ cot(φt) + ϕ)), (27) j = 1 H3 1 a d3a dt3 = 1 + 1 ζ2(e−2ϕt csc(φt)(3ζeϕt(φ cot(φt) + ϕ) + csc(φt)(2ϕφ cot(φt) + ϕ2 − φ2))), (28) and s = 1 H4 1 a d4a dt4 = − 1 3ζ(3ζeϕt + 2 csc(φt)(φ cot(φt) + ϕ))(2e−ϕt csc(φt)(csc(φt) (3ζϕeϕt sin(φt) + ϕ2 − φ2) + φ cot(φt)(3ζeϕt + 2ϕ csc(φt)))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' (29) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='1 shows the progression of the Hubble (left panel) and scale parameters (right panel) along the positive time axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Similarly, the development of jerk (left panel) and snap factors (right panel) are provided in the fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' The evolution of the deceleration parameter towards the negative value i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='e, q → −1, before the bouncing point, provided in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='6, shows the accelerating universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' 4 Energy Conditions under the EoS Parameter For a specific cosmology model, energy conditions play an important role to make its val- idation for the restricted free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' These energy conditions help to maintain the specifications of the certain cosmic model [51–55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Similarly, these energy conditions also work for the bouncing cosmology and provide a reasonable approach to validate the proce- dure for our toy bouncing model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' These conditions are described as Dominant energy condition (DEC)⇔ ρ ≥ 0 , ρ ± p ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Strong energy condition (SEC)⇔ ρ + 3p ≥ 0 , ρ + p ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Weak energy condition (WEC)⇔ ρ ≥ 0 , ρ + p ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' 9 H � 0 Bouncing H � H � 0 300 305 310 315 320 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='0 t a a�t� � 0 0 50 100 150 200 250 300 350 �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='0 t a Figure 1: The illustrations of Hubble parameter and scale factor with fixed values of ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='001 and φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' 280 290 300 310 320 330 340 �4 �2 0 2 4 t j 300 310 320 330 340 �1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='0 �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='0 t s 0 50 100 150 200 250 300 350 �1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='0 �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='0 t s Figure 2: The illustration of jerk and snap factors with fixed values of ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='001 and φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' 10 6F =1 2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='8 ■=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='6 M5=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='4 ■2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='2 0 100 200 300 400 500H 量5=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='8 拉2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='6 2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='4 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='2 0 100 200 300 400 500 600 t• Null energy condition (N EC)⇔ ρ + p ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Trace energy condition (T EC)⇔ ρ − 3p ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' The positivity of DEC, SEC and WEC passes on the validity and necessity of the bouncing concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' However, the violation of N EC has a major role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' This violation is different in the GR context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Universal bouncing scenario is one of those ideas that provides a chance to discuss the singularity-free universal beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Many proposals in the literature suggested avoiding this singularity through quantum aspects, but these don’t have such reliability to fit in the gravitational theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' So, at this point gravitational theories allow a specific mechanism to check the validity of the bounce model and as well its own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Null energy condition is one such tool to help achieve the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Also, it has been proved that in the context of GR, the violation of N EC is extremely difficult to be achieved for local-field models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' So, effective field theories provide a chance to recognize the violation of the N EC and to allow a non- singular bounce [56–59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' One such effective field is f(G, TαβT αβ) theory that provides a chance to study the quadratic nature of the energy terms i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='e, energy density and matter pressure [16, 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' However, it also allows getting a non-singular bounce for the assumed gravity model form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' For an excellent bouncing model, the value of H(t) turns out to be ˙H = −4Gρπ(1 + ω) > 0 for the formulation of GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' However, if the N EC gets violated, we have the surety to get a bouncing scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' To provide the mathematical formulation of the energy conditions, we consider Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' (21) and (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Also, the EoS parameter in the negative regime provides the present cosmic evolution [61–63] and becomes favorable in the bouncing context with ω(t) ≈ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' However, bouncing cosmology provides the possible geodesic evolution of the universe by avoiding the singularity along with the resolution of the horizon problem, flatness problem, entropy problem and many more [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' For the modified gravity, EoS parameter enables us to study the universal dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' In this study, we used EoS parameter [44] to obtain the possible chance of obtaining a bounce solution in f(G, T 2) as ω(t) = −k log(t + ǫ) t − 1, (30) here k is assumed to be a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' This particular form of the EoS parameters allows us to study the contracting and expanding behavior without involving the Hubble parameter as well as the scale factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Elizalde et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' [44] produces cosmological dynamics by considering R2 gravity and logarithmic trace terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' They checked the effects of the λ parameter in the gravity model f(R, T ) = R+λR2+2β ln(T ) along with the bouncing solution depending on the two EOS parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Our work first described the choice of Hubble parameter and its effects on the dynamical field equations and then involves the EOS parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' We only took one of the ω(t) value, because this state factor after the bouncing point remains negative and 11 becomes ω(t) ≈ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Also, the current cosmic expansion and Λ − CDM can be verified by this state factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' However, the dynamic properties are greatly affected under the influence of this EoS parameter form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Hence, the general forms of the Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' (21) and (22), under the influence of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' are presented as ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='ρ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='2λ(9ω2 + 8ω + 3)(κ2 + (κ4 − 12ζ2λ(9ω2 + 8ω + 3)e2ϕt sin2(φt)(2304αζ7e7ϕt sin4(φt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='(sin(φt)(ζeϕt sin(φt) + ϕ) + φ cos(φt))(4ζϕeϕt sin3(φt) + 2ζφeϕt sin(2φt) sin(φt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='(φ2 − 3ϕ2) sin2(φt) + 2φ2 cos2(φt) + 3ϕφ sin(2φt)) − 96αζ4e4ϕt sin2(φt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='(sin(φt)(ζeϕt sin(φt) + ϕ) + φ cos(φt))2 − 1)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='(31) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='2(3λω2 + λ)(κ2ω2 + (κ4ω4 + 4ζω2(3λω2 + λ)eϕt(18432αζ9e9tϕ(2ϕ(2ϕ + 1) − φ2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='sin10(tφt) + 4608αζ8e8tϕ(ϕ2(25ϕ + 22) − (13ϕ + 2)φ2) sin9(φt) + 9216αζ7φ3e7ϕt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='sin5(φt) cos3(φt)(7ζeϕt sin(φt)10ϕ + 8) + 288αζ7e7ϕt(144ϕ4 + 320ϕ3 − 16(9ϕ + 4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='ϕφ2 − 1) sin8(φt) + 576αζ6φ4e6ϕt sin4(2φt)(ζeϕt + 2 csc(φt)) + 576αζ6ϕe6ϕt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='(48ϕ3 − 16ϕφ2 − 1) sin7(φt) − 96αζ5ϕ(3ϕ − 8)e5ϕt sin6(φt) + 192αζ4e4ϕt(3ϕ2 − φ2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='sin5(φt) + 2 sin(φt)(5760αζ6ϕφ3e6ϕt sin3(2φt) + 48αζ4φ2e4ϕt sin2(2φt) − ϕ) + 288α ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='ζ5φ2e5ϕt sin4(φt) cos2(φt)(192ζ4e4ϕt sin4(φt) + 32ζ3(31ϕ + 10)e3ϕt sin3(φt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='+16ζ2e2ϕt(ϕ(45ϕ + 56) − 7φ2) sin2(φt) + 32ζeϕt(17ϕ2 − φ2) sin(φt) − 1) − 2φ cos(φt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='(−18432αζ9(4ϕ + 1)e9ϕt sin9(φt) − 2304αζ8e8ϕt(ϕ(75ϕ + 44) − 11φ2) sin8(φt) − 9216αζ7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='e7ϕt(3ϕ2(3ϕ + 5) − (4ϕ + 1)φ2) sin7(φt) − 288αζ6e6ϕt(192ϕ3 − 32ϕφ2 − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='sin6(φt) + 96αζ5(3ϕ − 4)e5ϕt sin5(φt) − 576αζ4ϕe4ϕt sin4(φt) + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='−3ζeϕt sin2(φt))) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='(32) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='Now,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' the profiles of energy density and pressure under the presence of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' (30), are provided in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' The plots indicate that the energy density suffers a positive behavior for the assumed values of free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Similarly, the negative behavior for the pressure term indicates that the universe is in the accelerated expansion phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' However, the positive density proves a strong validation for the verification of the energy conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Also, one can get the positive and alternate trends of the both terms for different time periods due to the oscillatory behavior of the assumed Hubble parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' We restrict our work for the positive density and negative pressure behavior to ascertain the energy conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' The evolutionary profiles of the energy conditions are provided in the figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' The N EC plot shows the violation with in the bouncing regime and confirms the major verification for the universe to attain 12 Ζ t Ρ Ζ t p Figure 3: The illustration of energy density and matter pressure with fixed values of α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='005, k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='5, ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='001, ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='001, φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='01, κ = 1 and λ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' a bounce with in the framework of FLRW spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' The violated WEC and SEC are given in the left plots of the figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' The violated SEC also maintains the recent observations for the accelerating universe [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' One important energy condition i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='e, T EC has also been given in this recent study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' The positive profiles for the DEC and T EC are given in the fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' The evolution of these energy conditions is strictly dependent on the values of the free parameters used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' However, one can get another configuration of these physical factors by implementing the different free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' The evolution of EoS parameter is provided in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='6 to encounter the negative value i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='e, ω(t) ≈ −1, for the current expansion phase of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' 5 Discussions This work involves the study of bouncing cosmology for an isotropic configuration of fluid Tαβ and FLRW metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' We comprehend this work under f(G, TαβT αβ) theory of gravitation by assuming a specific model i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='e, f(G, T 2) = G + αG2 + 2λT 2 with α and λ are constants, serving as free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' This is the first-ever attempt to cover bouncing cosmology in the f(G, TαβT αβ) theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' By the consideration of a specific functional form of the Hubble parameter, we discuss the evolution of cosmographic parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' The assumption of a well-known equation of state (EoS) parameter, ω(t) = −k log(t+ǫ) t − 1, is used as a direct implementation to represent the dynamical behavior of energy density, matter pressure, and energy conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' The free parameters are restricted to the special values provided in each 13 51 52 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='0 53 54 54 100 200 300 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='048 46 44 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='0 42 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='5 40 100 38 200 300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='0Ρ � p t Ζ Ρ � 3 p t Ζ Figure 4: The illustration of N EC and SEC with fixed values of α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='005, k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='5, ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='001, ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='001, φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='01, κ = 1 and λ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Ρ � p Ζ t Ρ � 3 p t Ζ Figure 5: The illustration of DEC and T EC with fixed values of α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='005, k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='5, ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='001, ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='001, φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='01, κ = 1 and λ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' 14 204 202 305 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='0 200 2005 195 198 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='5 100 196 200 300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='098 96- 96 94 100 200 300 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='0105 110 110 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='0 115 120 120 100 125 200 300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='0 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='5 15 100 200 00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='00 50 100 150 200 250 300 350 �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='0 �1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='5 �1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='0 �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='0 t Ω 290 300 310 320 330 �10 �5 0 5 10 t q 0 50 100 150 200 250 300 350 �10 �5 0 5 10 t q Figure 6: The illustration of EoS and deceleration parameters with fixed values of k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='5 and ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' graph plot and are used for H(t) to act as the bouncing solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' The viability of energy conditions is studied with the help of a graphical approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Following are the concluding remarks for this present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' The Hubble parameter H(t) used in this study is considered to have a trigonomet- ric functional form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' The evolutionary behavior of different cosmographic factors is described under the same form of H(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' This parameterized form of H(t) depends on the periodic values of the function sin(φt) and h(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' We considered this h(t) as a nonvanishing function for the periodic values of sin(φt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' A perfect bouncing model allows the Hubble parameter to show the contraction phase i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='e, H < 0, and when the universe expands it becomes H > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' During this expansion and contraction phase, there is the point in between, at where H(t) becomes zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' So, in order to produce such a scenario, we have arranged the constants (φ and ϕ) in the Hubble parameter (H(t) = ζ sin(φt) exp(ϕt)) to some specific values and notice the bounce at t = 313.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' However, t = 313 is significant in such a way that all the energy conditions necessary for the bounce, get satisfied accordingly till t = 313, depending on the values of φ and ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' One can also produce other values of t for bounce by restricting other values of φ and ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' The plot of H(t) is given in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' The Hubble parameter gives us the bounce at t = 313 which is the future singularity in the scale factor, see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' The mathematical forms of deceleration, jerk, and snap are evaluated with the same H(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' The deceleration parameter tends to have a negative trend i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='e, q(t) approaches −1, which can be seen in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Similarly, the trends of jerk and snaps are given in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='2 with j(t) approaches to 1 and s(t) approaches to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' All these values show a deflection 15 at the bouncing point, that fits in for the bouncing universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' We ensure the configuration of the bouncing cosmology by studying energy conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' These energy conditions are provided in terms of energy density and matter pressure derived from the modified field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' We assumed a specific EoS parameter in the form ω(t) = −k log(t+ǫ) t − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' This EoS parameter helped to maintain the positive and negative growth of energy density and matter pressure for the limited bouncing time period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' The profiles of ρ and p are provided in the fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' However, the mathematical expression for these terms is evaluated in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' (31) and (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Under the restricted values of the free parameters, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='005, k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='5, ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='001, ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='001, φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='01, κ = 1 and λ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='005, we get the violation of the N EC and SEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' The violated N EC derives the bouncing nature of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' However, the violated SEC and WEC provide the phase of cosmic expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' suitable with the observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' The left plots of figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='3 and 4 shows the violated SEC and WEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Similarly, the positive behavior of DEC and T EC assure that the assumed model configuration is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Figure 5 represents the illustration of DEC and T EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Also, the evolution of EoS can be seen in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='6, showing that ω(t) → −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' This value of ω(t) favors the current accelerated expansion phase of the universe [61–63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' The above discussion provides that the bouncing evolution of the universe, studied in the framework of f(G, T 2) = G + αG2 + 2λT 2 and agrees with the recent astronomical observations [64, 65] i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content='e, all the energy conditions are fully satisfied, a great negative pressure behavior had been observed and provided help to study the late time acceler- ated universe [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' However, this study can be used in the future for different models of the scale factors and Hubble parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' We finally conclude that the bouncing evolution of the universe can be studied effec- tively with the oscillating nature of the scale factor under the flat FLRW regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' References [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FRT4oBgHgl3EQfozcR/content/2301.13610v1.pdf'} +page_content=' Hogan, The little book of the 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a/89E3T4oBgHgl3EQfSAmG/content/tmp_files/2301.04428v1.pdf.txt b/89E3T4oBgHgl3EQfSAmG/content/tmp_files/2301.04428v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3d24fceda8ce3d4ddd9d66531e8a898e6fc459a5 --- /dev/null +++ b/89E3T4oBgHgl3EQfSAmG/content/tmp_files/2301.04428v1.pdf.txt @@ -0,0 +1,1133 @@ +arXiv:2301.04428v1 [math.QA] 11 Jan 2023 +THE PRIME SPECTRUM OF THE DRINFELD DOUBLE OF +THE JORDAN PLANE +K. A. BROWN AND J. T. STAFFORD +Abstract. The Hopf algebra D which is the subject of this paper can be +viewed as a Drinfeld double of the bosonisation of the Jordan plane. Its prime +and primitive spectra are completely determined. As a corollary of this anal- +ysis it is shown that D satisfies the Dixmier-Moeglin Equivalence, leading to +the formulation of a conjecture on the validity of this equivalence for pointed +Noetherian Hopf algebras. +1. Introduction +1.1. Throughout, k will denote an algebraically closed field of characteristic 0. +The Hopf k-algebra D of the title was defined and some initial properties were +derived in [2], with further results in [1, 3]. Our focus here is on the prime and +primitive spectra of D, which we completely determine. The Hopf algebra D is a +pointed affine noetherian domain of Gelfand-Kirillov dimension 6 whose definition +is recalled in §2.1. It is a beautiful algebra with a number of striking properties +which make it worthy of study from several perspectives, three of which we briefly +outline in §§1.3-1.5. First we summarise our results and explain where they are +located. +1.2. Results. It was proved in [1, Theorem 4.10], and explained in detail here +in Lemma 2.1 and Theorem 2.2(iv), that the centre Z(D) of D is generated by +elements z, ω and θ with zθ = ω2. Thus Maxspec(Z(D)) has one singular point, +namely m0 := ⟨z, ω, θ⟩. There is one other distinctive maximal ideal of the centre, +namely +m+ := Z(D) ∩ D+ = ⟨z − 16, ω + 16, θ − 16⟩, +where D+ is the augmentation ideal of D. Finally, let K denote the kernel of the +Hopf algebra surjection π : D −→ U(sl(2, k)), mentioned in §1.3, so K is a Hopf +ideal which is described in Theorem 2.2(i),(ii). +The main results of this paper are given by the following theorems and give a +complete description of the prime and primitive ideals of D. +Theorem 1.1. Retain the above notation. The primitive ideals of D are: +(I) the maximal ideals mD for m ∈ Maxspec(Z(D)), with m ̸= m0, m+; +(II) the primitive ideals containing m+D, namely m+D itself together with +P := π−1(Privspec(U(sl(2, k)))); +2010 Mathematics Subject Classification. Primary 16T05, 16D25; Secondary 16T20, 16S40, +17B37. +Both authors are partially supported by Leverhulme Emeritus Fellowships, respectively EM- +2017-081 and EM-2019-015. The first author thanks Nicolas Andruskiewitsch, Ivan Angiono and +Hector Pena Pollastri for helpful comments and for sharing early versions of their work. +1 + +2 +K. A. BROWN AND J. T. STAFFORD +(III) the unique prime ideal P0 containing m0D, which has m0D = (P0)2 ⊊ P0. +Theorem 1.2. In the above notation, the non-primitive prime ideals of D are: +(A) {0}, K; +(B) the principal prime ideals pD for every height one prime p of Z(D) except +p1 = ⟨z, ω⟩ and p2 = ⟨θ, ω⟩; +(C) height one primes P1, P2, with piD = P 2 +i ⊊ Pi for i = 1, 2, with each Pi +generated by a normal (but not central) element. Moreover, P1 + P2 = P0. +Theorem 1.3. Retain the above notation. +(a) Every non-primitive prime is completely prime. +Every primitive ideal is +completely prime, except the co-Artinian maximal ideals (other than the +counit), which form a subset of P. +(b) Every prime ideal P, apart from the co-Artinian maximal ideals and (pos- +sibly) P0, has D/P birationally equivalent to a Weyl algebra An(K), where +1 ≤ n ≤ 2 and K is a field of transcendence degree at most 2 over k. +(c) D satisfies the Dixmier-Moeglin Equivalence. +Theorem 1.1 is proved in Section 4, see in particular Subsection 4.5, while Theo- +rem 1.2 is proved in Theorem 5.3. Finally, Theorem 1.3 is proved in Subsection 5.2. +Some questions and a conjecture (Conjecture 5.5) are scattered through the paper. +1.2. Hopf algebras in duality. The full Drinfeld double D(H) = H ⊲⊳ H◦ of an +infinite dimensional Hopf algebra H may often be unwieldy due to H having “too +many” finite dimensional representations and thus leading to an unmanageably +large finite dual H◦. This has generated significant recent interest in constructing +doubles D(H) := H ⊲⊳ H′ where H′ is some suitable Hopf subalgebra of H◦; see +for example [9, 20] and the papers listed in §1.1. Much is at present unclear: for +example, what is an appropriate definition of a “suitable” Hopf subalgebra H′; and +does a suitable algebra H′ always exist? The double D of the Jordan plane is a test +case for these and other questions. In particular, some of the desirable properties +exhibited by D may form a paradigm for what one might aim for in defining doubles +in more general settings. +1.3. Representation theory. A striking feature of the representation theory of +D is the fact, proved in [1, Theorem 3.11] and recalled here in Theorem 2.2(iii), that +U(sl(2, k)) is a quotient Hopf algebra of D and the finite dimensional irreducible D- +modules are precisely the finite dimensional irreducible U(sl(2, k))-modules. This +immediately suggests a plethora of questions, some of which are addressed in [3], +where Verma D-modules are introduced. But many others remain untouched: for +instance, what can be said about the category of locally finite dimensional D- +modules, and for each primitive ideal P of D can one find a canonical irreducible +module (hopefully a factor of a Verma module) whose annihilator equals P? The +first step in these questions is to classify the primitive ideals of D, as we do here. +1.4. +Dixmier-Moeglin equivalence. +The validity of the Dixmier-Moeglin +Equivalence for an algebra R yields simultaneous representation-theoretic, alge- +braic and topological characterisations of the primitive ideals amongst the prime +ideals of R. This equivalence is a feature of some but not all Hopf k-algebras; see, +for example, [5,6] for discussions of when it holds for a noetherian (Hopf) algebra. +As we prove in Theorem 1.2, D satisfies the equivalence. See §5.2 for the details, + +THE PRIME SPECTRUM OF THE DRINFELD DOUBLE OF THE JORDAN PLANE +3 +where we also give a rather ambitious Conjecture 5.5, proposing a general result +encompassing all affine Noetherian pointed Hopf C-algebras. +Notation. Throughout, all vector spaces and all unadorned tensor products are +understood to be over the base field k. We denote the comultiplication of a Hopf +algebra H by ∆ and its augmentation ideal by H+. The Gelfand-Kirillov, or GK +dimension of an object X is denoted by GKdim(X), while the global (homological) +dimension, respectively injective dimension of a ring R is denoted by gldim(R), +respectively injdim(R). For precision, we specify that in the Ore extension T = +S[v; σ, ∂], multiplication is defined by +(1.1) +vs = sσv + ∂(s) +for s ∈ S. +It follows that ∂ is a σ-derivation in the sense that ∂(ab) = aσ∂(b) + ∂(a)b. This +follows the conventions of, for example, [17, p.34]. +2. Preliminaries +2.1. Definitions and notation. The following definitions and notation from [1]) +will remain in play throughout the paper. First, the Jordan plane is +J := k⟨x, y : [y, x] = − 1 +2x2⟩, +with bosonization +H := J#C∞ = J#⟨g±1⟩, where gxg−1 = x, gyg−1 = y + x. +Then the Drinfeld double of J is defined to be D := H⟨u, v, ζ⟩, with additional +relations as follows: +[u, v] = 1 +2u2; [ζ, v] = −v; [ζ, u] = −u; [u, y] = 1 − g; +[v, x] = 1 − g + xu; [v, y] = yu − gζ; [v, g] = gu; [ζ, y] = y; [ζ, x] = x; +[x, u] = [x, g] = [u, g] = [ζ, g] = 0. +The coalgebra structure, which will mostly not concern us here, is determined for +H by specifying that g is grouplike and x and y are (g, 1)−primitive; and then +extended to D by setting u and ζ to be primitive and ∆(v) = v ⊗ 1 + 1 ⊗ v + ζ ⊗ u. +Observe that K := k⟨u, v, ζ⟩ is a Hopf subalgebra of D and in fact, as one can +see from the PBW theorem for D as described in [2, Proposition 2.3(ii)], D = +J ⊗k K as vector spaces. As noted in [2, Lemma 2.2] there is a non-degenerate +skew pairing between J and K which yields the multiplication relations between +these subalgebras as in [14]. +2.2. Initial results. We gather together in Theorem 2.2 some of the main results +of [2] and [1]. We must first define some elements of D, as follows. Set +(2.1) +q := ux + 2(1 + g), +and s := xv + uy + (−1 +2ux + g − 1)ζ − 2(g + 1). +The following lemma is partly explicit, partly implicit, in [1, §4]. Given a k-algebra +automorphism σ of a k-algebra H, we say that the element h of H is σ-normal if +ha = σ(a)h for all a ∈ H. +Lemma 2.1. Keep the above notation. + +4 +K. A. BROWN AND J. T. STAFFORD +(i) q and s are both σ-normal, where σ is the automorphism of D defined by +σ(y) = y + 1 +2x, σ(v) = v − 1 +2u, +with σ acting as the identity on the other generators of D. +(ii) σ2 equals conjugation by g on D; that is, σ2(h) = ghg−1 for all h ∈ D. +(iii) The elements z := q2g−1, θ := s2g−1 and ω := qsg−1 are in the centre +Z(D) of D. +Proof. (i) and (ii) are easy checks, and (iii) is immediate from (i) and (ii). +□ +Theorem 2.2. Retain the notation introduced above. +(i) [2, Proposition 2.7(i)] O(G) := k⟨x, u, g±1⟩ is a normal commutative Hopf +subalgebra of D, with G = ((k, +) × (k, +)) ⋊ (k∗, ×). +(ii) [2, Proposition 2.7(ii)] DO(G)+ is a Hopf ideal of D, with an isomorphism +of Hopf algebras +(2.2) +D/DO(G)+ ∼= U(sl2(k)). +(iii) [1, Theorem 3.11] The finite dimensional irreducible D-modules are the finite +dimensional irreducible U(sl2(k))-modules given by the epimorphism (2.2). +(iv) [1, Theorem 4.10] With the notation from Lemma 2.1, the centre of D is +(2.3) +Z(D) = k⟨z, ω, θ : zθ = ω2⟩, +(v) [1, Remark 2.2] D is pointed. +□ +We’ll need the following labelling of the maximal ideals of Z(D). Note that here +there are two maximal ideals of Z(D) which require particular attention. +Notation 2.3. (i) By Theorem 2.2((iv), Maxspec(Z(D)) consists of +{m(α,γ) := ⟨z −α2γ−1, ω −α, θ −γ⟩ : α ∈ k, γ ∈ k∗} ˙∪ {mβ := ⟨z −β, ω, θ⟩ : β ∈ k}. +Note that m(α,γ) can be simplified to m(α,γ) = ⟨ω − α, θ − γ⟩, while mβ = ⟨z − β, ω⟩ +when β ̸= 0. +(ii) It is easy to calculate using the definition of the counit that +D+ ∩ Z(D) = O(G)+D ∩ Z(D) = m(−16,16). +We thus denote m(−16,16) by m+. +(iii) It is clear that the singular locus of Z(D) is {m0}. +3. Ring-theoretic preparations +In this section we assemble some properties needed in the analysis of the primitive +spectrum of D. The proofs are most easily approached by viewing D as an iterated +Hopf Ore extension starting not from the base field k but from the commutative +normal Hopf subalgebra O(G) = k⟨x, u, g±1⟩ of Theorem 2.2(i). More precisely: +Proposition 3.1. D is an iterated Ore extension +(3.1) +D = O(G)[y; δ1][ζ; δ2][v; τ, δ3], +where the derivations δ1 and δ2, the automorphism τ and the τ-derivation δ3 can +be read off from the defining relations of D given in Subsection 2.1. +In particular, D is a noetherian domain. + +THE PRIME SPECTRUM OF THE DRINFELD DOUBLE OF THE JORDAN PLANE +5 +Proof. Use the proof of [2, Proposition 1.6] to show that D has basis +{gaxbucydζevf : a ∈ Z, b, . . . , f ∈ N}. +Then the form of the relations (2.1) combined with [12, Theorem 1, p.438] show +that it is indeed an Ore extension. +□ +Although it will be not needed in this paper, the description (3.1) even describes +D as an Iterated Hopf Ore Extension (IHOE), in the sense that each extension in +that formula is itself a Hopf algebra. It also shows that, by setting +deg x = deg u = deg g = deg g−1 = 0; +deg y = deg ζ = deg v = 1, +one obtains a filtration F on D with associated graded algebra +(3.2) +grF D = O(G)[y, ζ, v]. +So grF D is a commutative polynomial algebra in 6 variables with one variable +inverted. +3.1. Homological properties. In this subsection we note that D has certain use- +ful homological properties, and we begin with the relevant definitions. A ring A +is called Auslander Gorenstein if it has finite injective dimension and satisfies the +Gorenstein condition: if p < q are non-negative integers and M is a finitely gener- +ated A-module, then Extp +A(N, A) = 0 for every submodule N of Extq +A(M, A). The +ring A is Auslander regular if it is Auslander Gorenstein of finite global dimension. +Set jA(M) = min{r : Extr +A(M, A) ̸= 0} for the homological grade of M. Then an +Auslander Gorenstein ring A of finite GK dimension is called GK-Cohen-Macaulay +(or just CM), provided that jA(M)+ GKdim(M) = GKdim(A) holds for each such +M. Obviously affine commutative regular rings are both Auslander regular and +CM. +Proposition 3.2. +(i) D is Auslander regular and CM. +(ii) D is AS regular in the sense of, say, [19]. +(iii) GKdim(D) = 6 = gldim(D). +(iv) GK dimension is an exact function on finitely generated D-modules. +Proof. (i) By [7, Remark, p.157] the filtration F is Zariskian and so the result +follows from [7, Theorems 3.8, 3.9 and Remark, p. 165]. +(ii) This is immediate from (i) and [11, Lemma 6.1]. +(iii) By [22, Corollary 1.4], GKdim(D) = GKdim(grF(D)) = 6. +By Proposition 3.1 and [21, Theorem 7.5.3(i)], we have gldim(D) ≤ 6. +By +[1, Theorem 3.11] D has a finite dimensional module, say M and the CM condition +implies that M has homological dimension ≥ 6. Hence gldim(D) = 6. +(iv) Since jD is exact on finitely generated D-modules by [19, Theorem 2.3], this +follows from the CM condition. +□ + +6 +K. A. BROWN AND J. T. STAFFORD +3.2. Key lemma. The following lemma will be crucial in our analysis of the prim- +itive spectrum of D. In its proof, given an ideal B of a noetherian ring S, we denote +by +√ +B the ideal of S such that +√ +B/B is the nilradical of S/B. +Lemma 3.3. Let M be a finitely generated (right or left) D-module such that either +Annk[x](M) ̸= 0 or Annk[u](M) ̸= 0. Then +(i) there exists r ≥ 1 such that +(m+D)r ⊆ (O(G)+D)r ⊆ AnnD(M); +(ii) GKdim(M) ≤ 3. +Proof. (i) Let I := AnnD(M), an ideal of D. Assume that I ∩ k[x] ̸= 0, the proof +in the other case being exactly similar, but with k⟨u, v⟩ replacing J. One easily +confirms that every non-zero prime ideal of the Jordan plane J = k⟨x, y⟩ contains x. +Therefore, since I ∩k[x] ̸= 0, there exists N ≥ 1 such that xN ∈ I ∩k[x] ⊆ I ∩O(G). +Since O(G) is commutative, +(3.3) +x ∈ +� +(I ∩ O(G)). +Since I is an ideal of D, [v, I] ⊆ I; moreover, from the defining relations of D and +the fact that O(G) = k⟨x, u, g±1⟩, [v, O(G)] ⊆ O(G). Therefore +(3.4) +[v, I ∩ O(G)] ⊆ I ∩ O(G). +Since k has characteristic 0 it follows from (3.4) and [17, Lemma 3.20] that +(3.5) +[v, +� +(I ∩ O(G))] ⊆ +� +(I ∩ O(G)). +By (3.3) and (3.5) +[v, x] = 1 − g + xu ∈ +� +(I ∩ O(G)), +so that (1 − g) ∈ +� +(I ∩ O(G)). Then +[v, g − 1] = [v, g] = gu ∈ +� +(I ∩ O(G)), +so that u ∈ +� +(I ∩ O(G)). Since O(G)+ is generated by x, u and g − 1 we deduce +that O(G)+D ⊆ +√ +I, proving (i). +(ii) By (i) M is a finitely generated D/(O(G)+D)r-module for some r ≥ 1. Since +D/O(G)+D ∼= U(sl(2, k) by Theorem 2.2(ii), and so has GK dimension 3, (ii) +follows from this and Proposition 3.2(iv). +□ +3.3. Ore localisations of D. To help in the analysis of its primitive spectrum we +need four Ore localisations of D. The first of these is described in [1, Theorem 4.8], +and the others are similar. These sets are described as follows: +Definition 3.4. Label the following four subsets of D: +A := {qi : i ≥ 0} ˙∪ {xj : j ≥ 0}, +B := {si : i ≥ 0} ˙∪ {xj : j ≥ 0}, +C := {qi : i ≥ 0} ˙∪ {uj : j ≥ 0}, +D := {si : i ≥ 0} ˙∪ {uj : j ≥ 0}. +Lemma 3.5. (1) The elements x and u act ad-locally-nilpotently on D. Conse- +quently, {xi : i ≥ 0} and {ui : i ≥ 0} are Ore sets in D. +(2) For each Ω ∈ {A, B, C, D} the set Ω is an Ore set of regular elements of D, +and we write the corresponding localisation as D(Ω). + +THE PRIME SPECTRUM OF THE DRINFELD DOUBLE OF THE JORDAN PLANE +7 +Proof. (1) For x this is proved in [1, Lemma 4.3(i)]. The claim for u is a similar +easy consequence of the defining relations of D. +(2) Localising at the powers of q is the same as localising at the powers of q2 or +even at the powers z = q2g−1, since g is a unit. Thus, for Ω = A or Ω = C and +appealing to Lemma 2.1(iii), we can replace q by the central element z. Similarly +in the other two cases we can replace s by the central element θ. Thus in each +case we wish to localise at one central and one locally ad-nilpotent element in the +domain D. Thus it is indeed an Ore set of regular elements. +□ +Thus each of the four rings D(Ω) is a subalgebra of the quotient division algebra +Q(D) of D that contains D. As we next show, each of these rings is a localisation +of the second Weyl algebra over a commutative ring. +Notation 3.6. (i) In D(A), set pA := −2q−1x−1y, +qA := q, +tA := qx−1, and +ηA := −xq−1ζ. +(ii) In D(B), set pB := −2s−1x−1y, qB := s, tB := sx−1. ηB := −xs−1ζ. +(iii) In D(C), set pC := 2q−1u−1v, qC := q, tC := q−1u−1, ηC := −uqζ. +(iv) In D(D), set pD := 2s−1u−1v, qD := s, tD := s−1u−1, ηD := usζ. +We further set zΩ := z for Ω = A and Ω = C but zΩ := θ when Ω = B, D. +The motivation behind the above definitions becomes clear from the following +lemma. For Ω = A, this was obtained in the proof of [1, Theorem 4.8]. The claims +regarding the other elements can be checked by a similar direct calculation. +Lemma 3.7. Let Ω ∈ {A, B, C, D}. Then we have the following relations in Q(D): +[pΩ, qΩ] = 1 = [ηΩ, tΩ], with all other brackets being zero; +□ +When Ω = A the following result is given in [1, Theorem 4.8], although we give +a proof that works for all 4 cases simultaneously. +Theorem 3.8. For each Ω ∈ {A, B, C, D}, the localisation D(Ω) is a localised Weyl +algebra over its centre. More precisely: +D(Ω) = A(Ω) +2 +(k) ⊗ S(Ω), +where A(Ω) +2 +(k) denotes the localisation of the second Weyl algebra over k with gen- +erators pΩ, q±1 +Ω , ηΩ, t±1 +Ω , while S(Ω) is the commutative ring S(Ω) = k[z±1 +Ω , ω]. +Proof. The generators z, ω and θ of Z(D) are given in Lemma 2.1(iii), from which +it follows that the subalgebra S(Ω) of Q(D) is contained in the centre Z(D(Ω)). +Therefore we can consider the subalgebra +(3.6) +E(Ω) := S(Ω)⟨pΩ, qΩ, tΩ, ηΩ⟩ ⊆ D(Ω). +We claim that the inclusion (3.6) is an equality. In order to prove this, check that +given generators of DΩ are contained in E(Ω). Thus, for example, when Ω = A, one +shows that {q−1, x±1, y, ζ, g±1, u, v} ⊂ E(A), and similarly in the other cases. Thus, +E(Ω) = D(Ω), as claimed. As noted in the proof of Lemma 3.5 the localisation of +D at Ω involves inverting one central and one ad-nilpotent element of D. Thus, +by Proposition 3.2(iii) and [18, Lemma 4.7], GKdim(D(Ω) = GKdim(D) = 6. We +conclude that GKdim(E(Ω)) = GKdim(D) = 6. +On the other hand, by Lemma 3.7 E(Ω) is a factor of the ring +V(Ω) := S(Ω) ⊗k A(Ω) +2 +(k), + +8 +K. A. BROWN AND J. T. STAFFORD +which is also a domain of GK-dimension 6. So if E(Ω) were a proper factor of V(Ω), +then [21, Corollary 8.3.6] would imply that GKdim(E(Ω)) < 6, giving a contradic- +tion. +So the only possibility is that E(Ω) ∼= V(Ω) = S(Ω) ⊗k A(Ω) +2 +(k), as required. +□ +4. The primitive spectrum of D +In this section we describe the primitive spectrum of D. This splits naturally +into several cases: +• the primitive ideals not containing m+ or m0; these are the generic ones; +• the ideal m+D, which is also primitive; +• the ideal m0D, for which √m0D is a unique prime ideal P0; +• finally, P0 is also maximal. +The details are given in the next four subsections with the results being combined +in Subsection 4.5. +In this section and in Section 5.1 we will without further reference use of the +yoga for prime ideals of Noetherian rings under Ore localisation as described in, +for example, [17, Theorems 10.18 and 10.20]. We use Notation 2.3 to describe the +maximal ideals of Z(D) and Definition 3.4 to define Ore sets in D. +4.1. The generic minimal primitives. We begin by looking at the generic case. +Theorem 4.1. Let m be a maximal ideal of Z(D) with m ̸= m+ and m ̸= m0. Then +the following are true. +(i) mD is a completely prime maximal ideal of D. +(ii) The localisation of D/mD at the powers of (the image of) either x or u is +isomorphic to a localised Weyl algebra A(Ω) +2 +(k), where Ω ∈ {A, B, C, D}. +(iii) GKdim(D/mD) = 4. +(iv) mD is generated by a central regular sequence of length 2. +(v) D/mD is CM and is Auslander Gorenstein with injdim(D/mD) < 4. +Proof. (i), (ii) By Notation 2.3, m = ⟨z − α, ω − β, θ − γ⟩ with α, β, γ ∈ k and +αγ = β2. Moreover, thanks to the hypothesis on m, either (a) α ̸= 0 or (b) γ ̸= 0. +Assume (a). We prove (ii) for the localisation at the powers of x. (The arguments +for powers of u are exactly similar, but using the Ore sets C and D rather than A +and B.) Using the notation of §3.3 and applying Theorem 3.8, we see that mD(A) +is a maximal ideal of D(A). Observe that, since A := {qi, xj : i, j ≥ 0} and +z = q2g−1 ≡ α ̸= 0 mod(mD), +A(A) +2 +(k) is isomorphic to the localisation of D/mD at the powers of x. Define +Pm := mD(A) ∩ D, +so that Pm is a completely prime ideal of D with mD ⊆ Pm. By definition of Pm, +(4.1) +mD(A) = PmD(A). +We claim that in fact +(4.2) +Pm = mD. + +THE PRIME SPECTRUM OF THE DRINFELD DOUBLE OF THE JORDAN PLANE +9 +Since D is (left) noetherian there exist e1, . . . , et ∈ Pm such that Pm = mD + +�t +i=1 Dei. By (4.1), for each i = 1, . . . , t there exist fi ∈ mD and si ∈ Z≥0 such +that +(4.3) +ei = fix−si. +Define s := max{si : 1 ≤ i ≤ t} ∈ Z≥0, and +I := {τ ∈ D : Pmτ ⊆ mD}. +Thus I is an ideal of D containing mD and, by (4.3), xs ∈ I. If s = 0 then I = D; +otherwise we see from Lemma 3.3 that (m+)r ⊂ I for some r ≥ 1. Since also m ⊆ I +and m ̸= m+ by hypothesis, it follows that I = D, and (4.2) is proved. +In case (a) it remains only to prove that Pm is a maximal ideal of D. Suppose +then that J is an ideal of D with Pm ⊊ J. Then JD(A) = D(A) by the maximality +of the ideal PmD(A) of D(A). Again using the fact that q + mD is a unit of D/mD +we see that xs ∈ J for some s ≥ 1. Then, as before, Lemma 3.3 implies that J = D. +Suppose that (b) holds rather than (a). Then the element s is a unit mod mD, +so we use the same argument as for (a), but working with D(C) rather than D(A). +(iii) By (ii) and [18, Example 3.7 and Theorem 4.9] the localisation of D/mD at +the powers of x has GK dimension 4. Since ad(x) acts nilpotently on D/mD by +Lemma 3.5, it follows from [18, Theorem 4.9] that GKdim(D/mD) = 4. +(iv) Again we assume (a) that z−α ∈ m for α ∈ k\{0}, the proof in case (b) being +similar. We can begin a regular central sequence in mD with z − α. Since D is CM +of GK-dimension 6 by Proposition 3.2(i, iii), it follows from [16, Theorem 7.2(b)] +that D/(z − α)D is CM of GK-dimension 5. Moreover, by [19, Remark 2.4] the +CM property ensures that D/(z − α)D is GK-homogeneous; that is, it contains no +non-zero ideal with GK-dimension strictly less than 5. Since Z(D)/(z − α)Z(D) is +a polynomial algebra we can choose y ∈ m such that m = ⟨z − α, y⟩. If y + (z − α)D +is a zero divisor in D/(z − α)D we obtain a non-zero ideal of D/(z − α)D killed +by mD, contradicting the GK-homogeneity of D/(z − α)D in view of (iii). Thus +{z − α, y} is a regular central sequence in mD. +(v) Since D is CM by Proposition 3.2(i), R = D/mD is CM with GKdim(R) = 4 +by (iv) and two applications of [16, Theorem 7.2(b)]. The Auslander Gorenstein +property is given by (iv) and [19, §3.4, Remark (3)]. As R is simple it cannot have a +finite dimensional module. Hence injdim(R) < 4 follows from the next lemma. +□ +The following observation is well-known. +Lemma 4.2. Let R be a noetherian, Auslander Gorenstein, CM ring and write +GKdim(R) = m. Then injdim(R) ≤ m. Moreover injdim(R) = m ⇐⇒ R has a +finite dimensional representation. +Proof. Let n = injdim(R) and pick a finitely generated R-module M such that +Extn +R(M, R) ̸= 0. By the Auslander condition and the spectral sequence [19, The- +orem 2.2] j(Enn(M)) = n for Enn = Extn(Extn(M, R), R). By the CM property +GKdim(Enn(M)) = m − n and the result follows easily. +□ + +10 +K. A. BROWN AND J. T. STAFFORD +4.2. Non-generic minimal primitives (I) - m+. The next case to consider is mD +for m = m+, as we do here. Recall from Notation 2.3(ii) that m+ = D+ ∩ Z(D) = +⟨ω + 16, θ − 16⟩. +We start with a subsidiary result, which works for any field k of characteristic +zero. +Theorem 4.3. D is a Jacobson ring that satisfies the Nullstellensatz, in other +words: +(i) every prime ideal of D is an intersection of primitive ideals; +(ii) for every simple D-module M, EndD(M) is algebraic over k. In particular, +every primitive ideal of D contains a maximal ideal of Z(D). +Proof. By (3.2), D has a filtration F such that the associated graded ring grF(D) is +a commutative affine ring. Hence by [22, Corollary 1.7] there is a second filtration +G by finite dimensional k-subspaces of D such that grF(D) is also a commutative +and affine ring. The result now follows from [4, Theorem 0.4]. +□ +Theorem 4.4. +(i) m+D is a completely prime, primitive ideal of D. +(ii) The localisation of D/m+D at the powers of x or the powers of u is a +localisation of the Weyl algebra A2(k) at powers of a generator. +(iii) m+D is generated by a regular central sequence of length 2. +(iv) D/m+D is Auslander Gorenstein and CM with +GKdim(D/m+D) = 4 = injdim(D/m+D). +(v) Every prime ideal P of D which strictly contains m+D satisfies +O(G)+D ⊆ P, +so the space of such primes P is homeomorphic to Spec(U(sl(2, k)). +Proof. Recall that qA = q. Since q2 ≡ 16g ̸≡ 0 mod(m+D), Theorem 3.8 implies +that m+D(A) is a maximal ideal of D(A), with D/m+D(A) ∼= A(A) +2 +(k). Therefore, +defining P+ := m+D(A) ∩ D, we deduce that P+ is a completely prime ideal of D +with +(4.4) +m+D ⊆ P+. +We will eventually show that (4.4) is an equality. +As in the proofs of Theorem 4.1(i),(ii), let I be the right annihilator in D of +P+/m+D. Then I contains m+D and a power of x, and hence, by Lemma 3.3, +(4.5) +(O(G)+D)r ⊆ I +for some r ∈ Z≥1, +In particular, GKdim(D/I) ≤ 3 by Lemma 3.3(ii). +Therefore, by [18, Proposi- +tion 5.1(d)] +(4.6) +GKdim(P+/m+D) ≤ 3. +Recall that GKdim(A(A) +2 +(k)) = 4 by [18, Example 7.3 and Theorem 4.9], so that +(4.7) +GKdim(D/P+) = 4 +by [18, Theorem 4.9]. Thus, from (4.6), (4.7) and Proposition 3.2(iv) it follows that +(4.8) +GKdim(D/m+D) = 4. + +THE PRIME SPECTRUM OF THE DRINFELD DOUBLE OF THE JORDAN PLANE +11 +By Proposition 3.2, D is CM and Auslander regular, with gldim(D) = 6 = +GKdim(D). It therefore follows from the CM property of D together with (4.8) +that +(4.9) +jD(D/m+D) = 6 − 4 = 2. +From (4.9) and [8, Proposition 3.6] we deduce that the maximum length of a reg- +ular sequence of elements of m+ on D is precisely 2; in particular any choice of +a generating pair of elements of m+, for example, {z − 16, ω + 16}, is a regular +sequence on D. Therefore, by two applications of [16, Theorem 7.2(b)], +(4.10) +D/m+D is CM of GK-dimension 4. +Similarly, two applications of [19, §3.4, Remark (3)] show that D/m+D is Auslander +Gorenstein. By Lemma 3.3(i) and Theorem 2.2(ii), U(sl(2, k)) ∼= D/DO(G)+ is a +factor of D/m+D and so D/m+D has a non-zero finite dimensional module, M. +Thus, by Lemma 4.2, injdim(D/m+D) = 4. +By [19, Remark 2.4], again, the CM property for D/m+D implies that D/m+D +is GK-homogeneous. Therefore we may conclude from (4.6) that P+ does indeed +equal m+D. +This proves (i) - (iv), with the exception of showing that m+D is +primitive. +(v) Let Q be a prime ideal of D with m+D ⊊ Q. As already noted, q is congruent +to a unit mod m+D. Then QD(A) = D(A) by (ii), so Q must contain a power of x. +Hence, by Lemma 3.3, O(G)+D ⊆ Q, as required. +Finally, to see that m+D is primitive note that (v) shows that it is locally closed. +Hence it is primitive by Theorem 4.3(i). +□ +4.3. Non-generic minimal primitives (II) - m0. In this subsection we begin +our study of the ideal m0D. +Recall the definition of q, s from (2.1) and, from +Notation 2.3(iii), that m0 := ⟨q2g−1, qsg−1, s2g−1⟩ is the unique singular point of +Maxspec(Z(D)). Clearly the right ideal +(4.11) +P0 := qD + sD +is a two-sided ideal of D since q and s are normal in D by Lemma 2.1. Moreover, +m0D = P 2 +0 ⊂ P0 ⊆ +� +m0D. +As part of the next proposition we see that P0 is completely prime, so the second +inclusion above is an equality. +In fact P0 is a maximal ideal, but this is more +difficult to prove, and is delayed until §4.4. +Proposition 4.5. Retain the above notation, and set T := D/P0. +(i) T is a localisation of a 4-step iterated Ore extension of k, namely +T = +� +(k[u, x]⟨(ux + 2)−1⟩)[y; ∂1] +� +[v; σ, ∂2], +where u and x commute, +∂1(u) = − 1 +2ux − 2, +∂1(x) = − 1 +2x2, +∂2(u) = − 1 +2u2, +∂2(x) = 3 +2ux + 2, +∂2(y) = 3 +2uy − 2, +and σ(y) = y + 1 +2x, with σ(x) = x and σ(u) = u. +(ii) {q, s} forms a regular normal sequence of generators of P0. +(iii) gldim(T ) ≤ 4 = GKdim(T ). + +12 +K. A. BROWN AND J. T. STAFFORD +(iv) T is CM and is an Auslander regular domain. +Proof. Throughout the proof we abuse notation by simply denoting the image in +T of an element ω of D by ω when no confusion seems likely. +(i),(ii) Since q := ux + 2(1 + g) and q ≡ 0 mod(P0), we can write +(4.12) +g ≡ − 1 +2(ux + 2) mod(P0), +so that +(4.13) +ux + 2 is a unit in T. +Using (4.12) we find that, mod(P0), +s := xv + uy + (− 1 +2ux + g − 1)ζ − 2(g + 1) ≡ xv + uy + 2gζ − 2g − 2, +so that, since s ∈ P0, +(4.14) +ζ ≡ − 1 +2g−1(ux + xv + uy) +mod(P0) +It follows from (4.12), (4.13) and (4.14) that +(4.15) +T = k⟨u, x, (ux + 2)−1, y, v⟩. +The relations for D given in §2.1 immediately imply the following relations for the +generators for T listed in (4.15) +[u, x] = 0, +[y, x] = − 1 +2x2, +[v, x] = 3 +2ux + 2, +[y, u] = − 1 +2ux − 2, +[v, u] = − 1 +2u2, +[v, y] = 3 +2uy + 1 +2xv − 2. +Clearly the iterated Ore extension of k[u, x]⟨(ux + 2)−1⟩ defined in (i), which we +temporarily label �T, satisfies precisely these relations, so there is an algebra epi- +morphism Φ from �T onto T . +We next show that Φ is an isomorphism, which we do by computing GKdim(T ). +First note that GKdim( �T) = 4 by [18, Theorem 12.3.1], since it is a PBW extension +in 2 variables of k[u, x]⟨(ux + 2)−1⟩. Thus, certainly GKdim(T ) ≤ 4. On the other +hand D is CM of GK-dimension 6 by Proposition 3.2(i, iii). Hence, because q is +a regular normal element of D by Lemma 2.1, D/qD is CM of GK-dimension 5 +by [16, Theorem 7.2(b) and its proof]. Moreover D/qD is GK-homogeneous by +[19, Remark 2.4]. Since GKdim(T ) ≤ 4, this ensures that +(4.16) +s cannot be a zero-divisor mod qD. +Since P0 := qD + sD, a second application of [16, Theorem 7.2(b) and its proof] +yields GKdim(T ) = 4 and also shows that +(4.17) +T is CM. +Since �T is a domain, the equality GKdim( �T) = 4 = GKdim(T ), combined with +[18, Proposition 3.15], shows that �T = T . Thus (i) is proved, with (ii) also following +thanks to (4.16). +(iii) By (i), T is a 2-step iterated Ore extension of k[u.x]⟨(ux + 2)−1⟩, and so two +applications of [21, Theorem 7.5.3(i)] gives gldim(T ) ≤ 4. +(iv) That T is a domain is clear from (i), while the CM property was proved in +(4.17). The Auslander Gorenstein property holds for D by Proposition 3.2(i). Thus, +by (ii) and two applications of [16, Theorem 7.2(a)], T is also Auslander Gorenstein +and it is then Auslander regular by (iii). +□ + +THE PRIME SPECTRUM OF THE DRINFELD DOUBLE OF THE JORDAN PLANE +13 +We remark that, by Lemma 4.2 and Theorem 2.2(iii) it follows that gldim(T ) < 4. +We do not know the exact value of gldim(T ). +4.4. Maximality of P0. Let T := D/P0 as in Proposition 5.3. Define also the +following subalgebras of T : +R := k⟨u, x, (ux + 2)−1⟩, and S := R[y; ∂1], +so that T = S[v; σ, ∂2]. It is important to note that, by the formulæ in Proposi- +tion 4.5, R is preserved by the σ-derivation ∂2. Moreover, since σ|R is the identity, +∂2 actually restricts to a derivation on R. +It is much easier to determine when an Ore extension is simple if the ring is a +differential operator ring, in the sense that the defining automorphism is actually +the identity. Thus we will reduce to that case. The idea follows from Lemma 2.1 +which shows that σ2 is given by the inner automorphism τg in the sense that +σ(s) = τg(s) = gsg−1 for suitable g ∈ S. We will therefore extend R, S and T by +√g and show that σ is then inner, and so can be removed. The details are given in +the next few results, culminating in Proposition 4.9. +Notation 4.6. In the algebraic closure of R, set h = (ux + 2)− 1 +2 . Write �R = +R⟨h⟩ = k⟨u, x, h, h−1⟩. We extend the ∂i to derivations on �R by the usual rules for +fractional powers: +∂(h) = (−1 +2)(ux + 2)−1h∂(ux + 2), +for ∂ = ∂1, ∂2. Set �S = �R[y; ∂1]. Finally, we can extend σ to �R and �S by setting +σ(h) = h. Then both σ and ∂2 are naturally defined on �S as an automorphism, +respectively σ-derivation and so �T = �S[v; σ, ∂2] is a well-defined Ore extension of +�S. +The following observation will prove useful. +Lemma 4.7. �S is a free left and right S module on basis {1, h}. Similarly, �T is a +free left and right T module on basis {1, h}. +Proof. As h2 = (ux + 2)−1 ∈ R, the construction of �R ensures that �R is a free left +and right R-module on basis {1, h}. We can then write +�S = +∞ +� +i=0 +�Ryi = +� +Ryi ⊕ +� +Rhyi = +� +Ryi ⊕ +� +Ryih. +Collecting terms shows that �S = S ⊕ Sh. As S is a domain this is necessarily a +direct sum of free modules. The same argument works for �T. +□ +Lemma 4.8. On �S, σ is the inner automorphism τh−1; thus σ(f) = h−1fh for +f ∈ �S. +Proof. Since �R is a commutative ring on which σ is the identity, the lemma holds +trivially on �R. It therefore just remains to check that the automorphisms agree on +y. To prove this, we rewrite h−1yh as follows. + +14 +K. A. BROWN AND J. T. STAFFORD +h−1yh = (ux + 2) +1 +2 y(ux + 2)− 1 +2 += (ux + 2) +1 +2 (ux + 2)− 1 +2 y + (ux + 2) +1 +2 · ∂1 +� +(ux + 2)− 1 +2 � += y + (ux + 2) +1 +2 (− 1 +2)(ux + 2)− 3 +2 · ∂1((ux + 2)) += y − +1 +2(ux + 2)−1� +(− 1 +2ux − 2)x − u( 1 +2x2) +� += y − +1 +2(ux + 2)−1� +−(ux + 2)x +� += y + 1 +2x = σ(y); +as required. +□ +Proposition 4.9. Set α = hv. Then �T is the Ore extension �T = �S[α; �∂2] where �∂2 +is the derivation of �S defined by �∂2(s) = h∂2(s) for s ∈ �S; thus +�∂2(u) = − 1 +2hu2, +�∂2(x) = h( 3 +2ux + 2) +and +�∂2(y) = h(( 3 +2uy − 2). +As such, �T is a noetherian domain. +Proof. This is a formal computation. Indeed, for s ∈ �S, Lemma 4.8 implies that +σ(s) = h−1sh. Equivalently, +(4.18) +αs = hvs = hσ(s)v + h∂2(s) += hh−1shv + h∂2(s) = sα + h∂2(s). +Therefore, since �T = �S[v; σ, ∂2] = � �Svi, we see that �T = � �Sαi. Since �T is a +domain, combining this with (4.18) and [12, Theorem 1, p.438] gives the desired +conclusion. +□ +Our next aim will be to show that the ring �T is a simple domain, after which it +is easy to prove the same conclusion for T . We start with some preparatory results. +Lemma 4.10. If there exists a non-zero (∂1, �∂2)-invariant ideal I in �R, then there +exists a non-zero (∂1, �∂2)-invariant prime ideal P in �R. +Proof. Using [17, Lemma 3.18(b)] twice, clearly I �S is a proper non-zero ideal of �S +and then I �T is a proper nonzero ideal of �T. Pick a prime ideal Q ⊇ I �T. Then, by +[17, Lemmata 3.18 and 3.21], twice, Q1 = Q ∩ �S is a �∂2-invariant prime ideal of �S +and hence Q2 = Q1 ∩ �R is a ∂1-invariant prime ideal of �R. However, since �R and +Q1 are both �∂2-invariant, so is Q2. Thus, P = Q2 is the desired prime ideal. +□ +Proposition 4.11. There is no proper, non-zero (∂1, �∂2)-invariant ideal I in �R. +Proof. Suppose that there exists such an ideal I. By Lemma 4.10 we can and will +assume that I is a prime ideal. Suppose, first, that (xu + λ) ∈ I, for some λ ∈ k. +Then +I ∋ ∂1(xu + λ) = (− 1 +2ux − 2)x − ( 1 +2x2)u = −(ux + 2)x +and +I ∋ �∂2(xu + λ) = h +� +− 1 +2u2x + ( 3 +2ux + 2)u +� += h +� +ux2 + 2u +� += h(ux + 2)u. + +THE PRIME SPECTRUM OF THE DRINFELD DOUBLE OF THE JORDAN PLANE +15 +As (xu + 2)−1 = h2 ∈ �R, clearly λ ̸= 2 and so the two equations imply that x ∈ I, +respectively u ∈ I. Thus, I = x �R + u �R. But, now I ∋ ∂1(u) = − 1 +2ux − 2 and so +I = �R, a contradiction. We conclude that +(4.19) +I ∩ C = ∅ +for C = {(xu + λ) : λ ∈ k∗} +Since I is a prime ideal it follows that C ⊆ C(I) and hence that IC is a proper prime +ideal of the localisation �RC. +Next, if IC ∋ f = f(u) for some f(u) ∈ k[u], then IC ∋ ∂1(f) = − 1 +2(ux + 4) df +du. +Hence df +du ∈ IC. By induction on deg f, this implies that IC = �RC, a contradiction. +Thus IC ∩ k[u]∗ = ∅ and so we can further localise at S = k[u]∗ and conclude that +ICS is a proper prime ideal of �RCS. Now consider �RCS. We have �R = k⟨u, x, h, h−1⟩ +and h−2 = (ux + 2) whence x = u−1(h−2 − 2). Thus �RCS = �RSC is a localisation +of k(u)[h, h−1]. +The advantage of working in �RCS is that we can simplify our derivation �∂2. On +�R and �RCS write ∂u = +∂ +∂u and ∂x = +∂ +∂x. Then, as derivations on either ring, +∂1 = −( 1 +2xu + 2)∂u − +1 +2x2∂x +while +�∂2 = − 1 +2hu2∂u + h( 3 +2ux + 2)∂x. +We now set µ := −hu2(ux + 4)−1 and take +�∂′ +2 := �∂2 + µ∂1 = +� +− 1 +2hu2 + µ(− 1 +2ux − 2) +� +∂u + +� +h( 3 +2ux + 2) − 1 +2x2µ +� +∂x. +This element µ has been chosen so that the coefficient of ∂u in �∂′ +2 is +− 1 +2(ux + 4)−1� +hu2(ux + 4) − (ux + 4)hu2� += 0. +Therefore, +�∂′ +2 = +� +h( 3 +2ux + 2) + +1 +2hx2u2(ux + 4)−1� +∂x += (ux + 4)−1h +� +( 3 +2ux + 2)(ux + 4) + 1 +2x2u2� +∂x += (ux + 4)−1h +� +2u2x2 + 8ux + 8 +� +∂x += β∂x +for +β := 2(xu + 4)−1(ux + 2)2h. +Since ICS is invariant under both ∂1 and �∂2, it is also invariant under �∂′ +2. Since +β is a unit in �RCS, it follows that +(4.20) +ICS is also invariant under β−1 �∂′ +2 = ∂x. +Thus, by (4.20) and the expression given above for ∂1, ICS is invariant under +( 1 +2ux + 2)∂u, and therefore under ∂u since 1 +2ux + 2 is a unit. So ICS is invariant +under ∂u and ∂x. Since �RCS is a localisation of k[u, x] this forces ICS = �RCS, giving +the required contradiction. +□ +In order to pass between T and �T we need: +Lemma 4.12. If �T is a simple ring then so is T . + +16 +K. A. BROWN AND J. T. STAFFORD +Proof. Suppose that T has a proper ideal J. Then X = �T/J �T is a (T, �T)-bimodule. +Moreover, by Lemma 4.7 �T is a finitely generated left T -module and so X is a +finitely generated left T -module; say X = �r +i=1 T xi. Then, as �T is an Ore domain, +ann � +T (X) = � +i ann � +T (xi) ̸= 0. Since �T is a simple ring this implies that ann � +T (X) = +�T and hence that X = 0. In other words, J �T = �T. +On the other hand, by Lemma 4.7, �T = T + T h is a free left T -module and so +J �T = J ⊕ Jh ̸= �T. This contradiction proves the lemma. +□ +We now put everything together and prove the main result of this subsection. +Theorem 4.13. T is a simple ring. +Proof. By Lemma 4.12 it suffices to prove that �T is simple. By [21, Theorem 1.8.4] +applied to �T = �S[α; �∂2], we need to prove +(a) �∂2 is not an inner derivation on �S, and +(b) �S has no proper �∂2-invariant ideals. +Now, as ∂1(x) = − 1 +2x2, the right ideal x�S is a proper two-sided ideal of �S. As +such, it is preserved by any inner derivation of �S. But �∂2(x) = h( 3 +2ux + 2) ̸∈ x�S, +this means �∂2 cannot be an inner derivation of �S and so (a) holds. +Suppose that �S has a proper �∂2-invariant ideal I. Then, by [17, Lemma 3.18], +K = I ∩ �R is a ∂1-invariant ideal of �R, while by [17, Lemma 3.19], K ̸= 0. Since +both I and �R are both �∂2-invariant, so is K. In other words, K is a proper (∂1, �∂2)- +invariant ideal of �R. This contradicts Proposition 4.11. Thus (b) holds and so +[21, Theorem 1.8.4] implies that �T is simple. +□ +Remark 4.14. We end the subsection by noting that �T is obviously birational to +the Weyl algebra A2. We do not know if the same is true for T itself. +4.5. The shape of the primitive spectrum of D. In this subsection we combine +the earlier results of this section to prove Theorem 1.1. By Theorem 4.3, every +primitive ideal P of D contains a maximal ideal of Z(D). Thus Privspec(D) is the +disjoint union +(4.21) +Privspec(D) = +˙� +m∈Maxspec(Z(D))V(m) +where V(m) = {P ∈ Privspec(D) : m ⊆ P}. There are thus 3 cases, corresponding +to §§4.1, 4.2 and 4.3. +(I) V(m), where m ∈ Maxspec(Z(D)) with m ̸= m+ and m ̸= m0. By Theorem 4.1, +V(m) = {mD} is a single generic maximal ideal of D. Moreover D/mD is bira- +tionally equivalent to the second Weyl algebra, with other properties as listed in +that theorem. +(II) V(m+). By Theorem 4.4, this consists of m+D, together with +V(O(G)+D) := {P ∈ Privspec(D) : O(G)+D ⊂ P}, +which is homeomorphic to Privspec(U(sl(2, k))) by Theorem 2.2(ii). +Recall that Privspec(U(sl(2, k))) is composed of the co-Artinian maximal ideals +{Mn : n ∈ Z≥1}, where Mn = Ann(Vn), Vn being the n-dimensional irreducible + +THE PRIME SPECTRUM OF THE DRINFELD DOUBLE OF THE JORDAN PLANE +17 +U(sl2(k))−module, together with the minimal primitives of U(sl(2, k)); that is, the +ideals (Ω − λ)U(sl(2, k)) : λ ∈ k}, where Ω is the Casimir element. Each Mn +contains one such minimal primitive and each minimal primitive is contained in +at most one Mn; the remaining minimal primitives are also maximal. Note that +O(G)+D is prime but not primitive since D/O(G)+D ∼= U(sl(2, k)) and this domain +satisfies the Nullstellensatz and has non-trivial centre k[Ω]. +(III) V(m0). This is the singleton {P0 = qD + sD = √m0}, by Proposition 4.5 and +Theorem 4.13. +5. Prime ideals and the Dixmier-Moeglin equivalence +In this section we prove Theorem 1.2 from the introduction, which describes the +prime ideals of D, and we discuss the Dixmier-Moeglin equivalence for D. +5.1. The prime spectrum of D. We need the following lemmas for the proof of +the main result, Theorem 5.3. +Lemma 5.1. Let P be a nonzero prime ideal of D. Then P ∩ Z(D) ̸= {0}. +Proof. If xi ∈ P for some i ≥ 0 then O(G)+D ⊆ P by Lemma 3.3 applied with +M = D/P, and therefore m+ = O(G)+D ∩ Z(D) ⊆ P, proving the lemma for P. +So we may assume that {xi : i ≥ 0} ∩ P = ∅. Similarly, we may assume that +{qj : j ≥ 0} ∩ P = ∅, since otherwise 0 ̸= qng−2n ∈ P ∩ Z(D) for some n ≥ 0 and +again the result follows for P. +Hence, using Notation 3.6 and Theorem 3.8, PD(A) survives as a non-zero proper +ideal of D(A) = D⟨q−1, x−1⟩ = A(A) +2 +(k) ⊗k S(A), where A(A) +2 +(k) is a localised Weyl +algebra and S(A) = k[z±1, ω]. In particular, +(5.1) +PD(A) = (PD(A) ∩ S(A))D(A). +By [17, Theorem 10.20] and the discussion in the first paragraph of this proof, +P = PD(A) ∩ D, and therefore +(5.2) +P ∩ Z(D) = PD(A) ∩ Z(D) = (PD(A) ∩ S(A)) ∩ Z(D). +Since the Z(D)-module S(A)/Z(D) is {zi}-torsion, that is {qig−2i}-torsion, it fol- +lows from (5.1) and (5.2) that P ∩ Z(D) ̸= {0} as required. +□ +Note that, since k is algebraically closed of characteristic 0, the defining relation +zθ = ω2 of Z(D) can be rewritten using a linear change of variables as the quadratic +form X2 + Y 2 = Z2. Thus a proof of the next result can be found at [15, p.51 and +Proposition 11.4]. +Lemma 5.2. All height one primes of Z(D) are principal except p1 := ⟨z, ω⟩ and +p2 := ⟨θ, ω⟩. +□ +Here is the main result of this section, using in (ii) the notation of Lemma 5.2. +This proves Theorem 1.2 from the introduction. +Theorem 5.3. Let P be a prime but not primitive ideal of D. +(i) There are the following three possibilities for P. +(a) P = {0}. +(b) P = O(G)+D. + +18 +K. A. BROWN AND J. T. STAFFORD +(c) P has height one and is minimal over (P ∩ Z(D))D for a height one +prime ideal P ∩ Z(D) of Z(D). +(ii) In case (c), if P ∩ Z(D) = pi for i = 1, resp. i = 2, then P = qD, resp. +P = sD. The remaining primes in case (c) are precisely the set +{P : P = fD}, +as f ranges through the equivalence classes of irreducible elements of Z(D) +other than the associates of z, ω, θ. +Proof. Note first that {0} is completely prime by Proposition 3.1, and is not prim- +itive, because D satisfies the Nullstellensatz by Theorem 4.3 and Z(D) ̸= k. This +covers case (a). +Let P be a non-zero prime but not primitive ideal of D. By Lemma 5.1, +{0} ̸= p := P ∩ Z(D). +If p = m+ then Theorem 4.4 together with the discussion at §4.5(II) shows that +the only possibility is P = O(G)+D, which is completely prime but is again not +primitive thanks to the Nullstellensatz, since Z(U(sl(2, k))) ̸= k. This is case (b). +If p = m0 then P = P0, which is maximal by Theorem 4.13, so this case can’t +happen. Similarly, p is any maximal ideal of Z(D) apart from m+ or m0, then P = +pD is a maximal ideal of D by Theorem 4.1(i), which again gives a contradiction. +So we are left with the case when p has height one. Assume first that p = fZ(D) +is principal. Then, by Lemma 5.2, z = q2g−1 /∈ P, and {xi : i ≥ 0} ∩ P = ∅ by +Lemma 3.2 Therefore, using Notation 3.6 and Theorem 3.8 +pD(A) = (P ∩ S(A))D(A) = PD(A). +We claim that P = pD. +To see this, note that pD = fD is principal, so that +D/pD is CM of GK-dimension 5, by [16, Theorem 7.2(b) and its proof], and GK- +homogeneous by [19, §3.4, Remark (3)]. Now P/pD it is killed by pD and by a +power of q or a power of x, and so has GK-dimension less than 5, respectively by +Theorem 4.1(iii) and 4.4(iv) or by Lemma 3.2. This forces P/pD = {0} and so +P = pD, as claimed. +Suppose finally that p = p1 or p = p2. In the first case, since q is a normal +element of D by Lemma 2.1, q ∈ √pD. Thus +(5.3) +qD ⊆ P. +We claim that (5.3) is an equality. To see this, note that s /∈ P, since otherwise +P ∩Z(D) = m0, which is ruled out by hypothesis. Moreover {xj : j ≥ 0}∩P = ∅ by +Lemma 3.2. So we can localise at the Ore set B = {sixj : i, j ≥ 0} of Definition 3.4 +and pass to the localised Weyl algebra D(B) = A(B) +2 +(k) ⊗ S(B) of Theorem 3.8. +However, PD(B) and qD(B) have the same intersection with the centre S(B), namely +ωθ−1S(B) = p1S(B). Therefore PD(B) = qD(B) since the ideals of D(B) are centrally +generated. Therefore P/qD is B-torsion, so, if it is not zero, it contains a nonzero +element which is either killed by q and by s, or by q and x. As in the previous +paragraph D/qD is GK-homogeneous of GK-dimension 5, and so has no such non- +zero torsion submodule, proving that (5.3) is an equality. +If p = p2 then the argument to show that P = sD is similar, but using the Ore +set A; it is left to the reader. +□ + +THE PRIME SPECTRUM OF THE DRINFELD DOUBLE OF THE JORDAN PLANE +19 +5.2. The Dixmier-Moeglin equivalence. The following gives evidence in favour +of [6, Conjecture 1.3], which proposes that an affine noetherian Hopf C-algebra of +finite GK dimension should satisfy the Dixmier-Moeglin equivalence. See [5,10] for +definitions and background. +Corollary 5.4. D satisfies the Dixmier-Moeglin equivalence. +Proof. We check first using the description of the primitive spectrum in §4.5 that +every primitive ideal is locally closed. For classes (I) and (III) this is clear since +all these primitive ideals are maximal. The primitive ideals in (II) are homeomor- +phic to the primitive spectrum of U(sl(2, k)), and the latter algebra satisfies the +equivalence by [23]. Thus, by [10, Lemma II.7.15], it only remains to show that +every rational prime ideal P is primitive, where P is rational if the centre of the +Goldie quotient algebra of D/P is k. The non-primitive prime ideals are listed in +Theorem 5.3 and it is easy to check case by case that none of them is rational. +□ +Corollary 5.4 proves Theorem 1.3(c). With one exception, parts (a) and (b) of +that theorem are proved in the results of the last two sections that describe the +prime ideals of D. The exception is the claim that all the completely prime factors +of D (with the possible exception of D/P0, as noted in Remark 4.14) are birationally +equivalent to Weyl algebras. For the primitive ideals P strictly containing O(G)+D +this follows from [13, Remarque 7.1]. For the other prime ideals, this is clear from +the description of the prime ideals in the last two sections. +Based on little more than the known results and counterexamples for group +algebras and enveloping algebras, the theorem [6] for the cocommutative case, the +recent work of Sierra and Walton on the noetherian property for enveloping algebras +[25], together with the above result and other isolated examples, we are tempted +to propose the following conjecture as a strengthening in the pointed setting of [6, +Conjecture 1.3], as much in the hope of stimulating the discovery of counterexamples +as in expectation of a positive result. +Conjecture 5.5. Let H be an affine noetherian pointed Hopf C-algebra. Then the +following are equivalent: +(1) GKdim(H) is finite. +(2) H satisfies the Dixmier-Moeglin Equivalence. +(3) The group G(H) of grouplikes of H is nilpotent-by-finite. +Thanks to a famous result of Roseblade [24] for group algebras, the implication +(2) =⇒ (3) fails when k is a finite field. +References +[1] N. Andruskiewitsch, F. Dumas, and H. M. Pena Pollastri, On the double of the Jordan plane, +Ark. Mat. 60 (2022), 213-229. +[2] N. Andruskiewitsch and H. M. Pena Pollastri, On the restricted Jordan plane in odd charac- +teristic, J. Algebra Appln. 20 (2021), no. 2140012. +[3] +, On the finite-dimensional representations of the double of the Jordan plane, +arXiv2211.01581 (2022). +[4] M. Artin, L. W. Small, and J. J. Zhang, Generic flatness for strongly noetherian rings, J. +Algebra 221 (1999), 579-610. +[5] J. Bell, On the importance of being primitive, Rev. Colombiana Mat. 53 (2019), 87-112. +[6] J. Bell and W. H. Leung, The Dixmier-Moeglin equivalence for cocommutative Hopf Algebras +of finite Gelfand-Kirillov Dimension, Alg. Rep. Theory 17 (2014), 1843-1852. + +20 +K. A. BROWN AND J. T. STAFFORD +[7] J.-E. Bjork, The Auslander condition on noetherian rings, Seminaire Malliavin, Lecture Notes +in Math. 1404 (1989), 137-173. +[8] K. A. Brown, Unruffled extensions and flatness over central subalgebras, J. Algebra 284 +(2005), 771-800. +[9] K. A. Brown, M. Couto, and A. Jahn, The finite dual of commutative-by-finite Hopf algebras, +Glasgow Math. J. (2022), 1-28. +[10] K. A. Brown and K. R. Goodearl, Lectures on Algebraic Quantum Groups, Advanced Courses +in Math. CRM Barcelona, Birkhauser, 2002. +[11] K. A. Brown and J. J. Zhang, Dualising complexes and twisted Hochschild (co)homology for +noetherian Hopf algebras, J. Algebra 320 (2008), 1814-1850. +[12] P. M. Cohn, Algebra, Vol. II, Wiley, 1977. +[13] J. Dixmier, Quotients simples de l’alg`ebre enveloppante de sl2, J. Algebra 24 (1973), 551-564. +[14] Y. Doi and M. Takeuchi, Multiplication alteration by two-cocycles - the quantum version, +Comm. in Algebra 22 (1994), 5715-5732. +[15] R. M. Fossum, The Divisor Class Group of a Krull Domain, Ergebnisse der Mathematik und +ihrer Grenzgebiete, vol. 74, Springer, 1973. +[16] K. R. Goodearl and T. L. Lenagan, Primitive ideals in quantum SL3 and GL3, Contemp. +Math. 562 (2012), 115-140. +[17] K. R. Goodearl and R. W. Warfield, An Introduction to Noncommutative Noetherian rings, +Second edition, London Math. Soc. Student Texts, vol. 61, Cambridge University Press, 2004. +[18] G. R. Krause and T. H. Lenagan, Growth of Algebras and Gelfand-Kirillov Dimension, Re- +vised Edition, Graduate Studies in Math., vol. 22, Amer. Math. Soc., 2000. +[19] T. Levasseur, Some properties of non-commutative regular graded rings, Glasgow Math. J. +34 (1992), 277-300. +[20] K. Li and G. Liu, The finite duals of affine prime regular Hopf algebras of GK-dimension +one, arXiv2103.00495 (2021). +[21] J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, Revised edition, Grad- +uate Studies in Mathematics, vol. 30, Amer. Math. Soc., Providence, RI, 2001. +[22] J. C. McConnell and J. T. Stafford, Gelfand-Kirillov dimension and associated graded mod- +ules, J. Algebra 125 (1989), 197-214. +[23] C. Moeglin, Id´eaux primitifs d´es alg`ebres enveloppantes, J. Math. Pures Appl. 59 (1980), +265-336. +[24] J. E. Roseblade, Group rings of polycyclic groups, J. Pure Appl. Algebra 3 (1973), 307-328. +[25] S. Sierra and C. Walton, The universal enveloping algebra of the Witt algebra is not noether- +ian, Adv. Math. 262 (2014), 239-260. +School of Mathematics and Statistics, University of Glasgow, Glasgow G12 8QQ, +Scotland +Email address: ken.brown@glasgow.ac.uk +School of Mathematics, The University of Manchester, Manchester M13 9PL, Eng- +land +Email address: Toby.Stafford@manchester.ac.uk + diff --git a/89E3T4oBgHgl3EQfSAmG/content/tmp_files/load_file.txt b/89E3T4oBgHgl3EQfSAmG/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..918863816b8dcbfeb7417c76db3ea74420b95db6 --- /dev/null +++ b/89E3T4oBgHgl3EQfSAmG/content/tmp_files/load_file.txt @@ -0,0 +1,1122 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf,len=1121 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='04428v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='QA] 11 Jan 2023 THE PRIME SPECTRUM OF THE DRINFELD DOUBLE OF THE JORDAN PLANE K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' BROWN AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' STAFFORD Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' The Hopf algebra D which is the subject of this paper can be viewed as a Drinfeld double of the bosonisation of the Jordan plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Its prime and primitive spectra are completely determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' As a corollary of this anal- ysis it is shown that D satisfies the Dixmier-Moeglin Equivalence, leading to the formulation of a conjecture on the validity of this equivalence for pointed Noetherian Hopf algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Throughout, k will denote an algebraically closed field of characteristic 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' The Hopf k-algebra D of the title was defined and some initial properties were derived in [2], with further results in [1, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Our focus here is on the prime and primitive spectra of D, which we completely determine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' The Hopf algebra D is a pointed affine noetherian domain of Gelfand-Kirillov dimension 6 whose definition is recalled in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' It is a beautiful algebra with a number of striking properties which make it worthy of study from several perspectives, three of which we briefly outline in §§1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='3-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' First we summarise our results and explain where they are located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' It was proved in [1, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='10], and explained in detail here in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='1 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2(iv), that the centre Z(D) of D is generated by elements z, ω and θ with zθ = ω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Thus Maxspec(Z(D)) has one singular point, namely m0 := ⟨z, ω, θ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' There is one other distinctive maximal ideal of the centre, namely m+ := Z(D) ∩ D+ = ⟨z − 16, ω + 16, θ − 16⟩, where D+ is the augmentation ideal of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Finally, let K denote the kernel of the Hopf algebra surjection π : D −→ U(sl(2, k)), mentioned in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='3, so K is a Hopf ideal which is described in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2(i),(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' The main results of this paper are given by the following theorems and give a complete description of the prime and primitive ideals of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Retain the above notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' The primitive ideals of D are: (I) the maximal ideals mD for m ∈ Maxspec(Z(D)), with m ̸= m0, m+;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (II) the primitive ideals containing m+D, namely m+D itself together with P := π−1(Privspec(U(sl(2, k))));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Primary 16T05, 16D25;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Secondary 16T20, 16S40, 17B37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Both authors are partially supported by Leverhulme Emeritus Fellowships, respectively EM- 2017-081 and EM-2019-015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' The first author thanks Nicolas Andruskiewitsch, Ivan Angiono and Hector Pena Pollastri for helpful comments and for sharing early versions of their work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' 1 2 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' BROWN AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' STAFFORD (III) the unique prime ideal P0 containing m0D, which has m0D = (P0)2 ⊊ P0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' In the above notation, the non-primitive prime ideals of D are: (A) {0}, K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (B) the principal prime ideals pD for every height one prime p of Z(D) except p1 = ⟨z, ω⟩ and p2 = ⟨θ, ω⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (C) height one primes P1, P2, with piD = P 2 i ⊊ Pi for i = 1, 2, with each Pi generated by a normal (but not central) element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Moreover, P1 + P2 = P0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Retain the above notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (a) Every non-primitive prime is completely prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Every primitive ideal is completely prime, except the co-Artinian maximal ideals (other than the counit), which form a subset of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (b) Every prime ideal P, apart from the co-Artinian maximal ideals and (pos- sibly) P0, has D/P birationally equivalent to a Weyl algebra An(K), where 1 ≤ n ≤ 2 and K is a field of transcendence degree at most 2 over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (c) D satisfies the Dixmier-Moeglin Equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='1 is proved in Section 4, see in particular Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='5, while Theo- rem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2 is proved in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Finally, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='3 is proved in Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Some questions and a conjecture (Conjecture 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='5) are scattered through the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Hopf algebras in duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' The full Drinfeld double D(H) = H ⊲⊳ H◦ of an infinite dimensional Hopf algebra H may often be unwieldy due to H having “too many” finite dimensional representations and thus leading to an unmanageably large finite dual H◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' This has generated significant recent interest in constructing doubles D(H) := H ⊲⊳ H′ where H′ is some suitable Hopf subalgebra of H◦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' see for example [9, 20] and the papers listed in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Much is at present unclear: for example, what is an appropriate definition of a “suitable” Hopf subalgebra H′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' and does a suitable algebra H′ always exist?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' The double D of the Jordan plane is a test case for these and other questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' In particular, some of the desirable properties exhibited by D may form a paradigm for what one might aim for in defining doubles in more general settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Representation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' A striking feature of the representation theory of D is the fact, proved in [1, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='11] and recalled here in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2(iii), that U(sl(2, k)) is a quotient Hopf algebra of D and the finite dimensional irreducible D- modules are precisely the finite dimensional irreducible U(sl(2, k))-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' This immediately suggests a plethora of questions, some of which are addressed in [3], where Verma D-modules are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' But many others remain untouched: for instance, what can be said about the category of locally finite dimensional D- modules, and for each primitive ideal P of D can one find a canonical irreducible module (hopefully a factor of a Verma module) whose annihilator equals P?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' The first step in these questions is to classify the primitive ideals of D, as we do here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Dixmier-Moeglin equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' The validity of the Dixmier-Moeglin Equivalence for an algebra R yields simultaneous representation-theoretic, alge- braic and topological characterisations of the primitive ideals amongst the prime ideals of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' This equivalence is a feature of some but not all Hopf k-algebras;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' see, for example, [5,6] for discussions of when it holds for a noetherian (Hopf) algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' As we prove in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2, D satisfies the equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' See §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2 for the details, THE PRIME SPECTRUM OF THE DRINFELD DOUBLE OF THE JORDAN PLANE 3 where we also give a rather ambitious Conjecture 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='5, proposing a general result encompassing all affine Noetherian pointed Hopf C-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Throughout, all vector spaces and all unadorned tensor products are understood to be over the base field k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' We denote the comultiplication of a Hopf algebra H by ∆ and its augmentation ideal by H+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' The Gelfand-Kirillov, or GK dimension of an object X is denoted by GKdim(X), while the global (homological) dimension, respectively injective dimension of a ring R is denoted by gldim(R), respectively injdim(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' For precision, we specify that in the Ore extension T = S[v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' σ, ∂], multiplication is defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='1) vs = sσv + ∂(s) for s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' It follows that ∂ is a σ-derivation in the sense that ∂(ab) = aσ∂(b) + ∂(a)b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' This follows the conventions of, for example, [17, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Preliminaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Definitions and notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' The following definitions and notation from [1]) will remain in play throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' First, the Jordan plane is J := k⟨x, y : [y, x] = − 1 2x2⟩, with bosonization H := J#C∞ = J#⟨g±1⟩, where gxg−1 = x, gyg−1 = y + x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Then the Drinfeld double of J is defined to be D := H⟨u, v, ζ⟩, with additional relations as follows: [u, v] = 1 2u2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' [ζ, v] = −v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' [ζ, u] = −u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' [u, y] = 1 − g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' [v, x] = 1 − g + xu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' [v, y] = yu − gζ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' [v, g] = gu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' [ζ, y] = y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' [ζ, x] = x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' [x, u] = [x, g] = [u, g] = [ζ, g] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' The coalgebra structure, which will mostly not concern us here, is determined for H by specifying that g is grouplike and x and y are (g, 1)−primitive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' and then extended to D by setting u and ζ to be primitive and ∆(v) = v ⊗ 1 + 1 ⊗ v + ζ ⊗ u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Observe that K := k⟨u, v, ζ⟩ is a Hopf subalgebra of D and in fact, as one can see from the PBW theorem for D as described in [2, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='3(ii)], D = J ⊗k K as vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' As noted in [2, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2] there is a non-degenerate skew pairing between J and K which yields the multiplication relations between these subalgebras as in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Initial results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' We gather together in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2 some of the main results of [2] and [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' We must first define some elements of D, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Set (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='1) q := ux + 2(1 + g), and s := xv + uy + (−1 2ux + g − 1)ζ − 2(g + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' The following lemma is partly explicit, partly implicit, in [1, §4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Given a k-algebra automorphism σ of a k-algebra H, we say that the element h of H is σ-normal if ha = σ(a)h for all a ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Keep the above notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' 4 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' BROWN AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' STAFFORD (i) q and s are both σ-normal, where σ is the automorphism of D defined by σ(y) = y + 1 2x, σ(v) = v − 1 2u, with σ acting as the identity on the other generators of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (ii) σ2 equals conjugation by g on D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' that is, σ2(h) = ghg−1 for all h ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (iii) The elements z := q2g−1, θ := s2g−1 and ω := qsg−1 are in the centre Z(D) of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (i) and (ii) are easy checks, and (iii) is immediate from (i) and (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' □ Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Retain the notation introduced above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (i) [2, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='7(i)] O(G) := k⟨x, u, g±1⟩ is a normal commutative Hopf subalgebra of D, with G = ((k, +) × (k, +)) ⋊ (k∗, ×).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (ii) [2, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='7(ii)] DO(G)+ is a Hopf ideal of D, with an isomorphism of Hopf algebras (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2) D/DO(G)+ ∼= U(sl2(k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (iii) [1, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='11] The finite dimensional irreducible D-modules are the finite dimensional irreducible U(sl2(k))-modules given by the epimorphism (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (iv) [1, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='10] With the notation from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='1, the centre of D is (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='3) Z(D) = k⟨z, ω, θ : zθ = ω2⟩, (v) [1, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2] D is pointed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' □ We’ll need the following labelling of the maximal ideals of Z(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Note that here there are two maximal ideals of Z(D) which require particular attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (i) By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2((iv), Maxspec(Z(D)) consists of {m(α,γ) := ⟨z −α2γ−1, ω −α, θ −γ⟩ : α ∈ k, γ ∈ k∗} ˙∪ {mβ := ⟨z −β, ω, θ⟩ : β ∈ k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Note that m(α,γ) can be simplified to m(α,γ) = ⟨ω − α, θ − γ⟩, while mβ = ⟨z − β, ω⟩ when β ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (ii) It is easy to calculate using the definition of the counit that D+ ∩ Z(D) = O(G)+D ∩ Z(D) = m(−16,16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' We thus denote m(−16,16) by m+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (iii) It is clear that the singular locus of Z(D) is {m0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Ring-theoretic preparations In this section we assemble some properties needed in the analysis of the primitive spectrum of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' The proofs are most easily approached by viewing D as an iterated Hopf Ore extension starting not from the base field k but from the commutative normal Hopf subalgebra O(G) = k⟨x, u, g±1⟩ of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' More precisely: Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' D is an iterated Ore extension (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='1) D = O(G)[y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' δ1][ζ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' δ2][v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' τ, δ3], where the derivations δ1 and δ2, the automorphism τ and the τ-derivation δ3 can be read off from the defining relations of D given in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' In particular, D is a noetherian domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' THE PRIME SPECTRUM OF THE DRINFELD DOUBLE OF THE JORDAN PLANE 5 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Use the proof of [2, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='6] to show that D has basis {gaxbucydζevf : a ∈ Z, b, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' , f ∈ N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Then the form of the relations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='1) combined with [12, Theorem 1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='438] show that it is indeed an Ore extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' □ Although it will be not needed in this paper, the description (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='1) even describes D as an Iterated Hopf Ore Extension (IHOE), in the sense that each extension in that formula is itself a Hopf algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' It also shows that, by setting deg x = deg u = deg g = deg g−1 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' deg y = deg ζ = deg v = 1, one obtains a filtration F on D with associated graded algebra (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2) grF D = O(G)[y, ζ, v].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' So grF D is a commutative polynomial algebra in 6 variables with one variable inverted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Homological properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' In this subsection we note that D has certain use- ful homological properties, and we begin with the relevant definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' A ring A is called Auslander Gorenstein if it has finite injective dimension and satisfies the Gorenstein condition: if p < q are non-negative integers and M is a finitely gener- ated A-module, then Extp A(N, A) = 0 for every submodule N of Extq A(M, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' The ring A is Auslander regular if it is Auslander Gorenstein of finite global dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Set jA(M) = min{r : Extr A(M, A) ̸= 0} for the homological grade of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Then an Auslander Gorenstein ring A of finite GK dimension is called GK-Cohen-Macaulay (or just CM), provided that jA(M)+ GKdim(M) = GKdim(A) holds for each such M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Obviously affine commutative regular rings are both Auslander regular and CM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (i) D is Auslander regular and CM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (ii) D is AS regular in the sense of, say, [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (iii) GKdim(D) = 6 = gldim(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (iv) GK dimension is an exact function on finitely generated D-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (i) By [7, Remark, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='157] the filtration F is Zariskian and so the result follows from [7, Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='8, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='9 and Remark, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' 165].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (ii) This is immediate from (i) and [11, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (iii) By [22, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='4], GKdim(D) = GKdim(grF(D)) = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='1 and [21, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='3(i)], we have gldim(D) ≤ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' By [1, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='11] D has a finite dimensional module, say M and the CM condition implies that M has homological dimension ≥ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Hence gldim(D) = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (iv) Since jD is exact on finitely generated D-modules by [19, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='3], this follows from the CM condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' □ 6 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' BROWN AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' STAFFORD 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Key lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' The following lemma will be crucial in our analysis of the prim- itive spectrum of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' In its proof, given an ideal B of a noetherian ring S, we denote by √ B the ideal of S such that √ B/B is the nilradical of S/B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Let M be a finitely generated (right or left) D-module such that either Annk[x](M) ̸= 0 or Annk[u](M) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Then (i) there exists r ≥ 1 such that (m+D)r ⊆ (O(G)+D)r ⊆ AnnD(M);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (ii) GKdim(M) ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (i) Let I := AnnD(M), an ideal of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Assume that I ∩ k[x] ̸= 0, the proof in the other case being exactly similar, but with k⟨u, v⟩ replacing J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' One easily confirms that every non-zero prime ideal of the Jordan plane J = k⟨x, y⟩ contains x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Therefore, since I ∩k[x] ̸= 0, there exists N ≥ 1 such that xN ∈ I ∩k[x] ⊆ I ∩O(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Since O(G) is commutative, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='3) x ∈ � (I ∩ O(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Since I is an ideal of D, [v, I] ⊆ I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' moreover, from the defining relations of D and the fact that O(G) = k⟨x, u, g±1⟩, [v, O(G)] ⊆ O(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Therefore (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='4) [v, I ∩ O(G)] ⊆ I ∩ O(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Since k has characteristic 0 it follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='4) and [17, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='20] that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='5) [v, � (I ∩ O(G))] ⊆ � (I ∩ O(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='3) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='5) [v, x] = 1 − g + xu ∈ � (I ∩ O(G)), so that (1 − g) ∈ � (I ∩ O(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Then [v, g − 1] = [v, g] = gu ∈ � (I ∩ O(G)), so that u ∈ � (I ∩ O(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Since O(G)+ is generated by x, u and g − 1 we deduce that O(G)+D ⊆ √ I, proving (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (ii) By (i) M is a finitely generated D/(O(G)+D)r-module for some r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Since D/O(G)+D ∼= U(sl(2, k) by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2(ii), and so has GK dimension 3, (ii) follows from this and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2(iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Ore localisations of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' To help in the analysis of its primitive spectrum we need four Ore localisations of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' The first of these is described in [1, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='8], and the others are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' These sets are described as follows: Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Label the following four subsets of D: A := {qi : i ≥ 0} ˙∪ {xj : j ≥ 0}, B := {si : i ≥ 0} ˙∪ {xj : j ≥ 0}, C := {qi : i ≥ 0} ˙∪ {uj : j ≥ 0}, D := {si : i ≥ 0} ˙∪ {uj : j ≥ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (1) The elements x and u act ad-locally-nilpotently on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Conse- quently, {xi : i ≥ 0} and {ui : i ≥ 0} are Ore sets in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (2) For each Ω ∈ {A, B, C, D} the set Ω is an Ore set of regular elements of D, and we write the corresponding localisation as D(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' THE PRIME SPECTRUM OF THE DRINFELD DOUBLE OF THE JORDAN PLANE 7 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (1) For x this is proved in [1, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='3(i)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' The claim for u is a similar easy consequence of the defining relations of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (2) Localising at the powers of q is the same as localising at the powers of q2 or even at the powers z = q2g−1, since g is a unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Thus, for Ω = A or Ω = C and appealing to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='1(iii), we can replace q by the central element z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Similarly in the other two cases we can replace s by the central element θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Thus in each case we wish to localise at one central and one locally ad-nilpotent element in the domain D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Thus it is indeed an Ore set of regular elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' □ Thus each of the four rings D(Ω) is a subalgebra of the quotient division algebra Q(D) of D that contains D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' As we next show, each of these rings is a localisation of the second Weyl algebra over a commutative ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Notation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (i) In D(A), set pA := −2q−1x−1y, qA := q, tA := qx−1, and ηA := −xq−1ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (ii) In D(B), set pB := −2s−1x−1y, qB := s, tB := sx−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' ηB := −xs−1ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (iii) In D(C), set pC := 2q−1u−1v, qC := q, tC := q−1u−1, ηC := −uqζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (iv) In D(D), set pD := 2s−1u−1v, qD := s, tD := s−1u−1, ηD := usζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' We further set zΩ := z for Ω = A and Ω = C but zΩ := θ when Ω = B, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' The motivation behind the above definitions becomes clear from the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' For Ω = A, this was obtained in the proof of [1, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' The claims regarding the other elements can be checked by a similar direct calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Let Ω ∈ {A, B, C, D}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Then we have the following relations in Q(D): [pΩ, qΩ] = 1 = [ηΩ, tΩ], with all other brackets being zero;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' □ When Ω = A the following result is given in [1, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='8], although we give a proof that works for all 4 cases simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' For each Ω ∈ {A, B, C, D}, the localisation D(Ω) is a localised Weyl algebra over its centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' More precisely: D(Ω) = A(Ω) 2 (k) ⊗ S(Ω), where A(Ω) 2 (k) denotes the localisation of the second Weyl algebra over k with gen- erators pΩ, q±1 Ω , ηΩ, t±1 Ω , while S(Ω) is the commutative ring S(Ω) = k[z±1 Ω , ω].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' The generators z, ω and θ of Z(D) are given in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='1(iii), from which it follows that the subalgebra S(Ω) of Q(D) is contained in the centre Z(D(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Therefore we can consider the subalgebra (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='6) E(Ω) := S(Ω)⟨pΩ, qΩ, tΩ, ηΩ⟩ ⊆ D(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' We claim that the inclusion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='6) is an equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' In order to prove this, check that given generators of DΩ are contained in E(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Thus, for example, when Ω = A, one shows that {q−1, x±1, y, ζ, g±1, u, v} ⊂ E(A), and similarly in the other cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Thus, E(Ω) = D(Ω), as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' As noted in the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='5 the localisation of D at Ω involves inverting one central and one ad-nilpotent element of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Thus, by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2(iii) and [18, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='7], GKdim(D(Ω) = GKdim(D) = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' We conclude that GKdim(E(Ω)) = GKdim(D) = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' On the other hand, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='7 E(Ω) is a factor of the ring V(Ω) := S(Ω) ⊗k A(Ω) 2 (k), 8 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' BROWN AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' STAFFORD which is also a domain of GK-dimension 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' So if E(Ω) were a proper factor of V(Ω), then [21, Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='6] would imply that GKdim(E(Ω)) < 6, giving a contradic- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' So the only possibility is that E(Ω) ∼= V(Ω) = S(Ω) ⊗k A(Ω) 2 (k), as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' The primitive spectrum of D In this section we describe the primitive spectrum of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' This splits naturally into several cases: the primitive ideals not containing m+ or m0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' these are the generic ones;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' the ideal m+D, which is also primitive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' the ideal m0D, for which √m0D is a unique prime ideal P0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' finally, P0 is also maximal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' The details are given in the next four subsections with the results being combined in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' In this section and in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='1 we will without further reference use of the yoga for prime ideals of Noetherian rings under Ore localisation as described in, for example, [17, Theorems 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='18 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' We use Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='3 to describe the maximal ideals of Z(D) and Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='4 to define Ore sets in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' The generic minimal primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' We begin by looking at the generic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Let m be a maximal ideal of Z(D) with m ̸= m+ and m ̸= m0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Then the following are true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (i) mD is a completely prime maximal ideal of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (ii) The localisation of D/mD at the powers of (the image of) either x or u is isomorphic to a localised Weyl algebra A(Ω) 2 (k), where Ω ∈ {A, B, C, D}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (iii) GKdim(D/mD) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (iv) mD is generated by a central regular sequence of length 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (v) D/mD is CM and is Auslander Gorenstein with injdim(D/mD) < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (i), (ii) By Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='3, m = ⟨z − α, ω − β, θ − γ⟩ with α, β, γ ∈ k and αγ = β2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Moreover, thanks to the hypothesis on m, either (a) α ̸= 0 or (b) γ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Assume (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' We prove (ii) for the localisation at the powers of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (The arguments for powers of u are exactly similar, but using the Ore sets C and D rather than A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=') Using the notation of §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='3 and applying Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='8, we see that mD(A) is a maximal ideal of D(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Observe that, since A := {qi, xj : i, j ≥ 0} and z = q2g−1 ≡ α ̸= 0 mod(mD), A(A) 2 (k) is isomorphic to the localisation of D/mD at the powers of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Define Pm := mD(A) ∩ D, so that Pm is a completely prime ideal of D with mD ⊆ Pm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' By definition of Pm, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='1) mD(A) = PmD(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' We claim that in fact (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2) Pm = mD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' THE PRIME SPECTRUM OF THE DRINFELD DOUBLE OF THE JORDAN PLANE 9 Since D is (left) noetherian there exist e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' , et ∈ Pm such that Pm = mD + �t i=1 Dei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='1), for each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' , t there exist fi ∈ mD and si ∈ Z≥0 such that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='3) ei = fix−si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Define s := max{si : 1 ≤ i ≤ t} ∈ Z≥0, and I := {τ ∈ D : Pmτ ⊆ mD}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Thus I is an ideal of D containing mD and, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='3), xs ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' If s = 0 then I = D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' otherwise we see from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='3 that (m+)r ⊂ I for some r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Since also m ⊆ I and m ̸= m+ by hypothesis, it follows that I = D, and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2) is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' In case (a) it remains only to prove that Pm is a maximal ideal of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Suppose then that J is an ideal of D with Pm ⊊ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Then JD(A) = D(A) by the maximality of the ideal PmD(A) of D(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Again using the fact that q + mD is a unit of D/mD we see that xs ∈ J for some s ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Then, as before, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='3 implies that J = D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Suppose that (b) holds rather than (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Then the element s is a unit mod mD, so we use the same argument as for (a), but working with D(C) rather than D(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (iii) By (ii) and [18, Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='7 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='9] the localisation of D/mD at the powers of x has GK dimension 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Since ad(x) acts nilpotently on D/mD by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='5, it follows from [18, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='9] that GKdim(D/mD) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (iv) Again we assume (a) that z−α ∈ m for α ∈ k\\{0}, the proof in case (b) being similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' We can begin a regular central sequence in mD with z − α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Since D is CM of GK-dimension 6 by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2(i, iii), it follows from [16, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2(b)] that D/(z − α)D is CM of GK-dimension 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Moreover, by [19, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='4] the CM property ensures that D/(z − α)D is GK-homogeneous;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' that is, it contains no non-zero ideal with GK-dimension strictly less than 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Since Z(D)/(z − α)Z(D) is a polynomial algebra we can choose y ∈ m such that m = ⟨z − α, y⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' If y + (z − α)D is a zero divisor in D/(z − α)D we obtain a non-zero ideal of D/(z − α)D killed by mD, contradicting the GK-homogeneity of D/(z − α)D in view of (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Thus {z − α, y} is a regular central sequence in mD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (v) Since D is CM by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2(i), R = D/mD is CM with GKdim(R) = 4 by (iv) and two applications of [16, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' The Auslander Gorenstein property is given by (iv) and [19, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='4, Remark (3)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' As R is simple it cannot have a finite dimensional module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Hence injdim(R) < 4 follows from the next lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' □ The following observation is well-known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Let R be a noetherian, Auslander Gorenstein, CM ring and write GKdim(R) = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Then injdim(R) ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Moreover injdim(R) = m ⇐⇒ R has a finite dimensional representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Let n = injdim(R) and pick a finitely generated R-module M such that Extn R(M, R) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' By the Auslander condition and the spectral sequence [19, The- orem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2] j(Enn(M)) = n for Enn = Extn(Extn(M, R), R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' By the CM property GKdim(Enn(M)) = m − n and the result follows easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' □ 10 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' BROWN AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' STAFFORD 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Non-generic minimal primitives (I) - m+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' The next case to consider is mD for m = m+, as we do here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Recall from Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='3(ii) that m+ = D+ ∩ Z(D) = ⟨ω + 16, θ − 16⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' We start with a subsidiary result, which works for any field k of characteristic zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' D is a Jacobson ring that satisfies the Nullstellensatz, in other words: (i) every prime ideal of D is an intersection of primitive ideals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (ii) for every simple D-module M, EndD(M) is algebraic over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' In particular, every primitive ideal of D contains a maximal ideal of Z(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2), D has a filtration F such that the associated graded ring grF(D) is a commutative affine ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Hence by [22, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='7] there is a second filtration G by finite dimensional k-subspaces of D such that grF(D) is also a commutative and affine ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' The result now follows from [4, Theorem 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' □ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (i) m+D is a completely prime, primitive ideal of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (ii) The localisation of D/m+D at the powers of x or the powers of u is a localisation of the Weyl algebra A2(k) at powers of a generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (iii) m+D is generated by a regular central sequence of length 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (iv) D/m+D is Auslander Gorenstein and CM with GKdim(D/m+D) = 4 = injdim(D/m+D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (v) Every prime ideal P of D which strictly contains m+D satisfies O(G)+D ⊆ P, so the space of such primes P is homeomorphic to Spec(U(sl(2, k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Recall that qA = q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Since q2 ≡ 16g ̸≡ 0 mod(m+D), Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='8 implies that m+D(A) is a maximal ideal of D(A), with D/m+D(A) ∼= A(A) 2 (k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Therefore, defining P+ := m+D(A) ∩ D, we deduce that P+ is a completely prime ideal of D with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='4) m+D ⊆ P+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' We will eventually show that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='4) is an equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' As in the proofs of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='1(i),(ii), let I be the right annihilator in D of P+/m+D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Then I contains m+D and a power of x, and hence, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='3, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='5) (O(G)+D)r ⊆ I for some r ∈ Z≥1, In particular, GKdim(D/I) ≤ 3 by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='3(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Therefore, by [18, Proposi- tion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='1(d)] (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='6) GKdim(P+/m+D) ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Recall that GKdim(A(A) 2 (k)) = 4 by [18, Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='3 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='9], so that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='7) GKdim(D/P+) = 4 by [18, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Thus, from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='6), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='7) and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2(iv) it follows that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='8) GKdim(D/m+D) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' THE PRIME SPECTRUM OF THE DRINFELD DOUBLE OF THE JORDAN PLANE 11 By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2, D is CM and Auslander regular, with gldim(D) = 6 = GKdim(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' It therefore follows from the CM property of D together with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='8) that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='9) jD(D/m+D) = 6 − 4 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' From (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='9) and [8, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='6] we deduce that the maximum length of a reg- ular sequence of elements of m+ on D is precisely 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' in particular any choice of a generating pair of elements of m+, for example, {z − 16, ω + 16}, is a regular sequence on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Therefore, by two applications of [16, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2(b)], (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='10) D/m+D is CM of GK-dimension 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Similarly, two applications of [19, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='4, Remark (3)] show that D/m+D is Auslander Gorenstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='3(i) and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2(ii), U(sl(2, k)) ∼= D/DO(G)+ is a factor of D/m+D and so D/m+D has a non-zero finite dimensional module, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Thus, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2, injdim(D/m+D) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' By [19, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='4], again, the CM property for D/m+D implies that D/m+D is GK-homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Therefore we may conclude from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='6) that P+ does indeed equal m+D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' This proves (i) - (iv), with the exception of showing that m+D is primitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (v) Let Q be a prime ideal of D with m+D ⊊ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' As already noted, q is congruent to a unit mod m+D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Then QD(A) = D(A) by (ii), so Q must contain a power of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Hence, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='3, O(G)+D ⊆ Q, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Finally, to see that m+D is primitive note that (v) shows that it is locally closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Hence it is primitive by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='3(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Non-generic minimal primitives (II) - m0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' In this subsection we begin our study of the ideal m0D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Recall the definition of q, s from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='1) and, from Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='3(iii), that m0 := ⟨q2g−1, qsg−1, s2g−1⟩ is the unique singular point of Maxspec(Z(D)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Clearly the right ideal (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='11) P0 := qD + sD is a two-sided ideal of D since q and s are normal in D by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Moreover, m0D = P 2 0 ⊂ P0 ⊆ � m0D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' As part of the next proposition we see that P0 is completely prime, so the second inclusion above is an equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' In fact P0 is a maximal ideal, but this is more difficult to prove, and is delayed until §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Retain the above notation, and set T := D/P0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (i) T is a localisation of a 4-step iterated Ore extension of k, namely T = � (k[u, x]⟨(ux + 2)−1⟩)[y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' ∂1] � [v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' σ, ∂2], where u and x commute, ∂1(u) = − 1 2ux − 2, ∂1(x) = − 1 2x2, ∂2(u) = − 1 2u2, ∂2(x) = 3 2ux + 2, ∂2(y) = 3 2uy − 2, and σ(y) = y + 1 2x, with σ(x) = x and σ(u) = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (ii) {q, s} forms a regular normal sequence of generators of P0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (iii) gldim(T ) ≤ 4 = GKdim(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' 12 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' BROWN AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' STAFFORD (iv) T is CM and is an Auslander regular domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Throughout the proof we abuse notation by simply denoting the image in T of an element ω of D by ω when no confusion seems likely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (i),(ii) Since q := ux + 2(1 + g) and q ≡ 0 mod(P0), we can write (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='12) g ≡ − 1 2(ux + 2) mod(P0), so that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='13) ux + 2 is a unit in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='12) we find that, mod(P0), s := xv + uy + (− 1 2ux + g − 1)ζ − 2(g + 1) ≡ xv + uy + 2gζ − 2g − 2, so that, since s ∈ P0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='14) ζ ≡ − 1 2g−1(ux + xv + uy) mod(P0) It follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='12), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='13) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='14) that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='15) T = k⟨u, x, (ux + 2)−1, y, v⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' The relations for D given in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='1 immediately imply the following relations for the generators for T listed in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='15) [u, x] = 0, [y, x] = − 1 2x2, [v, x] = 3 2ux + 2, [y, u] = − 1 2ux − 2, [v, u] = − 1 2u2, [v, y] = 3 2uy + 1 2xv − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Clearly the iterated Ore extension of k[u, x]⟨(ux + 2)−1⟩ defined in (i), which we temporarily label �T, satisfies precisely these relations, so there is an algebra epi- morphism Φ from �T onto T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' We next show that Φ is an isomorphism, which we do by computing GKdim(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' First note that GKdim( �T) = 4 by [18, Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='1], since it is a PBW extension in 2 variables of k[u, x]⟨(ux + 2)−1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Thus, certainly GKdim(T ) ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' On the other hand D is CM of GK-dimension 6 by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2(i, iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Hence, because q is a regular normal element of D by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='1, D/qD is CM of GK-dimension 5 by [16, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2(b) and its proof].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Moreover D/qD is GK-homogeneous by [19, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Since GKdim(T ) ≤ 4, this ensures that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='16) s cannot be a zero-divisor mod qD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Since P0 := qD + sD, a second application of [16, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2(b) and its proof] yields GKdim(T ) = 4 and also shows that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='17) T is CM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Since �T is a domain, the equality GKdim( �T) = 4 = GKdim(T ), combined with [18, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='15], shows that �T = T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Thus (i) is proved, with (ii) also following thanks to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (iii) By (i), T is a 2-step iterated Ore extension of k[u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='x]⟨(ux + 2)−1⟩, and so two applications of [21, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='3(i)] gives gldim(T ) ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (iv) That T is a domain is clear from (i), while the CM property was proved in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' The Auslander Gorenstein property holds for D by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Thus, by (ii) and two applications of [16, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2(a)], T is also Auslander Gorenstein and it is then Auslander regular by (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' □ THE PRIME SPECTRUM OF THE DRINFELD DOUBLE OF THE JORDAN PLANE 13 We remark that, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2(iii) it follows that gldim(T ) < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' We do not know the exact value of gldim(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Maximality of P0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Let T := D/P0 as in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Define also the following subalgebras of T : R := k⟨u, x, (ux + 2)−1⟩, and S := R[y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' ∂1], so that T = S[v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' σ, ∂2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' It is important to note that, by the formulæ in Proposi- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='5, R is preserved by the σ-derivation ∂2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Moreover, since σ|R is the identity, ∂2 actually restricts to a derivation on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' It is much easier to determine when an Ore extension is simple if the ring is a differential operator ring, in the sense that the defining automorphism is actually the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Thus we will reduce to that case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' The idea follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='1 which shows that σ2 is given by the inner automorphism τg in the sense that σ(s) = τg(s) = gsg−1 for suitable g ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' We will therefore extend R, S and T by √g and show that σ is then inner, and so can be removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' The details are given in the next few results, culminating in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Notation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' In the algebraic closure of R, set h = (ux + 2)− 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Write �R = R⟨h⟩ = k⟨u, x, h, h−1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' We extend the ∂i to derivations on �R by the usual rules for fractional powers: ∂(h) = (−1 2)(ux + 2)−1h∂(ux + 2), for ∂ = ∂1, ∂2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Set �S = �R[y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' ∂1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Finally, we can extend σ to �R and �S by setting σ(h) = h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Then both σ and ∂2 are naturally defined on �S as an automorphism, respectively σ-derivation and so �T = �S[v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' σ, ∂2] is a well-defined Ore extension of �S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' The following observation will prove useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' �S is a free left and right S module on basis {1, h}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Similarly, �T is a free left and right T module on basis {1, h}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' As h2 = (ux + 2)−1 ∈ R, the construction of �R ensures that �R is a free left and right R-module on basis {1, h}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' We can then write �S = ∞ � i=0 �Ryi = � Ryi ⊕ � Rhyi = � Ryi ⊕ � Ryih.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Collecting terms shows that �S = S ⊕ Sh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' As S is a domain this is necessarily a direct sum of free modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' The same argument works for �T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' On �S, σ is the inner automorphism τh−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' thus σ(f) = h−1fh for f ∈ �S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Since �R is a commutative ring on which σ is the identity, the lemma holds trivially on �R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' It therefore just remains to check that the automorphisms agree on y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' To prove this, we rewrite h−1yh as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' 14 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' BROWN AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' STAFFORD h−1yh = (ux + 2) 1 2 y(ux + 2)− 1 2 = (ux + 2) 1 2 (ux + 2)− 1 2 y + (ux + 2) 1 2 · ∂1 � (ux + 2)− 1 2 � = y + (ux + 2) 1 2 (− 1 2)(ux + 2)− 3 2 · ∂1((ux + 2)) = y − 1 2(ux + 2)−1� (− 1 2ux − 2)x − u( 1 2x2) � = y − 1 2(ux + 2)−1� −(ux + 2)x � = y + 1 2x = σ(y);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' □ Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Set α = hv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Then �T is the Ore extension �T = �S[α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' �∂2] where �∂2 is the derivation of �S defined by �∂2(s) = h∂2(s) for s ∈ �S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' thus �∂2(u) = − 1 2hu2, �∂2(x) = h( 3 2ux + 2) and �∂2(y) = h(( 3 2uy − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' As such, �T is a noetherian domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' This is a formal computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Indeed, for s ∈ �S, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='8 implies that σ(s) = h−1sh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Equivalently, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='18) αs = hvs = hσ(s)v + h∂2(s) = hh−1shv + h∂2(s) = sα + h∂2(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Therefore, since �T = �S[v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' σ, ∂2] = � �Svi, we see that �T = � �Sαi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Since �T is a domain, combining this with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='18) and [12, Theorem 1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='438] gives the desired conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' □ Our next aim will be to show that the ring �T is a simple domain, after which it is easy to prove the same conclusion for T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' We start with some preparatory results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' If there exists a non-zero (∂1, �∂2)-invariant ideal I in �R, then there exists a non-zero (∂1, �∂2)-invariant prime ideal P in �R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Using [17, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='18(b)] twice, clearly I �S is a proper non-zero ideal of �S and then I �T is a proper nonzero ideal of �T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Pick a prime ideal Q ⊇ I �T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Then, by [17, Lemmata 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='18 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='21], twice, Q1 = Q ∩ �S is a �∂2-invariant prime ideal of �S and hence Q2 = Q1 ∩ �R is a ∂1-invariant prime ideal of �R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' However, since �R and Q1 are both �∂2-invariant, so is Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Thus, P = Q2 is the desired prime ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' □ Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' There is no proper, non-zero (∂1, �∂2)-invariant ideal I in �R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Suppose that there exists such an ideal I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='10 we can and will assume that I is a prime ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Suppose, first, that (xu + λ) ∈ I, for some λ ∈ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Then I ∋ ∂1(xu + λ) = (− 1 2ux − 2)x − ( 1 2x2)u = −(ux + 2)x and I ∋ �∂2(xu + λ) = h � − 1 2u2x + ( 3 2ux + 2)u � = h � ux2 + 2u � = h(ux + 2)u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' THE PRIME SPECTRUM OF THE DRINFELD DOUBLE OF THE JORDAN PLANE 15 As (xu + 2)−1 = h2 ∈ �R, clearly λ ̸= 2 and so the two equations imply that x ∈ I, respectively u ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Thus, I = x �R + u �R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' But, now I ∋ ∂1(u) = − 1 2ux − 2 and so I = �R, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' We conclude that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='19) I ∩ C = ∅ for C = {(xu + λ) : λ ∈ k∗} Since I is a prime ideal it follows that C ⊆ C(I) and hence that IC is a proper prime ideal of the localisation �RC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Next, if IC ∋ f = f(u) for some f(u) ∈ k[u], then IC ∋ ∂1(f) = − 1 2(ux + 4) df du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Hence df du ∈ IC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' By induction on deg f, this implies that IC = �RC, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Thus IC ∩ k[u]∗ = ∅ and so we can further localise at S = k[u]∗ and conclude that ICS is a proper prime ideal of �RCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Now consider �RCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' We have �R = k⟨u, x, h, h−1⟩ and h−2 = (ux + 2) whence x = u−1(h−2 − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Thus �RCS = �RSC is a localisation of k(u)[h, h−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' The advantage of working in �RCS is that we can simplify our derivation �∂2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' On �R and �RCS write ∂u = ∂ ∂u and ∂x = ∂ ∂x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Then, as derivations on either ring, ∂1 = −( 1 2xu + 2)∂u − 1 2x2∂x while �∂2 = − 1 2hu2∂u + h( 3 2ux + 2)∂x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' We now set µ := −hu2(ux + 4)−1 and take �∂′ 2 := �∂2 + µ∂1 = � − 1 2hu2 + µ(− 1 2ux − 2) � ∂u + � h( 3 2ux + 2) − 1 2x2µ � ∂x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' This element µ has been chosen so that the coefficient of ∂u in �∂′ 2 is − 1 2(ux + 4)−1� hu2(ux + 4) − (ux + 4)hu2� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Therefore, �∂′ 2 = � h( 3 2ux + 2) + 1 2hx2u2(ux + 4)−1� ∂x = (ux + 4)−1h � ( 3 2ux + 2)(ux + 4) + 1 2x2u2� ∂x = (ux + 4)−1h � 2u2x2 + 8ux + 8 � ∂x = β∂x for β := 2(xu + 4)−1(ux + 2)2h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Since ICS is invariant under both ∂1 and �∂2, it is also invariant under �∂′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Since β is a unit in �RCS, it follows that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='20) ICS is also invariant under β−1 �∂′ 2 = ∂x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Thus, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='20) and the expression given above for ∂1, ICS is invariant under ( 1 2ux + 2)∂u, and therefore under ∂u since 1 2ux + 2 is a unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' So ICS is invariant under ∂u and ∂x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Since �RCS is a localisation of k[u, x] this forces ICS = �RCS, giving the required contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' □ In order to pass between T and �T we need: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' If �T is a simple ring then so is T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' 16 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' BROWN AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' STAFFORD Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Suppose that T has a proper ideal J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Then X = �T/J �T is a (T, �T)-bimodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Moreover, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='7 �T is a finitely generated left T -module and so X is a finitely generated left T -module;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' say X = �r i=1 T xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Then, as �T is an Ore domain, ann � T (X) = � i ann � T (xi) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Since �T is a simple ring this implies that ann � T (X) = �T and hence that X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' In other words, J �T = �T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' On the other hand, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='7, �T = T + T h is a free left T -module and so J �T = J ⊕ Jh ̸= �T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' This contradiction proves the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' □ We now put everything together and prove the main result of this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' T is a simple ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='12 it suffices to prove that �T is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' By [21, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='4] applied to �T = �S[α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' �∂2], we need to prove (a) �∂2 is not an inner derivation on �S, and (b) �S has no proper �∂2-invariant ideals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Now, as ∂1(x) = − 1 2x2, the right ideal x�S is a proper two-sided ideal of �S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' As such, it is preserved by any inner derivation of �S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' But �∂2(x) = h( 3 2ux + 2) ̸∈ x�S, this means �∂2 cannot be an inner derivation of �S and so (a) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Suppose that �S has a proper �∂2-invariant ideal I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Then, by [17, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='18], K = I ∩ �R is a ∂1-invariant ideal of �R, while by [17, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='19], K ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Since both I and �R are both �∂2-invariant, so is K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' In other words, K is a proper (∂1, �∂2)- invariant ideal of �R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' This contradicts Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Thus (b) holds and so [21, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='4] implies that �T is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' We end the subsection by noting that �T is obviously birational to the Weyl algebra A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' We do not know if the same is true for T itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' The shape of the primitive spectrum of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' In this subsection we combine the earlier results of this section to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='3, every primitive ideal P of D contains a maximal ideal of Z(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Thus Privspec(D) is the disjoint union (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='21) Privspec(D) = ˙� m∈Maxspec(Z(D))V(m) where V(m) = {P ∈ Privspec(D) : m ⊆ P}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' There are thus 3 cases, corresponding to §§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (I) V(m), where m ∈ Maxspec(Z(D)) with m ̸= m+ and m ̸= m0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='1, V(m) = {mD} is a single generic maximal ideal of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Moreover D/mD is bira- tionally equivalent to the second Weyl algebra, with other properties as listed in that theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (II) V(m+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='4, this consists of m+D, together with V(O(G)+D) := {P ∈ Privspec(D) : O(G)+D ⊂ P}, which is homeomorphic to Privspec(U(sl(2, k))) by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Recall that Privspec(U(sl(2, k))) is composed of the co-Artinian maximal ideals {Mn : n ∈ Z≥1}, where Mn = Ann(Vn), Vn being the n-dimensional irreducible THE PRIME SPECTRUM OF THE DRINFELD DOUBLE OF THE JORDAN PLANE 17 U(sl2(k))−module, together with the minimal primitives of U(sl(2, k));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' that is, the ideals (Ω − λ)U(sl(2, k)) : λ ∈ k}, where Ω is the Casimir element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Each Mn contains one such minimal primitive and each minimal primitive is contained in at most one Mn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' the remaining minimal primitives are also maximal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Note that O(G)+D is prime but not primitive since D/O(G)+D ∼= U(sl(2, k)) and this domain satisfies the Nullstellensatz and has non-trivial centre k[Ω].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (III) V(m0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' This is the singleton {P0 = qD + sD = √m0}, by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='5 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Prime ideals and the Dixmier-Moeglin equivalence In this section we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2 from the introduction, which describes the prime ideals of D, and we discuss the Dixmier-Moeglin equivalence for D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' The prime spectrum of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' We need the following lemmas for the proof of the main result, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Let P be a nonzero prime ideal of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Then P ∩ Z(D) ̸= {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' If xi ∈ P for some i ≥ 0 then O(G)+D ⊆ P by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='3 applied with M = D/P, and therefore m+ = O(G)+D ∩ Z(D) ⊆ P, proving the lemma for P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' So we may assume that {xi : i ≥ 0} ∩ P = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Similarly, we may assume that {qj : j ≥ 0} ∩ P = ∅, since otherwise 0 ̸= qng−2n ∈ P ∩ Z(D) for some n ≥ 0 and again the result follows for P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Hence, using Notation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='6 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='8, PD(A) survives as a non-zero proper ideal of D(A) = D⟨q−1, x−1⟩ = A(A) 2 (k) ⊗k S(A), where A(A) 2 (k) is a localised Weyl algebra and S(A) = k[z±1, ω].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' In particular, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='1) PD(A) = (PD(A) ∩ S(A))D(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' By [17, Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='20] and the discussion in the first paragraph of this proof, P = PD(A) ∩ D, and therefore (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2) P ∩ Z(D) = PD(A) ∩ Z(D) = (PD(A) ∩ S(A)) ∩ Z(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Since the Z(D)-module S(A)/Z(D) is {zi}-torsion, that is {qig−2i}-torsion, it fol- lows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='1) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2) that P ∩ Z(D) ̸= {0} as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' □ Note that, since k is algebraically closed of characteristic 0, the defining relation zθ = ω2 of Z(D) can be rewritten using a linear change of variables as the quadratic form X2 + Y 2 = Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Thus a proof of the next result can be found at [15, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='51 and Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' All height one primes of Z(D) are principal except p1 := ⟨z, ω⟩ and p2 := ⟨θ, ω⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' □ Here is the main result of this section, using in (ii) the notation of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' This proves Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2 from the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Let P be a prime but not primitive ideal of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (i) There are the following three possibilities for P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (a) P = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (b) P = O(G)+D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' 18 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' BROWN AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' STAFFORD (c) P has height one and is minimal over (P ∩ Z(D))D for a height one prime ideal P ∩ Z(D) of Z(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (ii) In case (c), if P ∩ Z(D) = pi for i = 1, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' i = 2, then P = qD, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' P = sD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' The remaining primes in case (c) are precisely the set {P : P = fD}, as f ranges through the equivalence classes of irreducible elements of Z(D) other than the associates of z, ω, θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Note first that {0} is completely prime by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='1, and is not prim- itive, because D satisfies the Nullstellensatz by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='3 and Z(D) ̸= k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' This covers case (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Let P be a non-zero prime but not primitive ideal of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='1, {0} ̸= p := P ∩ Z(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' If p = m+ then Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='4 together with the discussion at §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='5(II) shows that the only possibility is P = O(G)+D, which is completely prime but is again not primitive thanks to the Nullstellensatz, since Z(U(sl(2, k))) ̸= k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' This is case (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' If p = m0 then P = P0, which is maximal by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='13, so this case can’t happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Similarly, p is any maximal ideal of Z(D) apart from m+ or m0, then P = pD is a maximal ideal of D by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='1(i), which again gives a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' So we are left with the case when p has height one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Assume first that p = fZ(D) is principal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Then, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2, z = q2g−1 /∈ P, and {xi : i ≥ 0} ∩ P = ∅ by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2 Therefore, using Notation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='6 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='8 pD(A) = (P ∩ S(A))D(A) = PD(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' We claim that P = pD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' To see this, note that pD = fD is principal, so that D/pD is CM of GK-dimension 5, by [16, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2(b) and its proof], and GK- homogeneous by [19, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='4, Remark (3)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Now P/pD it is killed by pD and by a power of q or a power of x, and so has GK-dimension less than 5, respectively by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='1(iii) and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='4(iv) or by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' This forces P/pD = {0} and so P = pD, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Suppose finally that p = p1 or p = p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' In the first case, since q is a normal element of D by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='1, q ∈ √pD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Thus (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='3) qD ⊆ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' We claim that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='3) is an equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' To see this, note that s /∈ P, since otherwise P ∩Z(D) = m0, which is ruled out by hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Moreover {xj : j ≥ 0}∩P = ∅ by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' So we can localise at the Ore set B = {sixj : i, j ≥ 0} of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='4 and pass to the localised Weyl algebra D(B) = A(B) 2 (k) ⊗ S(B) of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' However, PD(B) and qD(B) have the same intersection with the centre S(B), namely ωθ−1S(B) = p1S(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Therefore PD(B) = qD(B) since the ideals of D(B) are centrally generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Therefore P/qD is B-torsion, so, if it is not zero, it contains a nonzero element which is either killed by q and by s, or by q and x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' As in the previous paragraph D/qD is GK-homogeneous of GK-dimension 5, and so has no such non- zero torsion submodule, proving that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='3) is an equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' If p = p2 then the argument to show that P = sD is similar, but using the Ore set A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' it is left to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' □ THE PRIME SPECTRUM OF THE DRINFELD DOUBLE OF THE JORDAN PLANE 19 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' The Dixmier-Moeglin equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' The following gives evidence in favour of [6, Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='3], which proposes that an affine noetherian Hopf C-algebra of finite GK dimension should satisfy the Dixmier-Moeglin equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' See [5,10] for definitions and background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' D satisfies the Dixmier-Moeglin equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' We check first using the description of the primitive spectrum in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='5 that every primitive ideal is locally closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' For classes (I) and (III) this is clear since all these primitive ideals are maximal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' The primitive ideals in (II) are homeomor- phic to the primitive spectrum of U(sl(2, k)), and the latter algebra satisfies the equivalence by [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Thus, by [10, Lemma II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='15], it only remains to show that every rational prime ideal P is primitive, where P is rational if the centre of the Goldie quotient algebra of D/P is k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' The non-primitive prime ideals are listed in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='3 and it is easy to check case by case that none of them is rational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' □ Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='4 proves Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='3(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' With one exception, parts (a) and (b) of that theorem are proved in the results of the last two sections that describe the prime ideals of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' The exception is the claim that all the completely prime factors of D (with the possible exception of D/P0, as noted in Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='14) are birationally equivalent to Weyl algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' For the primitive ideals P strictly containing O(G)+D this follows from [13, Remarque 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' For the other prime ideals, this is clear from the description of the prime ideals in the last two sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Based on little more than the known results and counterexamples for group algebras and enveloping algebras, the theorem [6] for the cocommutative case, the recent work of Sierra and Walton on the noetherian property for enveloping algebras [25], together with the above result and other isolated examples, we are tempted to propose the following conjecture as a strengthening in the pointed setting of [6, Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='3], as much in the hope of stimulating the discovery of counterexamples as in expectation of a positive result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Conjecture 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Let H be an affine noetherian pointed Hopf C-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Then the following are equivalent: (1) GKdim(H) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (2) H satisfies the Dixmier-Moeglin Equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' (3) The group G(H) of grouplikes of H is nilpotent-by-finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Thanks to a famous result of Roseblade [24] for group algebras, the implication (2) =⇒ (3) fails when k is a finite field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' References [1] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} +page_content=' Andruskiewitsch, F.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf'} diff --git a/8dE3T4oBgHgl3EQfqgpk/content/tmp_files/2301.04652v1.pdf.txt b/8dE3T4oBgHgl3EQfqgpk/content/tmp_files/2301.04652v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0616574124f1f4037f4c6dfa4d1c8af6f59b68e2 --- /dev/null +++ b/8dE3T4oBgHgl3EQfqgpk/content/tmp_files/2301.04652v1.pdf.txt @@ -0,0 +1,983 @@ +ESTIMATE DEFORMATION CAPACITY OF +NON-DUCTILE RC SHEAR WALLS USING +EXPLAINABLE BOOSTING MACHINE +Zeynep Tuna Deger (1*), Gulsen Taskin Kaya (1), John W. Wallace (2) +(1) Istanbul Technical University, (2) University of California, Los Angeles +(*) corresponding author +{zeynep.tuna@itu.edu.tr, gulsen.taskin@itu.edu.tr, wallacej@ucla.edu} +Abstract +Machine learning is becoming increasingly prevalent for tackling challenges in earthquake +engineering and providing fairly reliable and accurate predictions. However, it is mostly +unclear how decisions are made because machine learning models are generally highly +sophisticated, resulting in opaque black-box models. Machine learning models that are +naturally interpretable and provide their own decision explanation, rather than using an +explanatory, are more accurate in determining what the model actually computes. With +this motivation, this study aims to develop a fully explainable machine learning model +to predict the deformation capacity of non-ductile reinforced concrete shear walls based +on experimental data collected worldwide. The proposed Explainable Boosting Machines +(EBM)-based model is an interpretable, robust, naturally explainable glass-box model, yet +provides high accuracy comparable to its black-box counterparts. The model enables the +user to observe the relationship between the wall properties and the deformation capacity +by quantifying the individual contribution of each wall property as well as the correlations +among them. The mean coefficient of determination R2 and the mean ratio of predicted +to actual value based on the test dataset are 0.92 and 1.05, respectively. The proposed +predictive model stands out with its overall consistency with scientific knowledge, practicality, +and interpretability without sacrificing high accuracy. +Keywords: Explainable boosting machine, glass-box model, feature selection, general additive model, reinforced +concrete shear wall, deformation capacity, interpretability +1 +Introduction +Shear walls are typically utilized as the primary elements to resist lateral loads in reinforced concrete buildings. +Towards capacity design assumptions, shear walls are designed to exhibit ductile behavior by providing +adequate reinforcement and proper detailing. However, experimental studies have shown that walls with an +aspect ratio smaller than 1.5 (i.e., squat walls) and those with poor reinforcement and detailing, despite +their higher aspect ratio, end up showing brittle failure (e.g., diagonal tension, web crushing) [1, 2, 3]. Such +walls are often observed in buildings not designed according to modern seismic codes and are prone to severe +damage [4, 5]. As the performance-based design and assessment approach has gained importance concordant +arXiv:2301.04652v1 [cs.LG] 11 Jan 2023 + +Deger ZT, Kaya GT, Wallace JW. ESTIMATE DEFORMATION CAPACITY OF NON-DUCTILE RC +SHEAR WALLS USING EXPLAINABLE BOOSTING MACHINE,Preprint. +with hazard mitigation efforts, there has been an increasing need and demand for reliable models to predict +structural behavior under seismic actions. This objective is particularly important for walls that exhibit +shear behavior as the nonlinear deformation capacity of such walls is assumed to be zero, potentially leading +to technical and economical over-conservation. More realistic solutions can be achieved if their behavior is +accurately estimated and considered in seismic performance evaluation. +The prediction of structural behavior has been achieved through the use of predictive equations or models that +are developed based on available experimental data. Recently, machine learning (ML) methods have gained +significant attention structural/earthquake engineering field and have demonstrated promising results despite +the scarcity of data (compared to much larger data available in fields such as computer vision and image +processing). Black-box models, with their high complexity and nonlinearity, often represent the input-output +relationship better than the interpretable models in classification and regression applications. However, they +are not necessarily consistent with true physical behavior. There are examples of misleading conclusions of +black-box models in scientific and engineering applications [6, 7, 8]. Therefore, despite the high accuracy +they achieve, black-box models are not completely accepted in earthquake engineering society. To leverage +the advantages of the developments in artificial intelligence without ignoring the physical behavior, there has +been recent research efforts that incorporates black-box machine learning methods with physical knowledge +[8, 9, 10, 11]. This study takes this issue a step further and integrates an explainable machine learning +approach (versus black-box) with existing physics-based understanding of seismic behavior to estimate the +deformation capacity of non-ductile shear walls. +2 +Literature Review +Research efforts in the literature to estimate wall deformation capacity have produced empirical models, some +recently adopted by building codes [12]; however, they are relatively limited compared to other behavior +features such as shear strength or failure mode. Earlier models were mainly developed using a limited number +of experimental results [13, 14] or were trained using a single dataset; that is, they were not trained and tested +based on unmixed data [12, 15]. Over time, as machine learning is embraced in the earthquake engineering +field [16, 17, 18, 19, 20, 11] and new experiments are conducted, more advanced models have been developed. +Yet, two main issues are encountered: (i) Some models used simple approaches such as linear regression +for the sake of interpretability [21] and sacrificed overall accuracy (or had large dispersion). One might +think that accurate models that predict relatively complicated behavior attributes can only be achieved +by increasing model complexity; however, literature studies have shown that this may cause problems with +the structure and distribution of the data [22, 23]. More importantly, urging the model to develop complex +relationships to achieve higher performance typically leads to black-box models where internal mechanisms +include highly nonlinear, complex relations. (ii) Such black-box models achieve high overall accuracy at +the cost of explainability [18]. Researchers that acknowledge the significance of interpretability employed +model-agnostic, local or global explanation methods (e.g., SHapley Additive exPlanation, Local Interpretable +Model-agnostic Explanations) to interpret the decision mechanism of their models [11]. Such algorithms are +not fully verified [24, 25]; besides, they are approximate approaches. Moreover, despite their broadening +use and high accuracy, the black-box models are not entirely accepted in the earthquake engineering society +as their internal relations are opaque and, in some cases, not entirely reliable [26]. Therefore, it is critical +to understand how the model makes the decision/estimation to (i) verify that the model is physically +meaningful, (ii) develop confidence in the predictive model, and (iii) broaden existing scientific knowledge +with new insights.This study addresses this need and fills this important research gap by using domain-specific +knowledge to evaluate and validate the decisions made by ML methods. Unlike the existing ML-based +predictive models ([11, 18]), the proposed model aims particularly at the deformation capacity of non-ductile +shear walls and is naturally transparent and interpretable. +Concerns regarding the trustworthiness and transparency of the black-box models motivated the development +of a relatively new research area known as explainable artificial intelligence (XAI) [27, 28]. The XAI aims to +provide a set of machine learning (ML) techniques for building more comprehensible and understandable +models while maintaining a high level of learning performance. The strategies used in XAI are divided +2 + +Deger ZT, Kaya GT, Wallace JW. ESTIMATE DEFORMATION CAPACITY OF NON-DUCTILE RC +SHEAR WALLS USING EXPLAINABLE BOOSTING MACHINE,Preprint. +into two main categories: explaining existing black-box models (post-hoc explainability) and generating +glass-box (transparent) models. In the former, interpretability is confined to the usage of certain so-called +explanatory algorithms that are employed to explain a black-box model, while in the latter, a predictive +model is fully comprehensible and interpretable by humans. A model should have certain qualities to be +considered a transparent model such as decomposability, algorithmic transparency, and simulatability [29]. The +decomposability relates to the ability to explain each model component in terms of the inputs’ contributions +or correlations, whereas simulatability refers to the number of parameters (input) in the model representation +(the less is the more understandable). The algorithmically transparent models enable a clear comprehension +of the model’ behavior for predicting any given output from its input data. Therefore, transparent models +are highly needed approaches in fields where decisions are critical, but their performances are typically very +low. Machine learning models that can maintain the tradeoff between performance and explainability, i.e., +converging to the performance of black-box models while still providing explainability, would significantly +address the demands in earthquake engineering society. In this context, explainable boosting machine (EBM) +[30], a recently developed method belonging to the family of Generalized Additive Models (GAMs) [31], is a +highly accurate and transparent ML method delivering an explicit and fully explainable predictive model. +The EBM has been utilized in the literature to solve a variety of problems, including detecting common flaws +in data [32], diagnosing COVID-19 using blood test variables [33], predicting diseases such as Alzheimer [34], +or Parkinson [35], and has shown to outperform black-box models with the additional benefit of being an +inherently explainable predictive model. +In this study, the EBM is used for the first time in the earthquake engineering field to construct an EBM- +based predictive model for estimating deformation capacity on non-ductile RC shear walls. The inputs of +the predictive model are designated as the shear wall design properties (e.g., wall geometry, reinforcing +ratio), whereas the output is one of the constitutive components of the nonlinear wall behavior, that is, the +deformation capacity. The main contributions of this research are highlighted as follows: +• A fully transparent and interpretable predictive model is developed to estimate the deformation +capacity of RC shear walls that are failed in pure shear or shear-flexure interaction. +• The proposed model meets all desired properties, i.e., decomposability, algorithmic transparency, +and simulatability, without compromising high performance. +• This study integrates novel computational methods (i.e., EBM) and domain-based knowledge to +formalize complex engineering knowledge. The proposed model’s overall consistency with a physics- +based understanding of seismic behavior is verified. +3 +The RC Shear Wall Database +The experimental data used in this research is a sub-assembly of the wall test database utilized in Deger +and Taskin (2022) [19] with 30 additional data [36, 37]. As the main focus is to estimate the deformation +capacity of walls governed by shear or shear-flexure interaction, walls that did not show so-called shear failure +indications are excluded from the database, resulting in 286 specimens of use for this research. All specimens +were tested under quasi-static cyclic loading, whereas none was retrofitted and re-tested. The database +consists of wall design parameters, which are herein designated as the input variables of the machine learning +problem, namely: wall geometry (tw, lw, hw), shear span ratio (M/V lw), concrete compressive strength +(fc), longitudinal and transverse reinforcing ratios at web (fyl, fyt), longitudinal and transverse reinforcing +ratios at boundary elements (fybl, fysh), axial load ratio (P/Agfc), shear demand (or strength) at the section +(Vmax), cross-section type (rectangular, barbell-shape, or flanged), curvature type (single or double). It is +noted that single curvature and double curvature correspond to the end conditions of the specimen, i.e., +cantilever and fixed-fixed, respectively. Distributions of the input variables are presented in Fig.1 along with +their box plots (shown in blue). +The output variable of the ML problem, the deformation capacity, is taken directly as the reported ultimate +displacement prior to its failure if the specimen is tested until failure. Otherwise, it is assumed as the +displacement corresponding to 0.8Vmax as suggested by Park, 1989. It is noted that failure displacement +3 + +Deger ZT, Kaya GT, Wallace JW. ESTIMATE DEFORMATION CAPACITY OF NON-DUCTILE RC +SHEAR WALLS USING EXPLAINABLE BOOSTING MACHINE,Preprint. +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +MVlw +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +hw/lw +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +P/Agfc +0 +1 +2 +3 +4 +5 +6 +rol +0.5 +1.0 +1.5 +2.0 +2.5 +rot +0 +2 +4 +6 +8 +10 +rosh +0 +5 +10 +15 +20 +25 +robl +20 +40 +60 +80 +100 +120 +140 +fc +50 +100 +150 +200 +250 +300 +tw +0 +500 +1000 +1500 +2000 +2500 +Vmax +Figure 1: Distribution of the input variables in the database. +was taken as the total wall top displacement and was not separated into shear and flexural deformation +components. +4 +Explainable Boosting Machines +Explainable Boosting Machines (EBM) is a state-of-the-art machine learning technique designed as accurately +as random forests and boosted trees while also being simple to understand [30, 38]. The EBM delivers a +complete explainable learning model that belongs to the family of Generalized Additive Models (GAMs) [39]: +g(f(x1, . . . , xn)) = f0 + f1(x1) + f2(x2) + . . . , fn(xn) +(1) +where f0 is an intercept, and each fj is called a shape function, representing the individual effect of the +xj-th variable on the model output, f(x1, . . . , xn). The g is utilized as a link function, adapting the model to +different settings, e.g., identity function for regression and logistic function for classification. The intercept, f0, +is calculated as the mean response of all the outputs. Because the shape functions are trained independently +for each input variable, the model is additive (decomposable), allowing to separately analyze the effect of +each input variable on the model output. The EBM is designed to improve the performance of the standard +GAM while maintaining its interpretability. +Generalized Additive Models are more comprehensible than black-box models, but the analytical form of the +shape functions is typically unknown, making it unsuitable for machine learning purposes. Although other +analytical functions, such as splines or orthogonal polynomials, can be offered for defining shape functions, +they are frequently less accurate when representing a nonlinear model [40]. The EBM uses shallow trees to +construct the shape functions; therefore, it easily captures the nonlinearity of the data. Each input variable +(xi) is modeled with ensemble trees such as bagging and gradient boosting. As a result, rather than employing +the spline method, which is prevalent in traditional GAMs, the function associated with each input variable +or interaction is produced from a vast set of shallow trees. +The EBM offers both local and global explanations of the learning model as each variable importance is +estimated as the average absolute value of the predicted score. Moreover, each shape function can be visualized +(algorithmically transparent); therefore, it is possible to observe the effects of the particular feature at certain +intervals. In the inference phase, all the terms in Eq.1 are added up and passed through the link function to +g to compute the final prediction as shown in Fig. 2. In other words, individual predictions are generated +using the shape functions, fi, which act as a lookup table. +To demonstrate which feature had the largest impact on the individual forecast, the contributions can be +sorted and shown using the principles of additivity and modularity. +The EBM’s performance can be improved by including pairwise effects between variables in the model +representation. For better performance, additional interactions can be incorporated; however, this may result +in a more complex model with lower generalization performance due to the increased number of model +parameters to be trained. The pairwise interactions are included in GA2Ms [41], which is a second-order +4 + +Deger ZT, Kaya GT, Wallace JW. ESTIMATE DEFORMATION CAPACITY OF NON-DUCTILE RC +SHEAR WALLS USING EXPLAINABLE BOOSTING MACHINE,Preprint. +Inference Phase +Intercept + ++ ++ ++ +E B M P r e d i c t i o n M o d e l +f0 ++ +f1(x1) +f2(x2) +f3(x3) ++ ++ ++ +f12(x1, x5) +x1 +x2 +x3 +x1 +x5 +score +score +score +score +Figure 2: Inference phase of EBM. +additive model: +g(E[y])) = f0 + +n +� +i=1 +fi(xi) + +K +� +i=1 +fk(xk1, xk2) +(2) +where K pairs of features (k1, k2) are chosen greedily (FAST algorithm) [42]. The pairwise interaction +fij(xi, xj) could be rendered as a heatmap on the two-dimensional xi, xj - plane, still providing high +intelligibility. Even though adding more interactions does not affect the model’s explainability, the final +prediction model may be less comprehensible due to a large number of interactions (less simulatability). +5 +Development of the Predictive Model +5.1 +Overall Performance of EBM +To assess whether the method compromises accuracy for the sake of interpretability, the performance of the +EBM model is compared to three state-of-the-art black-box machine learning models, namely: XGBoost [43], +Gradient Boost [44], Random Forest [45], and two glass-box models, namely Ridge Linear Regression [46], +Decision Tree [47]. All the implementations are carried out in a Python environment. For all ML models, the +entire database, including all twelve input variables (ten variables from Fig.1 and two binary coded variables +for curvature type and cross-section type), is randomly split into training and test datasets with a ratio of +90% and 10%, respectively. +Tunning of the hyperparameters, such as learning rate, number of leaves, number of interactions, and so +on, typically affects the performance of the corresponding regression model. For hyperparameter tuning, a +10-fold cross-validation technique (Fig. 3) is used, wherein a subset of the data is kept as validation data, +and the model’s performance is evaluated using various hyperparameter settings on the validation set. This +method prevents the tuning from overfitting the training dataset. +Training Phase +Split Training dataset into k-folds +Validation +Training +Validation +Training +Validation +Training +Validation +Training +Validation +Training +Training +Training +Training +Hyperparameter tuning +Model Evalution on validation data +R^2, RE, PA +Figure 3: Illustration of k-fold cross-validation technique, where k is set to 5. +For performance evaluations, the following three metrics are used over “unseen” (i.e., not used in the training +process) test data sets of ten random train-test data splittings: coefficient of determination (R2), relative +5 + +60 +40 +Score +20 +0 +2 +m60 +40 +Score +20 +0 +-20 +50 +100 +150 +200 +25060 +40 +Score +20 +-20 +0 +0.1 +0.2 +0.3 +0.4 +0.5300 +60 +250 +40 +200 +20 +150 +0 +100 +-20 +50 +2 +4 +MVIWDeger ZT, Kaya GT, Wallace JW. ESTIMATE DEFORMATION CAPACITY OF NON-DUCTILE RC +SHEAR WALLS USING EXPLAINABLE BOOSTING MACHINE,Preprint. +Table 1: Mean performance scores based on the test datasets over ten random splittings. +Black-box +Glass-box +EBM +XGBoost +GB +RF +RLR +DT +R2 +0.83 +0.80 +0.79 +0.83 +0.41 +0.67 +RE (%) +0.41 +0.40 +0.32 +0.28 +0.67 +0.47 +PA +1.21 +1.17 +1.20 +1.15 +2.0 +1.9 +error (RE), and prediction accuracy (PA), as given in Eqs. 3, 4, and 5, respectively. +R2 += +1 − +� +i(yi − ˆyi)2 +� +i(yi − ¯y)2 +(3) +RE += +� +i | ˆyi − yi| +� +i |yi| +| × 100% +(4) +PA += +� +i +yi +ˆyi +(5) +where yi, ¯y, ˆyi, and m refer to the actual output, the mean value of yis, predicted output of corresponding +regression model, and a number of samples in the test dataset, respectively. +Mean performance scores of the ML models are summarized in Table 1, along with their dispersion demon- +strated in box plots in Fig. 4. The results indicate that EBM achieves comparable performance with its +black-box counterparts, with a correlation of determination of R2 = 0.83, a relative error of 0.41%, and a +PA = 1.21. As seen in Fig. 4, the low R2, RE, and PA deviations of EBM imply that reliable predictions can +be achieved regardless of the selected train-test splitting and verify the model’s robustness. Mean prediction +accuracy (PA) shows around 20% of overestimation for EBM and the black-box methods, suggesting that +some input variables are potentially noisy. Compared to transparent models, the EBM outperforms both +the Decision Tree (DT) and Ridge Linear Regression (RLR) across all three metrics, indicating that it is far +superior to the traditional glass-box approaches. +The most remarkable advantage of the EBM method over the others is that it provides full explainability +without sacrificing accuracy. Unlike other methods, EBM enables the user to understand how the prediction +is made and which parameters are essential in the decision-making process. Therefore, the EBM method is +selected as the baseline algorithm for the rest of the analysis to propose a prediction model for estimating the +deformation capacity based on the following criteria: developing a model with fewer input variables (high +simulatability), achieving high accuracy, and ensuring physical consistency. +5.2 +The Proposed EBM-based Predictive Model +The importance of the wall properties in predicting the deformation capacity is evaluated based on additive +term contributions visualized in Fig.5. Results reveal that tw and M/V lw (or hw/lw) have the greatest +impact on individual predictions. This is consistent with the mechanics of the behavior as walls with smaller +thickness are shown to be more susceptible to lateral stiffness degradation due to concrete spalling, leading +to a failure caused by lateral instabilities or out-of-plane buckling [48, 49]. The shear span ratio (or aspect +ratio), on the other hand, both have a significant impact on deformation capacity as the higher the shear +span ratio gets, the slender the wall is, and the higher deformations it typically can reach prior to its failure. +The least important wall parameters, on the other hand, are identified as curvature type, cross-section type, +and concrete compressive strength. +Another critical aspect considered in this study is to develop the predictive model with as few input variables +as possible. With that, the computational workload is aimed to be reduced, and a more practical and +interpretable model is proposed for potential users. To achieve this, knowledge-based-selected combinations +6 + +Deger ZT, Kaya GT, Wallace JW. ESTIMATE DEFORMATION CAPACITY OF NON-DUCTILE RC +SHEAR WALLS USING EXPLAINABLE BOOSTING MACHINE,Preprint. +Figure 4: Comparison of performance scores of ML methods for test samples based on ten random train +splittings. +Figure 5: EBM Global interpretation for twelve features included +of four-to-five features are exhaustively evaluated to reach performance scores as high as when twelve features +are included. +EBM can achieve similar performance scores using four features: M/V lw, P/Agfc, tw, and Vmax. Including +additional features (e.g., ρl, ρbl, ρsh) deemed impactful by EBM as well as experimental results [50, 51] +has only a modest effect on the overall performance. The performances of other methods are close to their +benchmark model (including twelve features), whereas the glass-box methods are affected by the reduction of +input size and show much lower performances. The mean R2 drops to 0.33 for the Ridge Linear Regression, +imparting that the input-output relation is not linear. +The proposed EBM-based predictive model is selected to achieve the highest R2 with a prediction accuracy +as close to 1.0 as possible. The correlation plots are presented in Fig. 6 for training and test data sets, where +7 + +Deger ZT, Kaya GT, Wallace JW. ESTIMATE DEFORMATION CAPACITY OF NON-DUCTILE RC +SHEAR WALLS USING EXPLAINABLE BOOSTING MACHINE,Preprint. +scattered data are concentrated along the y = x line, demonstrating that the proposed model can make +accurate predictions. It should be noted that the distribution of the residuals is concentrated around zero. +(a) +(b) +(a) +(b) +Training Dataset +Test Dataset +Prediction +Prediction +Actual +Actual +Figure 6: Correlations of the model outputs with the actual values for (a) training and (b) test datasets +As discussed above, the proposed model is an additive model in which each relevant feature is designated a +quantitative term contribution. The EBM allows the user to explore the contribution of each feature to the +model by plotting their shape functions (Fig.7a-d). As discussed above, the EBM method employs multiple +decision-tree learning models; therefore, inclines and declines are undertaken with jump-looking piece-wise +constant functions (versus smooth curves). The values, called scores, are read from these functions, and those +from heat maps (Fig.7e-f) representing pairwise interactions (i.e., between two features) are summed up to +calculate the prediction. The gray zones along the shape functions designate error bars that indicate the +model’s uncertainty and data sensitivity. This typically occurs in cases of sparsity or the presence of outliers +within the associated region. +The shape functions in Fig.7 also indicate their correlations with the output in a graphical representation. For +example, nonlinear patterns that can not be observed in linear approaches can be easily interpreted [52], which +provides new insights to broaden existing experimental-based knowledge. For example, the shear demand +Vmax (Fig.7d) reduces ductility; thus deformation capacity, as demonstrated by experimental results [53] and +suggested by ASCE 41-17 acceptance criteria. Yet, a highly nonlinear pattern is observed when relevant +experimental data are gathered [11, 21]. This nonlinearity can be observed in the shape function suggested +by the proposed method. Other input variables (tw, P/Agfc, M/V lw), on the other hand, demonstrate an +almost-linear trend. The interpretation of EBM for these variables is consistent with experimental results +8 + +Deger ZT, Kaya GT, Wallace JW. ESTIMATE DEFORMATION CAPACITY OF NON-DUCTILE RC +SHEAR WALLS USING EXPLAINABLE BOOSTING MACHINE,Preprint. +��� +� +��� +� +��� +� +��� +� +��� +� +�� +�� +�� +������������ +������������� +������������ +����������� +����������� +����������� +���������� +���������� +����������� +����������� +����������� +���������� +� +�� +�� +�� +���� +����� +������� +�� +��� +��� +��� +��� +��� +��� +� +�� +�� +�� +��������� +����������� +��������� +�������� +��������� +��������� +��������� +��������� +��������� +��������� +��������� +��������� +� +�� +�� +�� +�� +����� +������� +� +��� +��� +��� +��� +��� +��� +� +�� +�� +�� +���������� +��������������� +�������������� +������������� +������������� +������������� +������������� +������������� +������������� +������������� +������������� +������������� +����������� +� +�� +��� +������ +����� +������� +��� +���� +���� +���� +���� +��� +� +�� +�� +�� +���������� +��������� +��������� +��������� +����������� +������������ +������������ +������������� +������������ +������������ +������������� +������������� +� +�� +�� +�� +���� +����� +������� +� +� +� +� +�� +��� +��� +��� +��� +��� +��� +� +�� +�� +�� +��������� +���� +�� +� +� +� +� +��� +���� +���� +���� +���� +��� +� +�� +�� +�� +����������� +���� +���� +, +, +(a) +(b) +(c) +(d) +(e) +(f) +Figure 7: EBM shape functions (a-d) and pairwise interaction plots (e-f) for the proposed model. Note that +the intercept f0= 35.528. +in the literature, such that M/V lw (Fig.7a) and tw (Fig.7b) have a positive impact, as discussed above, +whereas P/Agfc (Fig.7c) has an adverse influence [54, 55]. The reason for M/V lw and tw (Fig.7b) suggesting +an inverse effect up to a certain point (M/V lw ≈ 1.2, tw ≈ 60 cm, P/Agfc ≈ 0.08) is because the model +has an intercept value (f0, Eq.2) and specimens with smaller deformation capacities (f0 less than 35.528) +are predicted adding up negative values. The unexpected jumps in tw are likely because there is an abrupt +accumulation of data at tw = 100 mm and tw = 200 mm (64 and 44 specimens, respectively), which probably +causes difficulty in decision making. +It is noted that the EBM method offers controllability over the structure of the model proposed by, for +instance, modifying the number of pairwise interactions. This allows the method to suggest more than one +9 + +Deger ZT, Kaya GT, Wallace JW. ESTIMATE DEFORMATION CAPACITY OF NON-DUCTILE RC +SHEAR WALLS USING EXPLAINABLE BOOSTING MACHINE,Preprint. +−30 +−20 +−10 +0 +10 +20 +30 +Vmax (421.58) +MVlw x tw +MVlw x P/Agfc +P/Agfc (0.00) +tw (200.00) +MVlw (2.08) +Intercept +Predicted (81.9) | Actual (81.8) +−30 +−20 +−10 +0 +10 +20 +30 +MVlw x tw +Vmax (445.60) +MVlw x P/Agfc +tw (130.00) +P/Agfc (0.50) +MVlw (2.00) +Intercept +Predicted (58.7) | Actual (67.2) +x +x +x +x +(a) +(b) +Figure 8: Variable contribution estimates for (a) well-predicted, (b) averagely-predicted samples. +model for the same input-output configuration for a particular train-test dataset. Reducing the number +of interactions brings simplicity to the model; however, it typically loses accuracy as EBM relies on its +automatically-determined interactions in the decision-making process. Given this trade-off, the number of +interactions is set to two for the proposed model. +5.3 +Sample-Based Explanation +This section presents the prediction of deformation capacity for two example specimens using the proposed +EBM-based predicted model. One specimen is predicted with excellent accuracy (almost zero error; Fig.8a), +whereas the other is predicted with around 15% error (Fig.8b). +Variable contribution estimates for each specimen are presented such that the intercept is constant and shown +in gray, the additive terms with positive impact are marked in orange, and additive terms decreasing the output +are shown in blue. Each contribution estimate is extracted from the shape functions and two-dimensional +heat maps (Fig.7) based on the input values of a specific specimen. Overall, the model is consistent with +physical knowledge, except Vmax has an unexpected positive impact on the output for the relatively worse +prediction (Fig.8b). This is an excellent advantage of EBM; that is, the user can prudently understand +how the prediction is made for a new sample and develop confidence in the predictive model (versus blind +acceptation in black-box models). +5.4 +Comparisons with Current Code Provisions +ASCE 41-17 and ACI 369-17 [56] provide recommended deformation capacities for nonlinear modeling purposes, +where shear walls are classified into the following two categories based on their aspect ratio: shear-controlled +(hw/lw > 1.5) and flexure-controlled (hw/lw > 3.0). The deformation capacity of shear-controlled walls is +identified as drift ratio such that ∆u/hw = 1.0 if the wall axial load level is greater than 0.5 and ∆u/hw = 2.0 +otherwise. +Deformation capacity predictions based on the proposed EBM model are compared to ASCE 41-17/ACI +369-17 provisions in Fig. 9. Predicted-to-actual ratios are 1.06 ± 0.49 and 6.42 ± 3.17 for EBM-based +model and code predictions, respectively. The results imply that traditional approaches may lead to the +overestimation of deformation capacities and cause unsafe assessments. +10 + +Deger ZT, Kaya GT, Wallace JW. ESTIMATE DEFORMATION CAPACITY OF NON-DUCTILE RC +SHEAR WALLS USING EXPLAINABLE BOOSTING MACHINE,Preprint. +0 +5 +10 +15 +20 +25 +30 +Specimen number +10 0 +10 1 +10 2 +10 3 +10 4 +10 5 +Ultimate displacement (mm) +Test (actual) +EBM-based model prediction +ASCE 41/ACI 369 prediction +Figure 9: Comparisons of EBM-based model predictions with code provisions. +6 +Conclusions +A fully transparent predictive model is developed to estimate the deformation capacity of reinforced concrete +shear walls that are failed in pure shear or shear-flexure interaction. To achieve this, a state-of-the-art machine +learning method, Explainable Boosting Machines (EBM), designed as accurately as random forests and +boosted trees, is utilized. The EBM provides an additive model such that each relevant feature is designated +a quantitative term contribution. The input-output configuration of the model is designated as the shear wall +design properties (e.g., wall geometry, axial load ratio) and ultimate wall displacement, respectively. The +conclusions derived from this study are summarized as follows: +• The importance of the wall properties in predicting the deformation capacity is evaluated based +on additive term contributions. tw and M/V lw (or hw/lw) have the greatest effect on individual +predictions, whereas the least relevant ones are identified as curvature type, cross-section type, and +concrete compressive strength. +• Compared to three black-box models (XGBoost, Gradient Boost, Random Forest), the EBM achieves +similar or better performance in terms of correlation of determination (R2), relative error (RE), and +prediction accuracy (PA; the ratio of predicted to the actual value). The EBM achieves a mean +R2 of 0.83 and a mean RE of 0.41% using twelve input variables based on ten random train-test +splittings. +• Compared to two glass-box methods (Decision Tree (DT) and Ridge Linear Regression (RLR)), the +EBM outperforms both methods across all three metrics. +• The dispersion of performance metrics of EBM is small, implying that the model is robust and the +performance is relatively less data-dependent. +• Compared to the developed model when all the available features are used, the EBM achieves +competitive performance scores using only four input variables: M/V lw, P/Agfc, tw, and Vmax. +Using these four features, the proposed EBM-based model achieves R2 of 0.92 and PA of 1.05 based +on the test dataset. Using fewer variables ensures that the model is less simulatable, more practical, +more comprehensible, and reduces the computational cost. +• It is important to note that the decision-making process developed by the proposed EBM-based model +has overall consistency with scientific knowledge despite several exceptions detected in sample-based +inferences. This is an excellent advantage of the proposed model; that is, the user can assess and +evaluate the prediction process before developing confidence in the result (versus blindly accepting as +in black-box models). +11 + +Deger ZT, Kaya GT, Wallace JW. ESTIMATE DEFORMATION CAPACITY OF NON-DUCTILE RC +SHEAR WALLS USING EXPLAINABLE BOOSTING MACHINE,Preprint. +• This model delivers exact intelligibility, i.e., there is no need to use local explanation methods (e.g., +SHAP, LIME) to interpret the learning model, which obviates the uncertainties associated with their +approximations. +The proposed EBM-based model is valuable in that it is simultaneously accurate, explainable, and consistent +with scientific knowledge. The EBM’s ability to provide interpretable and transparent results would allow +engineers to better understand the factors that affect the deformation capacity of non-ductile RC shear walls +and make informed design decisions. The use of the EBM to estimate deformation capacity would improve the +reliability and efficiency of structural analysis and design processes, leading to safer and more cost-effective +buildings. +12 + +Deger ZT, Kaya GT, Wallace JW. 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ACI (American Concrete Institute), Farmington Hills, MI, +2017. +15 + diff --git a/8dE3T4oBgHgl3EQfqgpk/content/tmp_files/load_file.txt b/8dE3T4oBgHgl3EQfqgpk/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..99095ee39a753b9a44914ab70398ba82bd9743a4 --- /dev/null +++ b/8dE3T4oBgHgl3EQfqgpk/content/tmp_files/load_file.txt @@ -0,0 +1,732 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf,len=731 +page_content='ESTIMATE DEFORMATION CAPACITY OF NON-DUCTILE RC SHEAR WALLS USING EXPLAINABLE BOOSTING MACHINE Zeynep Tuna Deger (1*), Gulsen Taskin Kaya (1), John W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Wallace (2) (1) Istanbul Technical University, (2) University of California, Los Angeles (*) corresponding author {zeynep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='tuna@itu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='tr, gulsen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='taskin@itu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='tr, wallacej@ucla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='edu} Abstract Machine learning is becoming increasingly prevalent for tackling challenges in earthquake engineering and providing fairly reliable and accurate predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' However, it is mostly unclear how decisions are made because machine learning models are generally highly sophisticated, resulting in opaque black-box models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Machine learning models that are naturally interpretable and provide their own decision explanation, rather than using an explanatory, are more accurate in determining what the model actually computes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' With this motivation, this study aims to develop a fully explainable machine learning model to predict the deformation capacity of non-ductile reinforced concrete shear walls based on experimental data collected worldwide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' The proposed Explainable Boosting Machines (EBM)-based model is an interpretable, robust, naturally explainable glass-box model, yet provides high accuracy comparable to its black-box counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' The model enables the user to observe the relationship between the wall properties and the deformation capacity by quantifying the individual contribution of each wall property as well as the correlations among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' The mean coefficient of determination R2 and the mean ratio of predicted to actual value based on the test dataset are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='92 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='05, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' The proposed predictive model stands out with its overall consistency with scientific knowledge, practicality, and interpretability without sacrificing high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Keywords: Explainable boosting machine, glass-box model, feature selection, general additive model, reinforced concrete shear wall, deformation capacity, interpretability 1 Introduction Shear walls are typically utilized as the primary elements to resist lateral loads in reinforced concrete buildings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Towards capacity design assumptions, shear walls are designed to exhibit ductile behavior by providing adequate reinforcement and proper detailing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' However, experimental studies have shown that walls with an aspect ratio smaller than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='5 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=', squat walls) and those with poor reinforcement and detailing, despite their higher aspect ratio, end up showing brittle failure (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=', diagonal tension, web crushing) [1, 2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Such walls are often observed in buildings not designed according to modern seismic codes and are prone to severe damage [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' As the performance-based design and assessment approach has gained importance concordant arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='04652v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='LG] 11 Jan 2023 Deger ZT, Kaya GT, Wallace JW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' ESTIMATE DEFORMATION CAPACITY OF NON-DUCTILE RC SHEAR WALLS USING EXPLAINABLE BOOSTING MACHINE,Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' with hazard mitigation efforts, there has been an increasing need and demand for reliable models to predict structural behavior under seismic actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' This objective is particularly important for walls that exhibit shear behavior as the nonlinear deformation capacity of such walls is assumed to be zero, potentially leading to technical and economical over-conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' More realistic solutions can be achieved if their behavior is accurately estimated and considered in seismic performance evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' The prediction of structural behavior has been achieved through the use of predictive equations or models that are developed based on available experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Recently, machine learning (ML) methods have gained significant attention structural/earthquake engineering field and have demonstrated promising results despite the scarcity of data (compared to much larger data available in fields such as computer vision and image processing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Black-box models, with their high complexity and nonlinearity, often represent the input-output relationship better than the interpretable models in classification and regression applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' However, they are not necessarily consistent with true physical behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' There are examples of misleading conclusions of black-box models in scientific and engineering applications [6, 7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Therefore, despite the high accuracy they achieve, black-box models are not completely accepted in earthquake engineering society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' To leverage the advantages of the developments in artificial intelligence without ignoring the physical behavior, there has been recent research efforts that incorporates black-box machine learning methods with physical knowledge [8, 9, 10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' This study takes this issue a step further and integrates an explainable machine learning approach (versus black-box) with existing physics-based understanding of seismic behavior to estimate the deformation capacity of non-ductile shear walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' 2 Literature Review Research efforts in the literature to estimate wall deformation capacity have produced empirical models, some recently adopted by building codes [12];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' however, they are relatively limited compared to other behavior features such as shear strength or failure mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Earlier models were mainly developed using a limited number of experimental results [13, 14] or were trained using a single dataset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' that is, they were not trained and tested based on unmixed data [12, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Over time, as machine learning is embraced in the earthquake engineering field [16, 17, 18, 19, 20, 11] and new experiments are conducted, more advanced models have been developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Yet, two main issues are encountered: (i) Some models used simple approaches such as linear regression for the sake of interpretability [21] and sacrificed overall accuracy (or had large dispersion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' One might think that accurate models that predict relatively complicated behavior attributes can only be achieved by increasing model complexity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' however, literature studies have shown that this may cause problems with the structure and distribution of the data [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' More importantly, urging the model to develop complex relationships to achieve higher performance typically leads to black-box models where internal mechanisms include highly nonlinear, complex relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' (ii) Such black-box models achieve high overall accuracy at the cost of explainability [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Researchers that acknowledge the significance of interpretability employed model-agnostic, local or global explanation methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=', SHapley Additive exPlanation, Local Interpretable Model-agnostic Explanations) to interpret the decision mechanism of their models [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Such algorithms are not fully verified [24, 25];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' besides, they are approximate approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Moreover, despite their broadening use and high accuracy, the black-box models are not entirely accepted in the earthquake engineering society as their internal relations are opaque and, in some cases, not entirely reliable [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Therefore, it is critical to understand how the model makes the decision/estimation to (i) verify that the model is physically meaningful, (ii) develop confidence in the predictive model, and (iii) broaden existing scientific knowledge with new insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='This study addresses this need and fills this important research gap by using domain-specific knowledge to evaluate and validate the decisions made by ML methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Unlike the existing ML-based predictive models ([11, 18]), the proposed model aims particularly at the deformation capacity of non-ductile shear walls and is naturally transparent and interpretable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Concerns regarding the trustworthiness and transparency of the black-box models motivated the development of a relatively new research area known as explainable artificial intelligence (XAI) [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' The XAI aims to provide a set of machine learning (ML) techniques for building more comprehensible and understandable models while maintaining a high level of learning performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' The strategies used in XAI are divided 2 Deger ZT, Kaya GT, Wallace JW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' ESTIMATE DEFORMATION CAPACITY OF NON-DUCTILE RC SHEAR WALLS USING EXPLAINABLE BOOSTING MACHINE,Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' into two main categories: explaining existing black-box models (post-hoc explainability) and generating glass-box (transparent) models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' In the former, interpretability is confined to the usage of certain so-called explanatory algorithms that are employed to explain a black-box model, while in the latter, a predictive model is fully comprehensible and interpretable by humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' A model should have certain qualities to be considered a transparent model such as decomposability, algorithmic transparency, and simulatability [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' The decomposability relates to the ability to explain each model component in terms of the inputs’ contributions or correlations, whereas simulatability refers to the number of parameters (input) in the model representation (the less is the more understandable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' The algorithmically transparent models enable a clear comprehension of the model’ behavior for predicting any given output from its input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Therefore, transparent models are highly needed approaches in fields where decisions are critical, but their performances are typically very low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Machine learning models that can maintain the tradeoff between performance and explainability, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=', converging to the performance of black-box models while still providing explainability, would significantly address the demands in earthquake engineering society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' In this context, explainable boosting machine (EBM) [30], a recently developed method belonging to the family of Generalized Additive Models (GAMs) [31], is a highly accurate and transparent ML method delivering an explicit and fully explainable predictive model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' The EBM has been utilized in the literature to solve a variety of problems, including detecting common flaws in data [32], diagnosing COVID-19 using blood test variables [33], predicting diseases such as Alzheimer [34], or Parkinson [35], and has shown to outperform black-box models with the additional benefit of being an inherently explainable predictive model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' In this study, the EBM is used for the first time in the earthquake engineering field to construct an EBM- based predictive model for estimating deformation capacity on non-ductile RC shear walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' The inputs of the predictive model are designated as the shear wall design properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=', wall geometry, reinforcing ratio), whereas the output is one of the constitutive components of the nonlinear wall behavior, that is, the deformation capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' The main contributions of this research are highlighted as follows: A fully transparent and interpretable predictive model is developed to estimate the deformation capacity of RC shear walls that are failed in pure shear or shear-flexure interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' The proposed model meets all desired properties, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=', decomposability, algorithmic transparency, and simulatability, without compromising high performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' This study integrates novel computational methods (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=', EBM) and domain-based knowledge to formalize complex engineering knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' The proposed model’s overall consistency with a physics- based understanding of seismic behavior is verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' 3 The RC Shear Wall Database The experimental data used in this research is a sub-assembly of the wall test database utilized in Deger and Taskin (2022) [19] with 30 additional data [36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' As the main focus is to estimate the deformation capacity of walls governed by shear or shear-flexure interaction, walls that did not show so-called shear failure indications are excluded from the database, resulting in 286 specimens of use for this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' All specimens were tested under quasi-static cyclic loading, whereas none was retrofitted and re-tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' The database consists of wall design parameters,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' which are herein designated as the input variables of the machine learning problem,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' namely: wall geometry (tw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' lw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' hw),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' shear span ratio (M/V lw),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' concrete compressive strength (fc),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' longitudinal and transverse reinforcing ratios at web (fyl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' fyt),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' longitudinal and transverse reinforcing ratios at boundary elements (fybl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' fysh),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' axial load ratio (P/Agfc),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' shear demand (or strength) at the section (Vmax),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' cross-section type (rectangular,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' barbell-shape,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' or flanged),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' curvature type (single or double).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' It is noted that single curvature and double curvature correspond to the end conditions of the specimen, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=', cantilever and fixed-fixed, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Distributions of the input variables are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='1 along with their box plots (shown in blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' The output variable of the ML problem, the deformation capacity, is taken directly as the reported ultimate displacement prior to its failure if the specimen is tested until failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Otherwise, it is assumed as the displacement corresponding to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='8Vmax as suggested by Park, 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' It is noted that failure displacement 3 Deger ZT, Kaya GT, Wallace JW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' ESTIMATE DEFORMATION CAPACITY OF NON-DUCTILE RC SHEAR WALLS USING EXPLAINABLE BOOSTING MACHINE,Preprint.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='5 P/Agfc 0 1 2 3 4 5 6 rol 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='5 rot 0 2 4 6 8 10 rosh 0 5 10 15 20 25 robl 20 40 60 80 100 120 140 fc 50 100 150 200 250 300 tw 0 500 1000 1500 2000 2500 Vmax Figure 1: Distribution of the input variables in the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' was taken as the total wall top displacement and was not separated into shear and flexural deformation components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' 4 Explainable Boosting Machines Explainable Boosting Machines (EBM) is a state-of-the-art machine learning technique designed as accurately as random forests and boosted trees while also being simple to understand [30, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' The EBM delivers a complete explainable learning model that belongs to the family of Generalized Additive Models (GAMs) [39]: g(f(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' , xn)) = f0 + f1(x1) + f2(x2) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' , fn(xn) (1) where f0 is an intercept, and each fj is called a shape function, representing the individual effect of the xj-th variable on the model output, f(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' , xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' The g is utilized as a link function, adapting the model to different settings, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=', identity function for regression and logistic function for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' The intercept, f0, is calculated as the mean response of all the outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Because the shape functions are trained independently for each input variable, the model is additive (decomposable), allowing to separately analyze the effect of each input variable on the model output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' The EBM is designed to improve the performance of the standard GAM while maintaining its interpretability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Generalized Additive Models are more comprehensible than black-box models, but the analytical form of the shape functions is typically unknown, making it unsuitable for machine learning purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Although other analytical functions, such as splines or orthogonal polynomials, can be offered for defining shape functions, they are frequently less accurate when representing a nonlinear model [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' The EBM uses shallow trees to construct the shape functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' therefore, it easily captures the nonlinearity of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Each input variable (xi) is modeled with ensemble trees such as bagging and gradient boosting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' As a result, rather than employing the spline method, which is prevalent in traditional GAMs, the function associated with each input variable or interaction is produced from a vast set of shallow trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' The EBM offers both local and global explanations of the learning model as each variable importance is estimated as the average absolute value of the predicted score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Moreover, each shape function can be visualized (algorithmically transparent);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' therefore, it is possible to observe the effects of the particular feature at certain intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' In the inference phase, all the terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='1 are added up and passed through the link function to g to compute the final prediction as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' In other words, individual predictions are generated using the shape functions, fi, which act as a lookup table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' To demonstrate which feature had the largest impact on the individual forecast, the contributions can be sorted and shown using the principles of additivity and modularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' The EBM’s performance can be improved by including pairwise effects between variables in the model representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' For better performance, additional interactions can be incorporated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' however, this may result in a more complex model with lower generalization performance due to the increased number of model parameters to be trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' The pairwise interactions are included in GA2Ms [41], which is a second-order 4 Deger ZT, Kaya GT, Wallace JW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' ESTIMATE DEFORMATION CAPACITY OF NON-DUCTILE RC SHEAR WALLS USING EXPLAINABLE BOOSTING MACHINE,Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Inference Phase Intercept + + + + E B M P r e d i c t i o n M o d e l f0 + f1(x1) f2(x2) f3(x3) + + + f12(x1, x5) x1 x2 x3 x1 x5 score score score score Figure 2: Inference phase of EBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' additive model: g(E[y])) = f0 + n � i=1 fi(xi) + K � i=1 fk(xk1, xk2) (2) where K pairs of features (k1, k2) are chosen greedily (FAST algorithm) [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' The pairwise interaction fij(xi, xj) could be rendered as a heatmap on the two-dimensional xi, xj - plane, still providing high intelligibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Even though adding more interactions does not affect the model’s explainability, the final prediction model may be less comprehensible due to a large number of interactions (less simulatability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' 5 Development of the Predictive Model 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='1 Overall Performance of EBM To assess whether the method compromises accuracy for the sake of interpretability, the performance of the EBM model is compared to three state-of-the-art black-box machine learning models, namely: XGBoost [43], Gradient Boost [44], Random Forest [45], and two glass-box models, namely Ridge Linear Regression [46], Decision Tree [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' All the implementations are carried out in a Python environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' For all ML models, the entire database, including all twelve input variables (ten variables from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='1 and two binary coded variables for curvature type and cross-section type), is randomly split into training and test datasets with a ratio of 90% and 10%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Tunning of the hyperparameters, such as learning rate, number of leaves, number of interactions, and so on, typically affects the performance of the corresponding regression model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' For hyperparameter tuning, a 10-fold cross-validation technique (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' 3) is used, wherein a subset of the data is kept as validation data, and the model’s performance is evaluated using various hyperparameter settings on the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' This method prevents the tuning from overfitting the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Training Phase Split Training dataset into k-folds Validation Training Validation Training Validation Training Validation Training Validation Training Training Training Training Hyperparameter tuning Model Evalution on validation data R^2, RE, PA Figure 3: Illustration of k-fold cross-validation technique, where k is set to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' For performance evaluations, the following three metrics are used over “unseen” (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=', not used in the training process) test data sets of ten random train-test data splittings: coefficient of determination (R2), relative 5 60 40 Score 20 0 2 m60 40 Score 20 0 20 50 100 150 200 25060 40 Score 20 20 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='5300 60 250 40 200 20 150 0 100 20 50 2 4 MVIWDeger ZT, Kaya GT, Wallace JW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' ESTIMATE DEFORMATION CAPACITY OF NON-DUCTILE RC SHEAR WALLS USING EXPLAINABLE BOOSTING MACHINE,Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Table 1: Mean performance scores based on the test datasets over ten random splittings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Black-box Glass-box EBM XGBoost GB RF RLR DT R2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='67 RE (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='47 PA 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='21 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='17 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='9 error (RE), and prediction accuracy (PA), as given in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' 3, 4, and 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' R2 = 1 − � i(yi − ˆyi)2 � i(yi − ¯y)2 (3) RE = � i | ˆyi − yi| � i |yi| | × 100% (4) PA = � i yi ˆyi (5) where yi, ¯y, ˆyi, and m refer to the actual output, the mean value of yis, predicted output of corresponding regression model, and a number of samples in the test dataset, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Mean performance scores of the ML models are summarized in Table 1, along with their dispersion demon- strated in box plots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' The results indicate that EBM achieves comparable performance with its black-box counterparts, with a correlation of determination of R2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='83, a relative error of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='41%, and a PA = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' As seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' 4, the low R2, RE, and PA deviations of EBM imply that reliable predictions can be achieved regardless of the selected train-test splitting and verify the model’s robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Mean prediction accuracy (PA) shows around 20% of overestimation for EBM and the black-box methods, suggesting that some input variables are potentially noisy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Compared to transparent models, the EBM outperforms both the Decision Tree (DT) and Ridge Linear Regression (RLR) across all three metrics, indicating that it is far superior to the traditional glass-box approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' The most remarkable advantage of the EBM method over the others is that it provides full explainability without sacrificing accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Unlike other methods, EBM enables the user to understand how the prediction is made and which parameters are essential in the decision-making process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Therefore, the EBM method is selected as the baseline algorithm for the rest of the analysis to propose a prediction model for estimating the deformation capacity based on the following criteria: developing a model with fewer input variables (high simulatability), achieving high accuracy, and ensuring physical consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='2 The Proposed EBM-based Predictive Model The importance of the wall properties in predicting the deformation capacity is evaluated based on additive term contributions visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Results reveal that tw and M/V lw (or hw/lw) have the greatest impact on individual predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' This is consistent with the mechanics of the behavior as walls with smaller thickness are shown to be more susceptible to lateral stiffness degradation due to concrete spalling, leading to a failure caused by lateral instabilities or out-of-plane buckling [48, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' The shear span ratio (or aspect ratio), on the other hand, both have a significant impact on deformation capacity as the higher the shear span ratio gets, the slender the wall is, and the higher deformations it typically can reach prior to its failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' The least important wall parameters, on the other hand, are identified as curvature type, cross-section type, and concrete compressive strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Another critical aspect considered in this study is to develop the predictive model with as few input variables as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' With that, the computational workload is aimed to be reduced, and a more practical and interpretable model is proposed for potential users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' To achieve this, knowledge-based-selected combinations 6 Deger ZT, Kaya GT, Wallace JW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' ESTIMATE DEFORMATION CAPACITY OF NON-DUCTILE RC SHEAR WALLS USING EXPLAINABLE BOOSTING MACHINE,Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Figure 4: Comparison of performance scores of ML methods for test samples based on ten random train splittings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Figure 5: EBM Global interpretation for twelve features included of four-to-five features are exhaustively evaluated to reach performance scores as high as when twelve features are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' EBM can achieve similar performance scores using four features: M/V lw, P/Agfc, tw, and Vmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Including additional features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=', ρl, ρbl, ρsh) deemed impactful by EBM as well as experimental results [50, 51] has only a modest effect on the overall performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' The performances of other methods are close to their benchmark model (including twelve features), whereas the glass-box methods are affected by the reduction of input size and show much lower performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' The mean R2 drops to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='33 for the Ridge Linear Regression, imparting that the input-output relation is not linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' The proposed EBM-based predictive model is selected to achieve the highest R2 with a prediction accuracy as close to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='0 as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' The correlation plots are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' 6 for training and test data sets, where 7 Deger ZT, Kaya GT, Wallace JW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' ESTIMATE DEFORMATION CAPACITY OF NON-DUCTILE RC SHEAR WALLS USING EXPLAINABLE BOOSTING MACHINE,Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' scattered data are concentrated along the y = x line, demonstrating that the proposed model can make accurate predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' It should be noted that the distribution of the residuals is concentrated around zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' (a) (b) (a) (b) Training Dataset Test Dataset Prediction Prediction Actual Actual Figure 6: Correlations of the model outputs with the actual values for (a) training and (b) test datasets As discussed above, the proposed model is an additive model in which each relevant feature is designated a quantitative term contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' The EBM allows the user to explore the contribution of each feature to the model by plotting their shape functions (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='7a-d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' As discussed above, the EBM method employs multiple decision-tree learning models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' therefore, inclines and declines are undertaken with jump-looking piece-wise constant functions (versus smooth curves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' The values, called scores, are read from these functions, and those from heat maps (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='7e-f) representing pairwise interactions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=', between two features) are summed up to calculate the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' The gray zones along the shape functions designate error bars that indicate the model’s uncertainty and data sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' This typically occurs in cases of sparsity or the presence of outliers within the associated region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' The shape functions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='7 also indicate their correlations with the output in a graphical representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' For example, nonlinear patterns that can not be observed in linear approaches can be easily interpreted [52], which provides new insights to broaden existing experimental-based knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' For example, the shear demand Vmax (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='7d) reduces ductility;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' thus deformation capacity, as demonstrated by experimental results [53] and suggested by ASCE 41-17 acceptance criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Yet, a highly nonlinear pattern is observed when relevant experimental data are gathered [11, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' This nonlinearity can be observed in the shape function suggested by the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Other input variables (tw, P/Agfc, M/V lw), on the other hand, demonstrate an almost-linear trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' The interpretation of EBM for these variables is consistent with experimental results 8 Deger ZT, Kaya GT, Wallace JW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' ESTIMATE DEFORMATION CAPACITY OF NON-DUCTILE RC SHEAR WALLS USING EXPLAINABLE BOOSTING MACHINE,Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='����������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' (a) (b) (c) (d) (e) (f) Figure 7: EBM shape functions (a-d) and pairwise interaction plots (e-f) for the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Note that the intercept f0= 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='528.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' in the literature, such that M/V lw (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='7a) and tw (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='7b) have a positive impact, as discussed above, whereas P/Agfc (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='7c) has an adverse influence [54, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' The reason for M/V lw and tw (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='7b) suggesting an inverse effect up to a certain point (M/V lw ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='2, tw ≈ 60 cm, P/Agfc ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='08) is because the model has an intercept value (f0, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='2) and specimens with smaller deformation capacities (f0 less than 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='528) are predicted adding up negative values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' The unexpected jumps in tw are likely because there is an abrupt accumulation of data at tw = 100 mm and tw = 200 mm (64 and 44 specimens, respectively), which probably causes difficulty in decision making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' It is noted that the EBM method offers controllability over the structure of the model proposed by, for instance, modifying the number of pairwise interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' This allows the method to suggest more than one 9 Deger ZT, Kaya GT, Wallace JW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' ESTIMATE DEFORMATION CAPACITY OF NON-DUCTILE RC SHEAR WALLS USING EXPLAINABLE BOOSTING MACHINE,Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' −30 −20 −10 0 10 20 30 Vmax (421.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='58) MVlw x tw MVlw x P/Agfc P/Agfc (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='00) tw (200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='00) MVlw (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='08) Intercept Predicted (81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='9) | Actual (81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='8) −30 −20 −10 0 10 20 30 MVlw x tw Vmax (445.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='60) MVlw x P/Agfc tw (130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='00) P/Agfc (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='50) MVlw (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='00) Intercept Predicted (58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='7) | Actual (67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='2) x x x x (a) (b) Figure 8: Variable contribution estimates for (a) well-predicted, (b) averagely-predicted samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' model for the same input-output configuration for a particular train-test dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Reducing the number of interactions brings simplicity to the model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' however, it typically loses accuracy as EBM relies on its automatically-determined interactions in the decision-making process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Given this trade-off, the number of interactions is set to two for the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='3 Sample-Based Explanation This section presents the prediction of deformation capacity for two example specimens using the proposed EBM-based predicted model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' One specimen is predicted with excellent accuracy (almost zero error;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='8a), whereas the other is predicted with around 15% error (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='8b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Variable contribution estimates for each specimen are presented such that the intercept is constant and shown in gray, the additive terms with positive impact are marked in orange, and additive terms decreasing the output are shown in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Each contribution estimate is extracted from the shape functions and two-dimensional heat maps (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='7) based on the input values of a specific specimen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Overall, the model is consistent with physical knowledge, except Vmax has an unexpected positive impact on the output for the relatively worse prediction (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='8b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' This is an excellent advantage of EBM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' that is, the user can prudently understand how the prediction is made for a new sample and develop confidence in the predictive model (versus blind acceptation in black-box models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='4 Comparisons with Current Code Provisions ASCE 41-17 and ACI 369-17 [56] provide recommended deformation capacities for nonlinear modeling purposes, where shear walls are classified into the following two categories based on their aspect ratio: shear-controlled (hw/lw > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='5) and flexure-controlled (hw/lw > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' The deformation capacity of shear-controlled walls is identified as drift ratio such that ∆u/hw = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='0 if the wall axial load level is greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='5 and ∆u/hw = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Deformation capacity predictions based on the proposed EBM model are compared to ASCE 41-17/ACI 369-17 provisions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Predicted-to-actual ratios are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='49 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='42 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='17 for EBM-based model and code predictions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' The results imply that traditional approaches may lead to the overestimation of deformation capacities and cause unsafe assessments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' 10 Deger ZT, Kaya GT, Wallace JW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' ESTIMATE DEFORMATION CAPACITY OF NON-DUCTILE RC SHEAR WALLS USING EXPLAINABLE BOOSTING MACHINE,Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' 0 5 10 15 20 25 30 Specimen number 10 0 10 1 10 2 10 3 10 4 10 5 Ultimate displacement (mm) Test (actual) EBM-based model prediction ASCE 41/ACI 369 prediction Figure 9: Comparisons of EBM-based model predictions with code provisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' 6 Conclusions A fully transparent predictive model is developed to estimate the deformation capacity of reinforced concrete shear walls that are failed in pure shear or shear-flexure interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' To achieve this, a state-of-the-art machine learning method, Explainable Boosting Machines (EBM), designed as accurately as random forests and boosted trees, is utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' The EBM provides an additive model such that each relevant feature is designated a quantitative term contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' The input-output configuration of the model is designated as the shear wall design properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=', wall geometry, axial load ratio) and ultimate wall displacement, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' The conclusions derived from this study are summarized as follows: The importance of the wall properties in predicting the deformation capacity is evaluated based on additive term contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' tw and M/V lw (or hw/lw) have the greatest effect on individual predictions, whereas the least relevant ones are identified as curvature type, cross-section type, and concrete compressive strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Compared to three black-box models (XGBoost, Gradient Boost, Random Forest), the EBM achieves similar or better performance in terms of correlation of determination (R2), relative error (RE), and prediction accuracy (PA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' the ratio of predicted to the actual value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' The EBM achieves a mean R2 of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='83 and a mean RE of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='41% using twelve input variables based on ten random train-test splittings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Compared to two glass-box methods (Decision Tree (DT) and Ridge Linear Regression (RLR)), the EBM outperforms both methods across all three metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' The dispersion of performance metrics of EBM is small, implying that the model is robust and the performance is relatively less data-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Compared to the developed model when all the available features are used, the EBM achieves competitive performance scores using only four input variables: M/V lw, P/Agfc, tw, and Vmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Using these four features, the proposed EBM-based model achieves R2 of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='92 and PA of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='05 based on the test dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' Using fewer variables ensures that the model is less simulatable, more practical, more comprehensible, and reduces the computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' It is important to note that the decision-making process developed by the proposed EBM-based model has overall consistency with scientific knowledge despite several exceptions detected in sample-based inferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' This is an excellent advantage of the proposed model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' that is, the user can assess and evaluate the prediction process before developing confidence in the result (versus blindly accepting as in black-box models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' 11 Deger ZT, Kaya GT, Wallace JW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' ESTIMATE DEFORMATION CAPACITY OF NON-DUCTILE RC SHEAR WALLS USING EXPLAINABLE BOOSTING MACHINE,Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' This model delivers exact intelligibility, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=', there is no need to use local explanation methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=', SHAP, LIME) to interpret the learning model, which obviates the uncertainties associated with their approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' The proposed EBM-based model is valuable in that it is simultaneously accurate, explainable, and consistent with scientific knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' The EBM’s ability to provide interpretable and transparent results would allow engineers to better understand the factors that affect the deformation capacity of non-ductile RC shear walls and make informed design decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE3T4oBgHgl3EQfqgpk/content/2301.04652v1.pdf'} +page_content=' The use of the EBM to estimate deformation capacity would improve the reliability and efficiency of structural analysis and design processes, leading to safer and more cost-effective buildings.' metadata={'source': 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b/BNFIT4oBgHgl3EQf_iwr/content/tmp_files/2301.11415v1.pdf.txt @@ -0,0 +1,1343 @@ +Approximate Bilevel Difference Convex Programming for Bayesian Risk +Markov Decision Processes +Yifan Lin 1 Enlu Zhou 1 +Abstract +We consider infinite-horizon Markov Decision +Processes where parameters, such as transition +probabilities, are unknown and estimated from +data. +The popular distributionally robust ap- +proach to addressing the parameter uncertainty +can sometimes be overly conservative. In this +paper, we utilize the recently proposed formu- +lation, Bayesian risk Markov Decision Process +(BR-MDP), to address parameter (or epistemic) +uncertainty in MDPs (Lin et al., 2022). To solve +the infinite-horizon BR-MDP with a class of con- +vex risk measures, we propose a computationally +efficient approach of approximate bilevel differ- +ence convex programming (ABDCP). The opti- +mization is performed offline and produces the +optimal policy that is represented as a finite state +controller with desirable performance guarantees. +We also demonstrate the empirical performance +of the infinite-horizon BR-MDP formulation and +proposed algorithms. +1. Introduction +In a Markov decision process (MDP), an agent must make +decisions in a sequence while facing uncertainty. In this +situation, some parameters of the MDP, such as the transi- +tion probabilities and costs, may be unknown and must be +estimated from available data. The problem then becomes +how to determine the best course of action, given the limited +or possibly absent data, in order to minimize the expected +total cost and optimize the decision-making process under +these uncertain parameters. +An alternative approach to addressing the epistemic uncer- +tainty in MDP is through the use of distributionally robust +MDPs (DR-MDPs, (Xu & Mannor, 2010)). This method +considers the unknown parameters as random variables and +assumes that their distributions belong to an ambiguity set +1H. Milton Stewart School of Industrial and Systems Engineer- +ing, Georgia Institute of Technology, Atlanta, GA, USA. Corre- +spondence to: Enlu Zhou . +Preliminary work. +determined by the available data. The optimal policy is +then found by minimizing the expected total cost using the +most adversarial distribution within this ambiguity set. How- +ever, these distributionally robust approaches may lead to +overly conservative solutions that do not perform well in +scenarios that are more likely to occur than the worst case. +Additionally, the DR-MDP framework does not explicitly +incorporate the dynamics of the problem, as the distribu- +tion of the unknown parameters does not depend on the +data process, and is therefore not time consistent, as noted +in (Shapiro, 2021). In light of these limitations, (Lin et al., +2022) proposes a Bayesian risk MDP (BR-MDP) framework +to address epistemic uncertainty in MDPs. This approach +stems from the static stochastic optimization literature (Wu +et al., 2018; Zhou & Xie, 2015) and involves using a nested +risk functional based on the Bayesian posterior distributions, +which are updated using all available data at each stage in +the process. However, the alpha-function approximation +algorithm proposed in (Lin et al., 2022) only applies to +finite-horizon MDPs and provides an upper bound on the +exact value, without any theoretical guarantee on the gap. +In this paper, we reformulate the considered problem as a +bilevel difference convex programming (DCP) such that we +can employ the powerful optimization methods for DCP to +solve infinite-horizon BR-MDP. Since the space of poste- +rior distributions (beliefs) is uncountably infinite, we ap- +proximate the bilevel DCP by considering only a subset of +posterior distributions. Although the DCP is approximate, +we show that its solution is a lower bound on the optimal +exact value function. Using the representation of a finite +state controller of the resulting policy, we further show an +upper bound on the optimal exact value function. We then +develop an iterative approach to reduce the gap between +upper and lower bounds by incrementally generating new +sets of posterior distributions, and show the convergence of +the proposed algorithm. +To summarize, the contributions of this paper are two folds. +First, we analyze the infinite-horizon MDP with epistemic +uncertainty under the framework of BR-MDP via a Bayesian +perspective and show the existence and uniqueness of sta- +tionary optimal policy. Second, we propose an approxi- +mate difference convex programming algorithm to solve +the proposed formulation, and show the convergence of the +arXiv:2301.11415v1 [eess.SY] 26 Jan 2023 + +Approximate Bilevel Difference Convex Programming for Bayesian Risk Markov Decision Processes +proposed algorithm. +The rest of the paper is organized as follows. We conduct +literature review and introduce the BR-MDP framework +in Section 2. We show the existence and uniqueness of a +stationary optimal policy to the infinite-horizon BR-MDP +in Section 3.1. We provide a bilevel DCP solution to the +infinite-horizon BR-MDP in Section 3.2. A computation- +ally efficient approximate DCP algorithm is then shown in +Section 3.3. We verify the theoretical results and demon- +strate the performance of our algorithms via numerical ex- +periments in Section 4. Finally, we conclude the paper in +Section 5. +2. Background +2.1. Related Literature +If the data used to estimate the true but unknown underlying +MDP are not sufficient, the estimated MDP may signifi- +cantly differ from the true MDP, leading to poor policy per- +formance. This discrepancy (between the estimated MDP +and the true MDP) can be seen tightly linked to the epistemic +uncertainty about the model. There have been numerous +approaches that address epistemic uncertainty in MDPs, +with robust MDP (Nilim & Ghaoui, 2004; Iyengar, 2005; +Delage & Mannor, 2010; Wiesemann et al., 2013; Petrik & +Russel, 2019) being one of the most widely used methods. +In robust MDPs, the optimal decisions are made based on +their performance under the most unfavorable conditions +within a known set of possible parameter values, known as +the ambiguity set. +In consideration of the overly conservativeness in the robust +MDP approach, risk-averse approach has been proposed to +address the epistemic uncertainty. Risk-averse approach is +originally proposed to address the aleatoric uncertainty that +is due to the inherent stochasticity of the underlying MDP +(Howard & Matheson, 1972; Ruszczy´nski, 2010; Petrik & +Subramanian, 2012; Osogami, 2012). It replaces the risk- +neutral expectation by some general risk measures, such as +conditional value-at-risk (CVaR, (Rockafellar & Uryasev, +2000)). However, most of the existing approaches assume +the agent has access to the true underlying MDP, and op- +timize some risk measures such as CVaR in that single +MDP (Chow & Ghavamzadeh, 2014; Tamar et al., 2015a;b; +Sharma et al., 2019). In this paper, we consider the offline +planning problem in MDPs, where we only have access to +a prior belief distribution over MDPs that is constructed +by the offline data. It should be noted that offline planning +problem has also been considered in (Duff, 2002), where the +author proposes a Bayes-adaptive MDP (BA-MDP) formu- +lation with an augmented state composed of the underlying +MDP state and the posterior distribution of the unknown +parameters. When the agent is equipped with the learned +optimal policy and placed in a real environment, it behaves +as if it is adapting to its surroundings. Mostly close to the +problem setting in this work are (Rigter et al., 2021; Lin +et al., 2022). (Rigter et al., 2021) optimizes a CVaR risk +functional over the total cost and simultaneously addresses +both epistemic and aleatoric uncertainty, while (Lin et al., +2022) considers a nested risk functional to ensure the time +consistency of the obtained optimal policy. +While there are many works proposing different models and +frameworks to address the epistemic uncertainty, developing +computationally efficient solutions is also of great interest. +In robust MDPs, with some mild conditions on the ambigu- +ity set such as rectangularity, the proposed formulation can +be solved by a second-order cone program when the horizon +is finite, or policy iteration when the horizon is infinite (Man- +nor & Xu, 2019). In BA-MDP and its variants, (Rigter et al., +2021) proposes an approximate algorithm based on Monte +Carlo tree search and Bayesian optimization. (Lin et al., +2022) develops an α-function approximation algorithm us- +ing the convexity of the CVaR risk measure. However, the +aforementioned works consider a finite-horizon MDP and +do not generalize well to the infinite-horizon setting. +Compared to standard MDPs, our considered problem has +two distinct features that make it difficult to apply value +iteration, policy iteration, or linear programming (Puterman, +2014). First is the resulting continuous-state MDP due to the +augmented belief state. We note that this continuous-state +MDP is similar to a belief-MDP, which is the equivalent way +to represent a partially observable MDP (POMDP) by treat- +ing the posterior distribution of the hidden state as a belief +state. Second is the risk measure taken with respect to the +unknown parameters in the MDPs. In this work, we propose +an optimization-based method to solve the infinite-horizon +BR-MDPs. It has been empirically shown in (Alagoz et al., +2015) that linear programming can efficiently solve a signif- +icant number of MDPs in comparison to standard dynamic +programming methods, such as value iteration and policy +iteration. Furthermore, linear programming requires less +memory and can handle MDPs with a larger number of +states and still achieve optimality. Works that are most re- +lated to our proposed optimization-based approach include +(Poupart et al., 2015) who proposes an approximate lin- +ear programming algorithm for the risk-neutral constrained +POMDPs, and (Ahmadi et al., 2021) who proposes a differ- +ence convex programming (DCP) for the constrained risk- +averse MDPs. Our approach for infinite-horizon BR-MDP +significantly differs from the above approaches in two as- +pects. First, compared to the linear programming approach +for risk-neutral POMDPs in (Poupart et al., 2015), we use +bilevel DCP, due to the additional risk measure that is used +for mitigating the epistemic uncertainty. Our considered risk +measure brings additional challenge to exactly evaluating +the policy, whereas policy evaluation can be easily solved + +Approximate Bilevel Difference Convex Programming for Bayesian Risk Markov Decision Processes +by a system of linear equations in (Poupart et al., 2015). +Second, compared to the DCP for the risk-averse MDP with +aleatoric uncertainty in (Ahmadi et al., 2021), the resulting +continuous-state MDP in our problem has an infinite number +of constraints, and thus requires appropriate approximation +to make the problem computationally feasible. +2.2. Preliminary: Bayesian Risk MDPs +Consider +an +infinite-horizon +MDP +defined +as +(S, A, P, C, γ), where S is the state space, A is the +action space, P is the transition probability with P(s′|s, a) +denoting the probability of transitioning to state s′ from +state s when action a is taken, C is the cost function with +C(s, a, s′) denoting the cost when action a is taken and +state transitions from s to s′, 0 ≤ γ < 1 is the discount +factor. We assume the state space and action space are finite. +A Markovian deterministic policy π is a function mapping +from S to A. Given an initial state s, the goal is to find an +optimal policy that minimizes the expected discounted total +cost: +min +π Eπ,P,C ��∞ +t=1 γt−1C (st, at, st+1) |s1 = s +� +, +where Eπ,P,C is the expectation with policy π when the +transition probability is P and the cost is C. In practice, P +and C are often unknown and estimated from data. +BR-MDP is a recently proposed framework that deals with +the epistemic uncertainty in MDPs (Lin et al., 2022). It is +assumed that the state transition is specified by the state +equation s′ = g(s, a, ξ′) with a known transition function +g, which involves state s ∈ S ⊆ Rs, action a ∈ A ⊆ Ra, +and randomness ξ ∈ Ξ ⊆ Rk, where s, a, k are the di- +mensions of the state, action, and randomness, respec- +tively. The state equation together with the distribution of ξ +uniquely determines the transition probability of the MDP, +i.e., P(s′ ∈ S′|s, a) = P({ξ ∈ Ξ : g(s, a, ξ) ∈ S′}|s, a), +where S′ is a measurable set in S. The cost is assumed to +be a function of state s, action a, and randomness ξ, i.e., +C(s, a, ξ). The distribution of ξ, denoted by f(·; θc), is +assumed to belong to a parametric family {f(·; θ)|θ ∈ Θ}, +where Θ ⊆ Rd is the parameter space, d is the dimension +of the parameter θ, and θc ∈ Θ is the true but unknown +parameter value. Many problems meet the requirement of +having a parametric assumption. For example, it is com- +monly assumed that the demand of customers follows a +Poisson distribution with an unknown arrival rate in inven- +tory control. +We begin by assuming a prior distribution, denoted by µ, +over the parameter space Θ. This prior accounts for the +uncertainty of the parameter estimate that comes from an +initial set of data, and it can also take expert opinions into +consideration. Then, given an observed realization of the +data process, we update the posterior distribution µ accord- +ing to the Bayes’ rule. Let the policy be a sequence of +mappings from state s and posterior µ to the action space, +i.e., π = {π : S × M → A}, where M is the space of +posterior distributions. Note that this representation im- +plies the policy is stationary. Now we present the BR-MDP +formulation below. +min +π +ρµ1Eθ1 +� +C1(s1, a1, ξ1) + · · · ++ γt−1ρµtEθt +� +Ct(st, at, ξt) + · · · +� +|s1 = s, µ1 = µ +� +(1) +s.t. st+1 = g(st, at, ξt), t = 1, 2, · · · ; +(2) +µt+1(θ) = +µt(θ)f (ξt; θ) +� +Θ µt(θ)f (ξt; θ) dθ , t = 1, 2, · · · , +(3) +where ρ is a risk measure, at = π(st, µt), θt is a random +vector following distribution µt, Eθt denotes the expectation +with respect to ξt ∼ f(·; θt) conditional on θt, and ρµt de- +notes a risk functional with respect to θt ∼ µt. Equation (2) +is the transition of the state st, and without loss of generality +we assume the initial state s1 takes a deterministic value s. +Equation (3) is the updating of the posterior µt. +2.3. Preliminary: Risk Measure +Let (Ω, F, P) be a probability space and Z be a linear space +of F-measurable functions Z : Ω → R. A risk measure is +a function ρ : Z → R which assigns to a random variable +Z a real number representing its risk. It is said that risk +measure ρ is convex if it possesses the properties of con- +vexity, monotonicity, and translation invariance (F¨ollmer & +Schied, 2002). In this paper we consider a class of convex +risk measures which can be represented in the following +parametric form: ρµ(Z) := infφ∈Φ Eµ[Ψ(Z, φ)],, where +Φ ⊂ Rm and Ψ : R × Φ → R is a real-valued func- +tion. There is a large class of risk measures which can +be represented in the parametric form. For example, con- +ditional value-at-risk (CVaR), defined as CVaRα(X) = +minφ∈R +� +φ + +1 +1−αE [(X − φ)+] +� +where (·)+ stands for +max(0, ·), is widely used (Rigter et al., 2021; Chow et al., +2015). Another example is risk measures constructed from +φ-divergence ambiguity sets (see Example 3 in (Guigues +et al., 2021)). We refer the readers to (Shapiro et al., 2021) +for a comprehensive discussion. +3. Algorithm and Theoretical Analysis +3.1. Bellman Equation and Optimality +We can write the value function under policy π of BR-MDP +in the following recursive forms. +V π(s, µ) = ρµEθ +� +C(s, a, ξ) + γV π(s′, µ′) +� +s.t. s′ = g(s, a, ξ), a = π(s, µ); +µ′(θ) = +µ(θ)f (ξ; θ) +� +Θ µ(θ)f (ξ; θ) dθ. +We refer the readers to (Lin et al., 2022) for a discussion + +Approximate Bilevel Difference Convex Programming for Bayesian Risk Markov Decision Processes +on the preference of dynamic risk measure over static risk +measure in consideration of time consistency and derivation +of the Bellman equation. For simplicity we only consider +deterministic policies, but all the analysis below can be ex- +tended to stochastic policies. As a consequence of Theorem +5.5.3b in (Puterman, 2014), it is sufficient to consider the +Markovian policy. The optimal value function is then de- +noted as V ∗(s, µ) = minπ∈ΠMD V π(s, µ), where ΠMD is +the set of Markovian deterministic policies. In the following, +we derive the intermediate results to show V ∗ is the unique +optimal value function to the infinite-horizon BR-MDP. +Definition 3.1 (Bellman Operator). Let B(S, M) be the +space of real-valued bounded measurable functions on (S × +M). For any bounded value function V ∈ B(S, M), define +an operator T : B(s, µ) → B(s, µ) as: +(T V )(s, µ) = min +a∈A ρµ [Eθ [C(s, a, ξ) + γV (s′, µ′)]] . +Also let T π : B(s, µ) → B(s, µ), where +(T πV )(s, µ) = ρµ [Eθ [C(s, π(s, µ), ξ) + γV (s′, µ′)]] . +The next two lemmas show the above Bellman operators are +monotonic and contraction mappings. Proofs can be found +in the appendix. +Lemma 3.2 (Monotonicity). The operators T π and T are +monotonic, in the sense that V ≤ V ′ implies T πV ≤ T πV ′ +and T V ≤ T V ′. +Lemma 3.3 (Contraction Mapping). The operators T π and +T are γ contraction for || · ||∞ norm. That is, for any two +bounded value functions V, V ′ ∈ B(S, M), we have +||T πV − T πV ′||∞ ≤ γ||V − V ′||∞. +The following proposition shows that sub-solutions and +super-solutions of the optimality equations V = T V pro- +vide lower and upper bounds on V ∗. As a result, when +a solution is obtained, both bounds are satisfied, meaning +that the solution must be equivalent to V ∗. Additionally, +this outcome serves as an important algorithmic tool for +optimization-based methods. +Proposition 3.4. For any V ∈ B(S, M), (i) if V ≥ T V , +then V ≥ V ∗; (ii) if V ≤ T V , then V ≤ V ∗. +According to Proposition 3.4, we have V ∗ = T V ∗. By +Banach fixed-point theorem, V ∗ is the unique optimal value +function to the infinite horizon BR-MDP. We also have that +the value V of a stationary policy π is the unique bounded +solution of the equation V = T πV . Similar analysis shows +the existence and uniqueness of the optimal stationary policy +π∗ that satisfies V ∗ = T π∗V ∗. +Applying the operator T on any initial value function V , we +have the value iteration algorithm for the infinite-horizon +BR-MDP problem. The following corollary of convergence +rate is similar to the standard with the contraction property. +Corollary 3.5. For any initial bounded value function +V , the convergence rate is shown to be ||(T kV )(s, µ) − +V ∗(s, µ)||∞ ≤ γk||V (s, µ) − V ∗(s, µ)||∞. +3.2. Bilevel Difference Convex Programming +The main challenge of executing the value iteration algo- +rithm (and similarly policy iteration algorithm) lies in the +continuous augmented state. In this work, we propose an +optimization-based method to solve the infinite-horizon BR- +MDPs. According to Proposition 3.4, the infinite-horizon +BR-MDP can be solved as follows: +max +V +� +s∈S,µ∈M +α(s, µ)V (s, µ) +s.t. V (s, µ) ≤ ρµEθ[C(s, a, ξ) + γV (s′, µ′)] +∀a ∈ A, s ∈ S, µ ∈ M, +where we choose α(s, µ) to be positive scalars which satisfy +� +s∈S,µ∈M α(s, µ) = 1. For the considered class of convex +risk measures, we can rewrite the above formulation as a +bilevel difference convex program: +min +V +− +� +s∈S,µ∈M +α(s, µ)V (s, µ) +(4) +s.t.V (s, µ) − min +φ Eµ[Ψ(Eθ[C(s, a, ξ) + γV (s′, µ′)], φ)] ≤ 0 +∀a ∈ A, s ∈ S, µ ∈ M. +Since Ψ(z, φ) is convex in (z, φ), it remains to be con- +vex in z after taking the minimum over φ. Thus, (4) is +a bilevel difference convex program (see (Horst & Thoai, +1999) for the definition of DCP). It should be noted that +(Ahmadi et al., 2021) shows that the minimum over φ can +be absorbed into the overall minimum problem, and φ is +treated as a single variable. However, it is clear that the +minimum is achieved at different φ for different augmented +state (s, µ), thus turning (4) into a bilevel optimization prob- +lem. When the lower-level problem is convex and satisfies +certain regularity conditions, we can use the Karush-Kuhn- +Tucker (KKT) conditions to reformulate the lower-level +optimization problem, which allows us to transform the +original bilevel optimization problem into a single-level +(constrained) optimization problem. +After being reduced to a single-level DCP problem, (4) +can be solved by the convex-concave procedure (see (Lipp +& Boyd, 2016) for such procedure), wherein the concave +terms are replaced by a convex upper bound. We employ +the method of disciplined convex-concave programming +(DCCP, (Shen et al., 2016), with Python package available +at https://github.com/cvxgrp/dccp), which converts a DCP +problem into a disciplined convex program and subsequently +into an equivalent cone program. However, one problem +remains to be solved: the number of constraints in (4) is + +Approximate Bilevel Difference Convex Programming for Bayesian Risk Markov Decision Processes +infinite, due to the continuous belief state. To tackle this +problem, we take a similar approach as (Poupart et al., 2015). +The main idea is to start with a finite posterior set (belief +space) ˆ +M, and then problem (4) can be solved efficiently +by DCCP, where the posterior distribution (belief point) not +in the set ˆ +M is replaced by some convex combination of +the points in ˆ +M. We then iteratively add to the posterior +set new posterior distributions that are reachable from the +current one and re-solve (4). We formally introduce the +approximate bilevel DCP algorithm in the next section. +3.3. Approximate Bilevel Difference Convex +Programming +Let ˆ +M be the current posterior set. Let µsas′ be the one-step +posterior distribution with observed randomness ξ indicated +by state transition s′ = g(s, a, ξ) and current posterior µ. +Initially the posterior set is constructed from corner (de- +generate) points. In case the parameter space Θ is finite, +the corner points are (1, 0, · · · , 0), (0, 1, 0, · · · ), · · · , and +(0, · · · , 0, 1). In case the parameter space is continuous, it +is impossible to express one-step posterior distribution (i.e., +µsas′) as a convex combination of those degenerate points. +Therefore, we assume the parameter space is finite, which is +practical in many real-world problems. It can also be viewed +as a discrete approximation of a continuous parameter set, +and the discretization can be chosen of any precision. +To interpolate all µsas′ that can be reached from some µi ∈ +ˆ +M in one step, we use some convex combination of points +µi in ˆ +M. Let w(µi, µsas′) be the weight wi associated with +µi when interpolating µsas′. We can use this interpolation +weight to define an approximate transition probability for +posterior as: +˜P(µ′|s, a, µ, θ) = +� +s′∈S +P(s′|s, a, θ)w(µ′, µsas′). +A sanity check that +˜P(µ′|s, a, µ, θ) is indeed a tran- +sition probability: +� +µ′∈ ˆ +M ˜P(µ′|s, a, µ, θ) += +1 and +˜P(µ′|s, a, µ, θ) ≥ 0. We choose the convex combination +that minimizes the weighted Euclidean norm of the differ- +ence between µ and each µi by solving the following linear +program: +min +w +� +i +wi||µi − µsas′||2 +(5) +s.t. +� +i +wiµi(θ) = µsas′(θ), ∀θ ∈ Θ +� +i +wi = 1, wi ≥ 0, ∀i. +With the approximation in the constraint in (4), we obtain +the following approximate bilevel DCP algorithm for a +given posterior set. For ease of notation, we denote by +C(s, a, θ) = Eθ[C(s, a, ξ)] the average cost at state s when +action a is taken, under the parameter value θ. +Algorithm 1 Approximate Bilevel DCP +input: posterior set ˆ +M +output: policy ˆπ∗, value function ˆV ∗ +1. Solve the following approximate bilevel DCP: +min +V +− +� +s∈S,µ∈M +α(s, µ)V (s, µ) +(6) +s.t. V (s, µ) ≤ min +φ +� +θ∈Θ +µ(θ)[Ψ(γ +� +µ′∈ ˆ +M,s′∈S +P(s′|s, a, θ) +w(µ′, µsas′)V (s′, µ′) + C(s, a, θ), φ)], ∀a ∈ A, s ∈ S, µ ∈ ˆ +M +where w(µ′, µsas′) is obtained by solving (5). +2. Obtain the policy +ˆπ∗(s, µ) = arg min +a∈A +ρµEθ[C(s, a, ξ) + γ ˆV ∗(s′, µ′)]. +Since the policy returned by Algorithm 1 is based on an ap- +proximate transition probability, there is a need to evaluate +the obtained policy. Next we show the approximate value +function obtained by Algorithm 1 is a lower bound on the +exact optimal value function V ∗. +Theorem 3.6. The approximate optimal value function ˆV ∗ +found by running Algorithm 1 is a lower bound on the exact +optimal value function V ∗. +We also develop an upper bound on the exact optimal value +function, using the obtained policy from Algorithm 1. The +obtained policy is a finite state controller (see (Hansen, +2013) for the definition of finite state controller). Let N be +the set of nodes in the controller such that we associate a +node ns,µ to each (s, µ) pair. The action chosen in node +ns,µ is determined by the policy ˆπ∗(a|s, µ). For a given +parameter θ, the transition probability to the next node +is P(ns′,µ′|ns,µ, a) = w(µ′, µsas′)P(s′|s, a). The value +function of the finite state controller can be computed by +ˆV ˆπ∗(ns,µ) = min +φ +� +θ∈Θ +µ(θ)[Ψ(c(s, a, θ) + γ +� +ns′,µ′∈N +w(µ′, µsas′)P(s′|s, a, θ) ˆV ˆπ∗(ns′,µ′), φ)]. +Similar to (Ahmadi et al., 2021), the value function can be +solved efficiently by DCP. It is also known from (Hansen, +2013) that the value function obtained by the finite state +controller ˆV ˆπ∗ serves as an upper bound for the optimal +value function. +Note that the inequality ˆV ∗ ≤ V ∗ ≤ ˆV ˆπ∗ provides infor- +mation about how well the optimal value function V ∗ is + +Approximate Bilevel Difference Convex Programming for Bayesian Risk Markov Decision Processes +approximated. As the posterior set ˆ +M gets closer to the true +one, the gap between the approximate value function and +the optimal value function gets smaller. +Next we incrementally add new posterior distributions to +the posterior set ˆ +M. Different methods can be employed to +produce new posterior distributions that are added to the set +ˆ +M at each iteration. We take a similar approach as (Poupart +et al., 2015), which is based on envelope techniques. It +considers the posterior distributions that can be reached in +one step from any posterior distribution in ˆ +M by executing +the policy ˆπ∗. As the number of posterior distributions to +be added might be excessive, we can prioritize them by +including the n reachable posterior distributions with the +largest weighted Euclidean distance to the posterior distri- +butions in ˆ +M, as determined by the interpolation outlined +in (5). Note that the point-based value iteration approach +in (Pineau et al., 2003) shares the similar idea, that is, to +include new posterior distribution that improves the worst- +case density as rapidly as possible, where density is defined +as the maximum distance from any posterior distribution to +ˆ +M. We summarize the new posterior set generation in the +following algorithm. +Algorithm 2 New Posterior Set Generation +input: policy ˆπ∗, posterior set ˆ +M, n +output: newly added posterior set ˆ +M′ +for each (s, µ) ∈ (S × ˆ +M) and s′ ∈ S do +µ′(θ) ∝ µ(θ)f(ξ|θ), where s′ = g(s, ˆπ∗(a|s, µ), ξ) +distµ′ ←− distance of µ′ to ˆ +M � ˆ +M′ +if distµ′ > 0 (i.e., µ′ not in ˆ +M � ˆ +M′) then +ˆ +M′ ←− ˆ +M′ �{µ′} +end if +if | ˆ +M′| > n (to reduce the size of ˆ +M′) then +for each µ′ ∈ ˆ +M′ do +distµ′ ←− distance of µ′ to ˆ +M � ˆ +M′\{µ′} +ˆ +M′ ←− ˆ +M′\{arg minµ′∈ ˆ +M′ distµ′} +end for +end if +end for +Algorithm 3 Approximate Bilevel DCP for Infinite-horizon +BR-MDPs +input: threshold ϵ, n, initial augmented state (s1, µ1) +output: policy ˆπ∗ +initialization: ˆ +M ←− {degenerate beliefs} �{µ1} +repeat +obtain (ˆπ∗, ˆV ∗) by running Algorithm 1 +evaluate policy ˆπ∗ by solving a DCP and obtain ˆV ˆπ∗ +ˆ +M ←− ˆ +M � ˆ +M′ generated by Algorithm 2 +until ˆV ˆπ∗(s1, µ1) − ˆV ∗(s1, µ1) ≤ ϵ +Combining Algorithm 1 and Algorithm 2, we now present +the full algorithm below (ABDCP), which iteratively add +to the new posterior set and solve a bilevel difference con- +vex program at each iteration. We are now ready to show +Algorithm 3 converges to a near-optimal policy. +Theorem 3.7. Algorithm 3 converges to a near-optimal +policy ˆπ∗, i.e., V ˆπ∗(s1, µ1) − V ∗(s1, µ1) ≤ ϵ, where ϵ is +the desired threshold. +4. Numerical Experiments +We illustrate the performance of the infinite-horizon BR- +MDP formulation with different choices of risk measures +and the proposed approximate bilevel DCP algorithm with +two offline planning problems. Code for the experiments is +included in the supplementary material. All algorithms are +implemented in Python and run on a 1.4 GHz Intel Core i5 +processor with 8 GB memory. Implementing details can be +found in the appendix. +• Path Planning +In the offline path planning prob- +lem, an autonomous car (agent) navigates a two- +dimensional terrain map represented by a 10 by 10 grid +along roads to the destination. The agent chooses from +four actions {up, down, left, right}. There are four +types of roads: {highway, main road, street, lane}. The +traffic time ξT +i in each type of road is assumed to be +independent and follows exponential distribution with +different rate, denoted by θT +i , i = 1, · · · , 4, where T +stands for traffic time. The parameter value is assumed +to be within the finite set. ξA +i ∈ {0, 1}, i = 1, · · · , 4 +denotes whether there is car accident in each type of +road, where A stands for accident. The probability +of car accident happening in each type of road is also +assumed to be independent, denoted by θA +i . The param- +eter value is assumed to be within the finite set. When +there is an car accident, the agent receives a constant +cost TA and makes no transition. Otherwise, the agent +transitions to the next road depending on the action it +takes and receives the cost, which is the traffic time for +traversing that type of road. The agent stops when it +reaches the destination. The discount factor γ = 0.95. +The agent is given a historical dataset H0 containing +past traffic times and car accident logs. +• Multi-item Inventory Control +In the offline multi- +item inventory control problem, the warehouse man- +ager decides how much to replenish from the set +{0, 1, · · · , Si − si} for each item i ∈ [K] at each time +stage, where Si is the storage capacity for item i, si is +the current inventory level for item i. The customer de- +mand is a random vector ξ = (ξ1, · · · , ξK) with each +ξi following a Poisson distribution with parameter θi. +The state transition is given by st+1 = max(st + at − +ξt, 0), the cost function is given by C(st, at, ξt) = +h · max(st + at − ξt, 0) + p · max(ξt − st − at, 0), + +Approximate Bilevel Difference Convex Programming for Bayesian Risk Markov Decision Processes +where h is the holding cost and p is the penalty cost. +The discount factor γ = 0.95. The warehouse manager +is given a historical dataset consisting of past customer +demands. +We adapt two methods to our offline planning problem and +evaluate their performances. +The first method (CALP) +comes from (Poupart et al., 2015) with a risk-neutral +POMDP formulation. +The second method (DR-MDP) +comes from (Xu & Mannor, 2010) with a distributionally +robust MDP formulation. Note that the BPO approach from +(Lee et al., 2018) solves a risk-neutral BA-MDP formula- +tion, where two separate encoders for the physical state and +belief state are designed to deal with the continuous latent +parameter space. It could have been a good benchmark if +its encoder design were made available. Apart from the two +benchmarks, we also compare with the nominal approach +(MLE), where a maximal likelihood estimator for the param- +eter is computed from the given dataset and then a policy is +obtained by solving the MDP with the plugged-in parameter +value. In our proposed algorithm (ABDCP) for the infinite- +horizon BR-MDP formulation, we consider two particular +risk measures, namely expectation and CVaR with different +risk levels α. It should be noted that when the considered +risk measure is expectation, our algorithm can be modified +and reduced to CALP. Similar observation is verified in +(Poupart et al., 2006), where the BA-MDP formulation is +transformed into a POMDP formulation. +For each of the considered algorithms, we obtain the cor- +responding optimal policy with the same dataset. It should +be noted that the calculations are carried out offline. The +obtained policy is then applied for risk-averse path planning +and evaluated on the true system, i.e., MDP with the true +parameter. This is referred to as one replication, and we +repeat the experiments for 200 replications on different in- +dependent datasets. Results for the path planning problem +can be found in Table 1 and Table 2, with different data +size N = 10 and N = 1000. Results for the multi-item +inventory control problem can be found in Table 3 and Ta- +ble 4, with different data size N = 10 and N = 1000. +The columns report the running time, expected performance +(cost), and the CVaR performance (cost) of our proposed al- +gorithm and benchmarks over the 200 replications. ABDCP- +EXP stands for our proposed algorithm ABDCP with ex- +pectation as the risk measure. ABDCP-CVaR stands for our +proposed algorithm ABDCP with CVaR as the risk measure. +We also show the histogram of the actual performance over +200 replications for our proposed algorithm and the nominal +benchmark on the path planning problem in Figure 1. We +summarize the main observations for the path planning prob- +lem. Similar observations can be made for the multi-item +inventory control problem. We include more observations +in the appendix. +BR-MDP hedges against epistemic uncertainty: in each +replication, data points are randomly sampled from the true +distribution. While facing the epistemic uncertainty, BR- +MDP formulation optimizes over a dynamic risk measure +that provides robustness. Table 1 shows that our proposed +ABDCP algorithm is the most robust in the sense of balanc- +ing the mean and variability of the actual cost. The CVaR +cost of our proposed algorithm is also lower than the other +benchmarks, showing that it avoids large costs. In contrast, +the nominal approach performs badly when the data size is +small, e.g. N = 5, indicating that it is not robust against the +epistemic uncertainty and suffers from the scarcity of data. +On the other hand, DR-MDP is overly conservative, even +though it has the smallest variability. This conservativeness +comes from two aspects. First, it always chooses to optimize +over the worst-case scenario, which rarely happens in the +true system. Second, the static worst-case risk measure pre- +vents it from adapting to the data realizations, which is one +of the motivations for the dynamic risk measure considered +in the BR-MDP formulation. +Convergence of ABDCP: the running time for a single +replication on the path planning problem using our pro- +posed ABDCP algorithm is affordable, and the proposed +algorithm solves the infinite-horizon BR-MDP in finite time. +In contrast, the infinite-horizon BR-MDP is intractable with +standard value iteration or policy iteration. +5. Conclusions +In this paper, we consider the offline planning problem +in MDPs with epistemic uncertainty, where we only have +access to a prior belief distribution over MDPs that is con- +structed by the offline data. We consider the infinite-horizon +BR-MDP that produces a time-consistent formulation and +provides the robustness against epistemic uncertainty. We +develop an efficient optimization-based approximation algo- +rithm that converges to the optimal policy. Our experiment +results demonstrate the efficiency of the proposed approxi- +mate algorithm, and show the robustness and the adaptivity +to future data realization of the infinite-horizon BR-MDP +formulation. +One of the future directions is to study the sample complex- +ity of the proposed algorithm. In its current form, we show +the convergence of the proposed algorithm without analysis +of the convergence rate. Another interesting direction is to +utilize function approximation to improve the scalability of +the proposed approach to more complex domains. Separate +encoders for the physical state and belief state have been +proposed in (Lee et al., 2018) to reduce the dimension of +the considered BA-MDP formulation, and adaptation from +the risk-neutral BA-MDP formulation to our risk averse BR- +MDP formulation with the designed policy network could +be interesting. + +Approximate Bilevel Difference Convex Programming for Bayesian Risk Markov Decision Processes +Approach +time (sec) +expected cost +CVaR (α = 0.95) cost +CVaR (α = 0.8) cost +ABDCP-EXP (CALP) +969.13(0.18) +70.06(0.51) +85.72 +82.06 +ABDCP-CVaR (α = 0.95) +2639.38(0.22) +67.51(0.24) +75.67 +73.72 +ABDCP-CVaR (α = 0.8) +2545.74(0.24) +66.02(0.38) +79.97 +75.50 +DR-MDP +62.34(0.11) +79.43(0.15) +81.64 +80.60 +Nominal +61.44(0.08) +82.59(0.59) +94.10 +92.46 +Table 1. Results for path planning problem. Running time for each replication, expected cost, and CVaR cost at different risk levels α are +reported for different algorithms. Standard errors are reported in parentheses. Number of data points is N = 10. +Approach +time (sec) +expected cost +CVaR (α = 0.95) cost +CVaR (α = 0.8) cost +ABDCP-EXP (CALP) +967.25(0.17) +64.15(0.05) +66.34 +65.97 +ABDCP-CVaR (α = 0.95) +2642.26(0.21) +65.18(0.03) +66.14 +65.76 +ABDCP-CVaR (α = 0.8) +2643.48(0.25) +65.17(0.04) +66.26 +65.84 +DR-MDP +63.15(0.09) +65.22(0.03) +66.43 +66.01 +Nominal +62.47(0.08) +64.31(0.12) +67.55 +65.59 +Table 2. Results for path planning problem. Running time for each replication, expected cost, and CVaR cost at different risk levels α are +reported for different algorithms. Standard errors are reported in parentheses. Number of data points is N = 1000. +Approach +time (sec) +expected cost +CVaR (α = 0.95) cost +CVaR (α = 0.8) cost +ABDCP-EXP (CALP) +1374.59(0.24) +3478.92(15.03) +4363.56 +4025.23 +ABDCP-CVaR (α = 0.95) +4109.21(0.35) +3072.84(10.24) +3651.75 +3472.44 +ABDCP-CVaR (α = 0.8) +4087.14(0.32) +2831.12(12.37) +3782.04 +3517.30 +DR-MDP +140.76(0.13) +3963.56(9.21) +4424.69 +4231.12 +Nominal +139.89(0.08) +3987.50(18.39) +4974.57 +4611.16 +Table 3. Results for multi-item inventory control problem. Running time for each replication, expected cost, and CVaR cost at different +risk levels α are reported for different algorithms. Standard errors are reported in parentheses. Number of data points is N = 10. +Approach +time (sec) +expected cost +CVaR (α = 0.95) cost +CVaR (α = 0.8) cost +ABDCP-EXP (CALP) +1377.27(0.22) +1806.45(0.57) +1825.06 +1823.71 +ABDCP-CVaR (α = 0.95) +4114.93(0.34) +1819.92(0.16) +1823.63 +1821.70 +ABDCP-CVaR (α = 0.8) +4002.62(0.33) +1817.21(0.19) +1824.97 +1822.39 +DR-MDP +138.26(0.11) +1826.03(0.12) +1828.80 +1827.62 +Nominal +136.38(0.10) +1802.34(1.28) +1836.04 +1828.99 +Table 4. Results for multi-item inventory control problem. Running time for each replication, expected cost, and CVaR cost at different +risk levels α are reported for different algorithms. Standard errors are reported in parentheses. Number of data points is N = 1000. +(a) ABDCP-EXP +(b) ABDCP-CVaR(α = 0.95) +(c) ABDCP-CVaR(α = 0.8) +(d) Nominal +Figure 1. Histogram of the actual performance over 200 replications for different algorithms. Number of data points is set to N = 10. + +25 +Mean: 70.06 +CVaR: 85.72 +requency +20 +15 +10 +5 +60 +7075 +8 +85 +costMean: 67.51 +CVaR: 75.67 +50 +30 +20 +10 +62.5 +每.067.570.072.575.077.5 +costMean: 66.02 +CVaR: 79.97 +40 +10 +60 +70 +75 +80 +cost.597 +OT' +25 +: 82. +CVaR: 94. +-ue +uanbr +15 +10 +5 +70 +80 +90 +costApproximate Bilevel Difference Convex Programming for Bayesian Risk Markov Decision Processes +References +Ahmadi, M., Rosolia, U., Ingham, M. D., Murray, R. M., +and Ames, A. D. Constrained risk-averse Markov deci- +sion processes. In Proceedings of the AAAI Conference +on Artificial Intelligence, volume 35, pp. 11718–11725, +2021. +Alagoz, O., Ayvaci, M. U., and Linderoth, J. T. Optimally +solving markov decision processes with total expected +discounted reward function: Linear programming revis- +ited. Computers & Industrial Engineering, 87:311–316, +2015. +Chow, Y. and Ghavamzadeh, M. 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(eds.), +Proceedings of the 2015 Winter Simulation Conference, +pp. 3714–3724, 2015. + +Approximate Bilevel Difference Convex Programming for Bayesian Risk Markov Decision Processes +A. Technical Proof +Proof of Lemma 3.2. Note that +(T πV )(s, µ) = ρµ1Eθ1[C(s1, a1.ξ1) + γρµ2Eθ2[C(s2, a2, ξ2) + · · · + γV (sk, µk)]] +for all positive integer k. As the terminal value function V (sk, µk) ≤ V ′(sk, µk) for all sk ∈ S, µk ∈ M, and using the +monotonicity of the convex risk measure ρ(X) ≤ ρ(Y ) if X(ω) ≤ Y (ω), ∀ω ∈ Ω, we have (T πV )(s, µ) ≤ (T πV ′)(s, µ). +Same analysis works for operator T . +Proof of Lemma 3.3. Let Cmax = maxs∈S,µ∈M |V (s, µ) − V ′(s, µ)|, we have +V (s, µ) − Cmax ≤ V ′(s, µ) ≤ V (s, µ) + Cmax +(7) +Applying T π on inequality (7), we have +(T πV )(s, µ) − γCmax ≤ (T πV ′)(s, µ) ≤ (T πV )(s, µ) + γCmax, +which is justified by the translation invariance of the convex risk measure. Then we have +max +s∈S,µ∈M |(T πV )(s, µ) − (T πV ′)(s, µ)| ≤ γ +max +s∈S,µ∈M |V (s, µ) − V ′(s, µ)|, +i.e., ||T πV − T πV ′||∞ ≤ γ||V − V ′||∞. Same analysis works for operator T . +Proof of Proposition 3.4. (i) Let π be the policy for which +V ≥ T πV. +(8) +Note that such policy exists as one can choose π that yields low current cost. Applying operator T πV to both sides of +inequality (8) and using Lemma 3.2, we have V ≥ (T π)tV , t = 1, 2, · · · . Note that the right hand side of the above +inequality represents the cost of a finite horizon problem with stationary policy π and with final value function V . Also note +that +(T π)tV = ρµ1Eθ1[C(s1, a1, ξ1) + · · · + γV (st+1, µt+1)] +≥ ρµ1Eθ1[C(s1, a1, ξ1) + · · · + γρµtEθt[C(st, at, ξt)]]. +Let t → ∞, we get V ≥ V π, where V π is the value function under policy π. Since V ∗ = minπ V π, we have V ≥ V ∗. +(ii) Consider an arbitrary policy π and a finite horizon problem with terminal cost V (st+1, µt+1). We have under the policy +π, +ρµ1Eθ1[C(s1, a1, ξ1) + · · · + γV (st+1, µt+1)] = ρµ1Eθ1[C(s1, a1, ξ1) + · · · + γρµtEθt[C(st, at, ξt) + γV (st+1, µt+1)]]. +Note that ρµtEθt[C(st, at, ξt) + γV (st+1, µt+1)] ≥ T V (st, µt) ≥ V (st, µt). Therefore, we have +ρµ1Eθ1[C(s1, a1, ξ1) + · · · + γV (st+1, µt+1)] ≥ ρµ1Eθ1[C(s1, a1, ξ1) + · · · + ρµt−1Eθt−1[C(st−1, at−1, ξt−1) + γV (st, µt)]]. +Continuing, we have +ρµ1Eθ1[C(s1, a1, ξ1) + · · · + γV (st+1, µt+1)] ≥ V (s, µ). +Let Cmax be an upper bound on |V (s, µ)|, ∀s ∈ S, µ ∈ M, we have +ρµ1Eθ1[C(s1, a1, ξ1) + · · · + γρµtEθt[C(st, at, ξt)]] ≥ V (s, µ) − Cmaxγt. +Passing to the limit t → ∞, we have for any policy π, V π(s, µ) ≥ V (s, µ). Therefore, the infimum over all policy π is +bounded from below by V (s, µ), i.e., V ∗ ≥ V . + +Approximate Bilevel Difference Convex Programming for Bayesian Risk Markov Decision Processes +Proof of Theorem 3.6. We first show V π(s, µ) is concave in µ for any policy π. Note that +V π(s, µ) = min +φ Eµ[Ψ(Eθ[C(s, a, ξ) + γV π(s′, µ′)], φ)]. +For 0 ≤ t ≤ 1 and any µ1, µ2 ∈ M, +V π(s, tµ1 + (1 − t)µ2) += min +φ Etµ1+(1−t)µ2[Ψ(Eθ[C(s, a, ξ) + γV π(s′, µ′)], φ)] += min +φ {Eµ1[Ψ(Eθ[C(s, a, ξ) + γV π(s′, µ′)], φ)] + (1 − t)Eµ2[Ψ(Eθ[C(s, a, ξ) + γV π(s′, µ′)], φ)]} +≥t min +φ1 {Eµ1[Ψ(Eθ[C(s, a, ξ) + γV π(s′, µ′)], φ1)]} + (1 − t) min +φ2 {Eµ2[Ψ(Eθ[C(s, a, ξ) + γV π(s′, µ′)], φ2)]} +=tV π(s, µ1) + (1 − t)V π(s, µ2). +The same analysis works for the optimal value function V ∗. Consider running Algorithm 1 with posterior set ˆ +M and the +entire posterior set M. Now applying Jensen’s inequality and by Theorem 12 in (Hauskrecht, 2000), we have ˆV ∗ ≤ V ∗. Note +that originally in (Hauskrecht, 2000), the proof is based on the fact that value function in partially observable Markov decision +process is convex in belief and the linear programming formulation has constraint V (b) ≥ R(s, a) + γ � +b′ P(b′|b, a)V (b′), +where R is the reward function. Since V is convex and by linear interpolation, applying Jensen’s inequality to the right +hand side of the constraint leads to ˆV (b) greater than V (b). Now we are in an opposite direction, by Jensen’s inequality and +concavity of V , we have ˆV ∗ ≤ V ∗. +Proof of Theorem 3.7. First we show Algorithm 3 terminates in finite time. Suppose not, i.e., ˆV ˆπ∗(s1, µ1)− ˆV ∗(s1, µ1) > ϵ. +As the number of iterations increases, +ˆ +M will contain an increasing number of reachable posterior distributions, since +Algorithm 3 is guaranteed to generate new reachable posterior distributions unless the current approximate optimal policy ˆπ∗ +is evaluated accurately. As the number of iterations goes to infinity, ˆ +M will eventually contain enough posterior distributions +to accurately evaluate all policies ˆπ∗ that Algorithm 3 produces infinitely often. Since Algorithm 3 terminates as soon as +Algorithm 1 produces a policy that is evaluated accurately, we reach a contradiction. +Nest we show the algorithm converges to V ∗. Suppose that the algorithm terminates, but it converges to a suboptimal policy +˜π. By Theorem 3.6, we know that ˆV ∗ ≤ V ∗, since V ∗ ≤ ˆV ˜π and the algorithm terminates when ˆV ˜π − ˆV ∗ ≤ ϵ. Then we +have ˆV ˜π − V ∗ ≤ ϵ, which reaches a contradiction. Thus the algorithm must converge to V ∗. +B. Implementation Details +B.1. Offline Path Planning +Figure 2. Path planning terrain +map. Colors indicate the road +types as follows–blue: high- +way, red: main road, orange: +street, green: lane. +An autonomous car (agent) navigates a two-dimensional terrain map represented by a +10 by 10 grid along roads to the destination, as shown in Figure 2. The agent chooses +from four actions {up, down, left, right}, as long it remains on the road. There are +four types of roads: {highway, main road, street, lane}. The traffic time ξT +i in each +type of road is assumed to be independent and follows exponential distribution with dif- +ferent rate, denoted by θT +i , i = 1, · · · , 4, where T stands for traffic time. Specifically, +the true rates are θT +1 = 1, θT +2 = 0.5, θT +3 = 0.2, and θT +4 = 0.1 but unknown to the +agent. We view the parameter as a random variable, whose value is assumed to be within +the following finite set {0.05, 0.1, 0.2, 0.25, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1, 1.5, 2, 2.5}. +ξA +i +∈ {0, 1}, i = 1, · · · , 4 denotes whether there is car accident in each type of road, +where A stands for accident. The probability of car accident happening in each type of road +is also assumed to be independent, denoted by θA +i . Specifically, the true probabilities are +θA +1 = 0.3, θA +1 = 0.2, θA +1 = 0.1 and θA +1 = 0.05 but unknown to the agent. We view the +parameter as a random variable, whose value is assumed to be within the following finite set +{0.05, 0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.45, 0.5}. When there is an car accident, the agent +receives a constant cost TA = 10 and makes no transition. Otherwise, the agent transitions +to the next road depending on the action it takes and receives the cost, which is the traffic +time for traversing that type of road. The agent stops when it reaches the destination. The + +origin +destinationApproximate Bilevel Difference Convex Programming for Bayesian Risk Markov Decision Processes +discount factor γ = 0.95. The agent is given a historical dataset H0 of size N containing past traffic times and car accident +logs, and uses the given dataset to construct the prior for the transition rate and probability of car accident. Other parameters +are as follows: number of points to be added at each iteration n = 20, threshold ϵ = 0.1. +B.2. Multi-item Inventory Control +The warehouse manager (agent) decides how much to replenish from the set {0, 1, · · · , Si − si} for each item i ∈ [K] +at each time stage, where K = 5 is the number of different items, Si = 100 is the storage capacity for each item i, si +is the current inventory level for each item i. The customer demand is a random vector ξ = (ξ1, · · · , ξK) with each ξi +following a Poisson distribution with parameter θi. The true parameter is θ1 = 10, θ2 = 15, θ3 = 20, θ4 = 25, and +θ5 = 30 but unknown to the agent. We view the parameter as a random variable, whose value is assumed to be within +the following finite set {5, 6, 7, · · · , 33, 34, 35}. The state transition is given by st+1 = max(st + at − ξt, 0), where at +is the amount of inventory to be replenished. Inventory level is not allowed to drop below zero (no backlog). When the +customer demand is higher than the supply, there is a penalty cost p for each unit of unsatisfied demand. When the customer +is lower than the supply, there is a holding cost h for each unit of overstock. In particular, for different items, p1 = 4, +p2 = 5, p3 = 6, p4 = 7, p5 = 8, h1 = 2, h2 = 3, h3 = 4, h4 = 5, h5 = 6. The cost function at each stage is then given by +C(st, at, ξt) = hT · max(st + at − ξt, 0) + pT · max(ξt − st − at, 0). The discount factor γ = 0.95. The agent starts with +0 inventory and is given a historical dataset H0 of size N containing past customer demands for different items, and uses the +given dataset to construct the prior for the rate parameter. Other parameters are as follows: number of points to be added at +each iteration n = 20, threshold ϵ = 0.1. +B.3. DR-MDP Details +The DR-MDP approach, or Distributionally Robust Markov Decision Process, is a method for decision making under +uncertainty where the ambiguity set, or the set of possible distributions for the uncertain parameters, is constructed using +prior knowledge about the probabilistic information. However, this prior knowledge is not always readily available from a +given data set, making the construction of the ambiguity set difficult in some cases. +We note that the Bayesian Risk Optimization (BRO) approach has a distributionally robust optimization (DRO) interpretation. +In particular, for a static stochastic optimization problem, it has been shown in (Wu et al., 2018) that the BRO formulation +with the risk functional taken as Value-at-Risk (VaR) with a confidence level of 100% is equivalent to a DRO formulation with +the ambiguity set constructed for the uncertain parameter, θ. This means that BRO and DRO can be used interchangeably, +depending on the problem at hand and the level of uncertainty and prior knowledge about the parameters. Therefore, for a +given problem when prior knowledge about the probabilistic information is not readily available, we adapt DR-MDP to +our considered problem as follows: we use samples of the uncertain parameter, θ, drawn from the posterior distribution +computed from a given data set. This allows us to construct an ambiguity set for θ using the available data, instead of relying +on prior knowledge. Once we have samples of θ, we can obtain the optimal policy that minimizes the total expected cost +under the most adversarial θ among the samples. +B.4. Bilevel Optimization +We show the bilevel DCP can be reduced to a single-level DCP. Specifically, we show this transformation for the exact +bilevel DCP in (4), and the same technique can be applied to the approximate algorithm. +Consider a general bilevel optimization problem: +min +xu,xl F(xu, xl) +(9) +s.t. xl ∈ arg min +xl +{f(xu, xl) : g(xu, xl) ≤ 0} +G(xu, xl) ≤ 0 +where xu is the upper-level variable, xl is the lower-level variable, G denotes the upper-level constraints, g denotes the +lower-level constraints, F denotes the upper-level objective function, f denotes the lower-level objective function. The +Karush-Kuhn-Tucker (KKT) conditions are a set of necessary and sufficient conditions for a solution to be optimal in a +convex optimization problem. When the lower-level problem in a bilevel optimization problem is convex and sufficiently +regular, the KKT conditions can be used to reformulate the problem as a single-level constrained optimization problem, + +Approximate Bilevel Difference Convex Programming for Bayesian Risk Markov Decision Processes +which is typically easier to solve. The general bilevel optimization problem 9 can then be reduced to the following +single-level optimization: +min +xu,xl F(xu, xl) +s.t. G(xu, xl) ≤ 0 +∇xlL(xu, xl, λ) = 0 +g(xu, xl) ≤ 0 +λg(xu, xl) = 0 +λ ≥ 0 +where L(xu, xl, λ) = f(xu, xl) + λg(xu, xl) is the Lagrangian function. In the bilevel DCP (4), V is the upper-level +variable and φ is the lower-level variable. The constraint in (4) can be rewritten as: +φ ∈ arg min +φ∈Φ +Eµ[Ψ(Eθ[C(s, a, ξ) + γV (s′, µ′)], φ)] +V (s, µ) − Eµ[Ψ(Eθ[C(s, a, ξ) + γV (s′, µ′)], φ)] ≤ 0. +Since the lower-level problem is convex, we can reformulate the bilevel DCP (4) as a single-level DCP problem. +min +V +− +� +s∈S,µ∈M +α(s, µ)V (s, µ) +s.t.V (s, µ) − Eµ[Ψ(Eθ[C(s, a, ξ) + γV (s′, µ′)], φ)] ≤ 0 +∇φEµ[Ψ(Eθ[C(s, a, ξ) + γV (s′, µ′)], φ)] = 0 +∀a ∈ A, s ∈ S, µ ∈ M. +B.5. Additional Observations +We show additional observations for the path planning problem on Table 1 and Table 2. +Larger data size reduces epistemic uncertainty: when there are more data, the posterior distribution used in BR-MDP +formulation and the MLE estimator used in the nominal approach converges to the true parameter, which reduces to solving +an MDP with known transition probability and cost function. Therefore, the optimal policies and the actual costs tend to be +the same. +Effect of risk measures: although both risk measures (expectation and CVaR) result in time-consistent optimal policy +for each considered formulation, they provide different levels of robustness. Even though the expectation case is faster to +compute, it provides the least robustness, especially when the data size is small. For the CVaR risk measure, different risk +level α also affects the robustness. As α increases, the agent is more risk-averse, and the CVaR cost is smaller since it avoids +more severe costs, as is shown in Figure 1(b) and Figure 1(c). But this comes with a price: its expected cost is higher. It is +intuitive: even though the agent avoids severe costs, it also forfeits a chance to traverse a path that is likely to have less +traffic, even though the likelihood is small. This is shown as a right-shift of the actual performance distribution from Figure +1(c) and Figure 1(b). + diff --git a/BNFIT4oBgHgl3EQf_iwr/content/tmp_files/load_file.txt b/BNFIT4oBgHgl3EQf_iwr/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3836441c340cceaad581e49fb5c4c124cb297943 --- /dev/null +++ b/BNFIT4oBgHgl3EQf_iwr/content/tmp_files/load_file.txt @@ -0,0 +1,1012 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFIT4oBgHgl3EQf_iwr/content/2301.11415v1.pdf,len=1011 +page_content='Approximate Bilevel Difference Convex Programming for Bayesian Risk Markov Decision Processes Yifan Lin 1 Enlu Zhou 1 Abstract We consider infinite-horizon Markov Decision Processes where parameters, such as transition probabilities, are unknown and estimated from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFIT4oBgHgl3EQf_iwr/content/2301.11415v1.pdf'} +page_content=' The popular distributionally robust ap- proach to addressing the parameter uncertainty can sometimes be overly conservative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFIT4oBgHgl3EQf_iwr/content/2301.11415v1.pdf'} +page_content=' In this paper, we utilize the recently proposed formu- lation, Bayesian risk Markov Decision Process (BR-MDP), to address parameter (or epistemic) uncertainty in MDPs (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFIT4oBgHgl3EQf_iwr/content/2301.11415v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFIT4oBgHgl3EQf_iwr/content/2301.11415v1.pdf'} +page_content=' To solve the infinite-horizon BR-MDP with a class of con- vex risk measures, we propose a computationally efficient approach of approximate bilevel differ- ence convex programming (ABDCP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFIT4oBgHgl3EQf_iwr/content/2301.11415v1.pdf'} +page_content=' The opti- mization is performed offline and produces the optimal policy that is represented as a finite state controller with desirable performance guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFIT4oBgHgl3EQf_iwr/content/2301.11415v1.pdf'} +page_content=' We also demonstrate the empirical performance of the infinite-horizon BR-MDP formulation and proposed algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFIT4oBgHgl3EQf_iwr/content/2301.11415v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFIT4oBgHgl3EQf_iwr/content/2301.11415v1.pdf'} +page_content=' Introduction In a Markov decision process (MDP), an agent must make decisions in a sequence while facing uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFIT4oBgHgl3EQf_iwr/content/2301.11415v1.pdf'} +page_content=' In this situation, some parameters of the MDP, such as the transi- tion probabilities and costs, may be unknown and must be estimated from available data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFIT4oBgHgl3EQf_iwr/content/2301.11415v1.pdf'} +page_content=' The problem then becomes how to determine the best course of action, given the limited or possibly absent data, in order to minimize the expected total cost and optimize the decision-making process under these uncertain parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFIT4oBgHgl3EQf_iwr/content/2301.11415v1.pdf'} +page_content=' An alternative approach to addressing the epistemic uncer- tainty in MDP is through the use of distributionally robust MDPs (DR-MDPs, (Xu & Mannor, 2010)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFIT4oBgHgl3EQf_iwr/content/2301.11415v1.pdf'} +page_content=' This method considers the unknown parameters as random variables and assumes that their distributions belong to an ambiguity set 1H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFIT4oBgHgl3EQf_iwr/content/2301.11415v1.pdf'} +page_content=' Milton Stewart School of Industrial and Systems Engineer- ing, Georgia Institute of Technology, Atlanta, GA, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFIT4oBgHgl3EQf_iwr/content/2301.11415v1.pdf'} +page_content=' Corre- spondence to: Enlu Zhou 5: +4 +def fd(model, test, train): +5 +assert len(test) < 500 +6 +model.fit(train, test) +7 +fitit = fd +8 +9 +else: +10 +fitit = lambda model, test, train: +model.fit(train, test) +↩→ +11 +15 +from sklearn.svm import LinearSVC +16 +17 +from pystruct.models import MultiClassClf +18 +from pystruct.learners import (NSlackSSVM, +OneSlackSSVM, SubgradientSSVM, +FrankWolfeSSVM) +↩→ +↩→ +19 +20 +digits = load_digits() +21 +X, y = digits.data, digits.target +22 +X = X / 16. +23 +X_train, X_test, y_train, y_test = +train_test_split(X, y) +↩→ +24 +25 +# we add a constant 1 feature for the bias +26 +X_train_bias = np.hstack([X_train, +np.ones((X_train.shape[0], 1))]) +↩→ +41 +42 +fw_bc_svm = FrankWolfeSSVM(model, C=.1, +max_iter=50) +↩→ +71 +72 +libsvm = +LinearSVC(multi_class='crammer_singer', +C=.1) +↩→ +↩→ +73 +start = time() +74 +fitit(libsvm, X_train, y_train) +75 +time_libsvm = time() - start +76 +print("Score with sklearn and libsvm: %f +(took %f seconds)" % +(libsvm.score(X_test, y_test), +time_libsvm)) +↩→ +↩→ +↩→ +77 +78 +start = time() +79 +fitit(fw_bc_svm, X_train_bias, y_train) +Figure 1. A Running example +see in Figure 3, is not possible in general in Python. +Hence, we combine the two sets into a single one. +The framework paper defines relevant program features +at the top of page 694, which we excerpt here in Figure 2. +We need two minor changes: +InstVariables is taken to be the set of strings possibly +used as field names, rather than a set of declared field +names, which it is in the original framework. While +2 + +Serenity: Library Based Python Code Analysis for Code Completion and Automated Machine Learning +, , +field names can be defined in Python, this is entirely +optional so we ignore such definitions. +NewSites becomes the same as the set CallSites to effect +the second item above; that is, there is one set that +combines all possible call sites and creation sites. This +represents the fact that every site can potentially see +both classes and functions. +Class all class declarations in the program +InstVariable all instance variable declara- +tions of the program +Procedure all procedure declarations of the +program +Variable all variable names used in the pro- +gram +CallSite all call sites in the program +NewSite all new sites in the program +LoadSite all loads of instance variables in +the program +StoreSite all stores to instance variables in +the program +Figure 2. Program features from [18] +3.2 +Language modeling +Figure 3 illustrates the kind of dynamism with which anal- +ysis of Python must contend, in this case 5 different options +for the meaning of X() on line 45 based on the value supplied +as sys.argv[1] and sometimes sys.argv[2]: +class1 class X (line 4) defines an ordinary class named +X, of which line 45 creates an instance. +class2 class X (line 11) defines a class named X that +redefines the new operator, so line 45 just returns 0. +def1 def X (line 16) defines a function named X, and +calling it at line 45 returns 1. +def2 X = lambda. . . (line 20) creates a closure and +assigns it to X; calling the closure at line 45 returns 2. +import The module X (line 23) overrides default module +behavior to become callable and return 3 at line 45. +method static X (line 32) is assigned the static method +s of class X (line 28) which returns 5 at line 45. +method instance X (line 41) gets a bound instance method +(i.e. a closure over y) i of class X (list 35), returning 4 +at line 45). +Note that all of these definitions of X can flow to the same call +at line 45, so there is literally no syntactic distinction between +different kinds of allocations, calls, and even modules in some +cases. And class and function names are all first class. Thus +analysis must handle these basic operations in a dynamic +manner, unlike e.g. Java, where calls, allocations and imports +have clear syntactic distinctions. Note further that even basic +method calls require closures to handle line 41. +1 +import sys +2 +3 +if sys.argv[1] == "class1" or sys.argv[1] == "inst": +4 +class X: +5 +pass +6 +7 +if sys.argv[1] == "inst": +8 +X = X() +9 +10 +elif sys.argv[1] == "class2": +11 +class X: +12 +def __new__(*args): +13 +return 0 +14 +15 +elif sys.argv[1] == "def1": +16 +def X(): +17 +return 1 +18 +19 +elif sys.argv[1] == "def2": +20 +X = lambda: 2 +21 +22 +elif sys.argv[1] == "import": +23 +import X +24 +25 +elif sys.argv[1] == "method": +26 +27 +if sys.argv[2] == "static": +28 +class X: +29 +def s(): +30 +return 5 +31 +32 +X = X.s +33 +34 +elif sys.argv[2] == "instance": +35 +class X: +36 +v = 4 +37 +def i(self): +38 +return self.v +39 +40 +y = X() +41 +X = y.i +42 +43 +44 +print(str(X)) +45 +print(str(X())) +Figure 3. Dynamic code examples +The X() at line 45 is a call on X, and this allows us to +use standard dynamic dispatch to model all of this behavior, +using synthetic "methods" where needed to handle language +semantics. We will use a similar "dispatch" at field accesses +to handle the difference between class and instance fields, +which again can only be known from the object accessed. We +shall make use of these indirections to define our framework +model in Section 3.3. +3 + +, , +Wenting Zhao, Ibrahim Abdelaziz, Julian Dolby, Kavitha Srinivas, Mossad Helali, and Essam Mansour +We adopt the terminology of Grove et al. [18] to present +our work as extensions to standard object-oriented call graph +construction. To fit our dynamic Python context, we make +a few changes to the core definitions of that work. These +changes reflect that Python does not require that fields be +declared in order to be used, and it makes no syntactic dis- +tinction between calls and allocations. Furthermore, as is +standard for representing first-class entities in an object- +oriented framework, we have one class for each first-class +entity. As Figure 3 shows, classes, functions, methods and +modules are all first-class, so our set of classes for analysis +includes the following: +𝐶𝑐𝑙𝑎𝑠𝑠 a class representing program class C +𝐶𝑖𝑛𝑠𝑡 a class representing instances of class C +𝑀𝑖𝑛𝑠𝑡 a class representing instances of module M +𝐷𝑖𝑛𝑠𝑡 a class representing instances of function D +𝑆𝑖𝑛𝑠𝑡 a class representing the instance of script S +Now most of the irregularities of Python calls and creations +are handled by treating every call site as a CallSite for each +receiver type ∗𝑖𝑛𝑠𝑡 and as a NewSite for every receiver type +∗𝑐𝑙𝑎𝑠𝑠. The site on line 45 in Figure 3 would have some types +handled by each mechanism. Fields are also handled seam- +lessly: on line 32, X is a 𝐶𝑐𝑙𝑎𝑠𝑠, and on line 41 X is a 𝐶𝑖𝑛𝑠𝑡, so +static and instance state are handled by making static fields +be instance fields of the class object. +Call graph construction starts with a root stub that creates +an instance of the main script 𝑆𝑖𝑛𝑠𝑡 and calls it. +3.3 +Framework modeling +In many situations, it is difficult or impossible to find actual +code for Python imports: there is no fixed relationship be- +tween names in import statements and locations of actually +source code. Even if there were, the structure of Python li- +braries is such that large amounts of the code is native and +hence a Python analysis framework is not applicable. Even +if it were possible to find Python code, many libraries are +large enough to make precise analysis challenging. In our +case, we are interested in the behavior of application code +rather than library internals, so we minimize these issues by +largely not analyzing framework code. +Our model, called Turtles3, abstracts Python frameworks +to capture how the framework interacts with user code and +to ignore all of its internal details. Specifically, we model +four aspects, all using the indirections of Section 3.2: +1. We model import statements as returning a new frame- +work, denoted by the name of the imported module. +The framework is an opaque object with no function- +ality beyond implementing the model. +2. Calls to framework functions and methods typically +return something, which is then possibly used by the +user code. We model every call to the framework as +3from "turtles all the way down". This phrase is of unknown origin, see +https://en.wikipedia.org/wiki/Turtles_all_the_way_down +returning a new object from it; this model is transitive, +so calls on those objects return further new objects +from the framework. We label these objects with the +path by which they are accessed. +3. Accesses to fields of framework objects have little +meaning in our model since we do not model the frame- +work state at all. However, user code typically expects +that a field access return something, so we model all +such field accesses as returning the container object. +4. Arguments to turtle methods are mostly ignored, since +we do not model what the framework does to them; +however, sometimes functions are passed as parame- +ters, and we assume that the framework might call it. +Since we do not model internal framework state, the +model invokes callbacks from where they are passed +as arguments. +The framework of Grove et al. [18] provides the customiza- +tion support needed to implement this model. We start by +introducing a new type of class, 𝑇𝑝𝑎𝑡ℎ, that represents a tur- +tle, i.e. an opaque model object. Item 1 is implemented by +modeling import M statements as a call to a synthetic import +procedure with M as its argument. This call is modeled as +returning a 𝑇𝑀. Item 2 is implemented as a Procedure Key +Selection Function (PKS) which takes the receiver of a type +𝑇𝑝𝑎𝑡ℎ and the name 𝑛 of the called procedure and returns a +new turtle of 𝑇𝑝𝑎𝑡ℎ.𝑛. Item 3 is implemented by simply re- +turning self when reading any field of any 𝑇𝑝𝑎𝑡ℎ type. Item 4 +is implemented as a PKS that generates calls for every argu- +ment that is of a function type (this is not illustrated in our +example). +3.4 +Inheritance from Turtles +One wrinkle in our data is that application classes often +inherit from turtle classes, meaning that method calls on +self should logically be turtle methods when the method +read is never assigned. That is, if a read of self is to a field +or method that is never assigned and the class inherits from +a turtle, the read should return a new turtle object to capture +unknown superclass behavior. However, this is tricky to do +because, since methods and fields can be assigned anywhere +in the code, it is not in general possible to know if one will not +be assigned until analysis terminates. What we need to do +is record such reads and, when analysis terminates, process +them as turtle reads and restart analysis. This restarting itself +may need to be repeated, since reading one turtle could make +more code reachable. +3.5 +Analysis of running example +When this analysis is applied to the running example (Fig- +ure 1), the result is the dataflow graph shown in Figure 4. To +illustrate our framework model, observe the import call of +LinearSVC on line 15; as an import, this returns an object of +type 𝐿𝑖𝑛𝑒𝑎𝑟𝑆𝑉𝐶𝑖𝑛𝑠𝑡, that is, an instance of the module. When +4 + +Serenity: Library Based Python Code Analysis for Code Completion and Automated Machine Learning +, , +load_digits +digits.data +digits.target +X +y +X/16 +train_test_split +X_train +X_test +y_train +y_test +hstack +fit (fd) +fit (fd) +LinearSVC +FrankWolfeSSVM +Invocations +arg 0, flow +arg > 0 +Reads +fit (lambda) +fit (lambda) +Figure 4. Dataflow graph for the running example +this is called (line 72), it returns a turtle of type 𝑇𝐿𝑖𝑛𝑒𝑎𝑟𝑆𝑉𝐶, +illustrated by the green node labeled LinearSVC. When fit +is called on this object in the fitit functions (line 74), +item 2 means it returns a derived turtle of type𝑇𝐿𝑖𝑛𝑒𝑎𝑟𝑆𝑉𝐶.𝑓 𝑖𝑡, +shown as a green node labeled fit. Since fit is called on +LinearSVC, a black data flow edge connects them. On the +other hand, the other non-self arguments to fit are shown +with red arrows. Other turtle functions are shown similarly: +load_digits, train_test_split, hstack, FrankWolfeSSVM. +Note that analysis has no idea what these functions do, just +that they pass data. Note that fitit is a variable holding one +of two first-class functions, and it is called for both of the ML +models created. To get the precise results shown in Figure 4 +requires analysis infrastructure that handles first-class func- +tions and also does context-sensitive analysis. In particular, +the model objects and the data flow to both the normal and +debugging functions assigned to fit, since both potentially +flow to fitit. In the figure, the nodes are distinguished with +labels of the function in which they occur. +Other nodes in Figure 4 represent local dataflow. The top- +most two blue nodes represent reads of the data and target +fields of data, so they have edges from the load_digits call +and edges to their respective variables X and y. X is scaled +by 16, shown by the nodes labeled X/16. X/16 and y then +flow to train_test_split with red edges since they are +arguments. +This graph focuses on data flow, which captures patterns +of how the various turtle APIs are used across programs. +This allows us to learn patterns that enable our applications. +3.6 +Implementation +Our analysis is implemented using WALA and its support +for both Python 2 and Python 3 using the Jython system. +WALA is built to be extensible, and we used several features +to ease our implementation work. +The main extension is for handling turtles. For item 1, we +override the model function that handles import to return a +synthetic object with a turtle type named for the given mod- +ule. For item 2, we override the selection of called methods +for turtle classes so that any call goes to a synthetic method +that creates and returns a turtle with the appropriate ex- +tended turtle name. For item 2, this synthetic method mostly +ignores its arguments, except generating a call to each one to +handle callbacks. For item 3, we override the code handling +field reads to simply return the container if it is of turtle +type. +The other configuration is to add aggressive context sen- +sitivity for all turtle types. Since the synthetic methods are +trivial anyway, it is cheap to ensure that every call site is +analyzed separately. +4 +Code Completion Application +The core research question we ask is how useful Serenity’s +analysis is and whether it can help other applications, de- +spite the challenges in modeling dynamic languages such as +Python accurately. As a first application, we examine a code +completion use case, which we cover below in detail. By code +completion, we refer to the problem where, when given a +snippet of a program, the problem is to predict a function +call, analogous to what an IDE does for method suggestions. +We do not refer to code generation given natural language +descriptions of code requirements, as in the Codex model +that powers GitHub Co-Pilot [8] or even models that gener- +ate entire functions in a generative style based on function +signatures or snippets of code, such as CodeT5 [41]. Our +observation is that for code completion, the analysis require- +ment is that the methods be callable from a specific type, and +so analysis for code completion is focused on detecting the +types of objects. For languages such as Python, type infer- +ence is hard, but our hypothesis is that code completion can +benefit significantly from the data flow analysis that Serenity +produces, simply because data flow can provide a focused +context for code completion. +Recently, there have been a plethora of neural models of +code such as [13], [19], [22], [41] trained with the objective +of either predicting randomly masked tokens in code, or +predicting the very next token, which one might assume is +consistent with the task of code completion. Our research +question is whether one can leverage the extensive training +of these models on millions of programs to perform code +completion. Specifically, we asked whether data flow analy- +sis provided by Serenity can improve code completion when +combined with these neural models. If data flow analysis +does provide any signal from Serenity, it should improve per- +formance on code completion task even with the extensive +training these language models already had. We therefore +5 + +, , +Wenting Zhao, Ibrahim Abdelaziz, Julian Dolby, Kavitha Srinivas, Mossad Helali, and Essam Mansour +1 +print("Score with pystruct subgradient +ssvm: %f (took %f seconds)" % +(np.mean(y_pred == y_test), +time_subgradient_svm)) +↩→ +↩→ +↩→ +2 +3 +# the standard one-vs-rest multi-class +4 +# would probably be as good and faster +5 +# but solving a different model +6 +libsvm = +LinearSVC(multi_class='crammer_singer', +C=.1) +↩→ +↩→ +7 +start = time() +8 +libsvm.fit(X_train, y_train) +9 +time_libsvm = time() - start +10 +print("Score with sklearn and libsvm: %f +(took %f seconds)" % +(libsvm.score(X_test, y_test), +time_libsvm)) +↩→ +↩→ +↩→ +11 +12 +13 +start = time() +14 +fw_bc_svm.? +Figure 5. Code snippet used for prediction +1 +from sklearn.cross_validation import +train_test_split +↩→ +2 +from pystruct.models import MultiClassClf +3 +from pystruct.learners import (NSlackSSVM, +OneSlackSSVM, +↩→ +4 +digits = load_digits() +5 +digits.data +6 +digits.target +7 +X = X / 16. +8 +train_test_split(X, y) +9 +X_train_bias = np.hstack([X_train, +np.ones((X_train.shape[0], 1))]) +↩→ +10 +model = +MultiClassClf(n_features=X_train_bias.shape[1], +n_classes=10) +↩→ +↩→ +11 +fw_bc_svm = FrankWolfeSSVM(model, C=.1, +max_iter=50) +↩→ +12 +fw_bc_svm.? +Figure 6. Code snippet corresponding to a slice from the +analysis graph +modeled code completion as a fine tuning task, and varied +the training inputs of fine tuning to be one of the three +conditions shown below: +• All code as text prior to the function call +• A slice of the code restricted to source expressions that +are relevant to a function call in data flow +• Both code as text, as well as the slice, separated by a +token to distinguish the two inputs. +For all text code prior to a function call, there are limits on +how many tokens modern language models can fit. That is, +when the code goes beyond the limit, truncation is needed +in order for the models to run. A widely-used truncation +strategy is to only keep 𝑛 tokens prior to the prediction point, +where 𝑛 is the maximum sequence length, which can lead to +fairly local information, as shown in Figure 5 for our running +example shown in Figure 1. The key prediction in Figure 5 +is to predict what method will be called on fw_bc_svm, but +notice that the construction of fw_bc_svm is out of the scope +of the truncation4. +For obtaining the slice restricted purely to dataflow, given +a program and its corresponding dataflow graph, to predict +the function call, we start at a node that we would like to +predict, reverse all edges coming into the node, and find +all reachable nodes. Each node in the reachability set corre- +sponds to a source expression in the original program, and +we only include the expressions that are not sub-expressions +of any other expressions as features. Then, we order these +expressions according to their positions in the source files, +and add in variable names from the analysis artifacts so the +code looks more or less like real code that the language mod- +els have been trained on. Figure 6 shows an example of such +a dataflow based slice looks for the code in Figure 1. Here we +start the fw_bc_svm.fit call in Figure 4, reverse all edges +coming into the node, and perform a reachability analysis, +to gather the slice, adding variable names such as digits = +load_digits(). In this example, dataflow analysis does give +important information useful for predicting the function call, +because the slice brings in non-local but relevant code such +as the definition of fw_bc_svm into the scope of text that +can be fed to a neural model. +4.1 +Dataset +We used the popular benchmark of ETH150K [35], which +comes with 100K programs used for training, and the re- +maining 50K used as a testing set. ETH150K was analyzed +using Serenity, and 147,288 of 150,000 files were successfully +analyzed. For the analyzed files, we parsed each file with a +Python AST parser, and gathered all function calls. For each +function call identified by the AST, we examined whether +we could find the function in the analysis output, and if it +was found in the output, we checked if the source location of +the call matched that in the AST. Our observation has been +that the Jython source mappings can be wrong sometimes, +so we used both metrics to measure the completeness of the +analysis. The analysis found 58.77% of function calls in the +AST with matching source locations, and 67.36% of function +calls when the requirements to match source was relaxed. +Manual inspection on a few cases where source locations did +not match indicated that the problem was indeed mapping +4In this example, truncation was set to 1024 tokens, as per the requirements +of one of the CuBERT models [22] +6 + +Serenity: Library Based Python Code Analysis for Code Completion and Automated Machine Learning +, , +1 +def f_Hp(self, pars, p, inpt, target): +2 +eps = 1E-6 +3 +deriv = self.fprime(pars, inpt, +target) +↩→ +4 +offseted = self.fprime(pars + p * +eps, inpt, target) +↩→ +5 +return (offseted - deriv) / eps +Figure 7. Example of code where a leaf node is an expression +being incorrect in Jython. Further investigation revealed that +many of the missing calls are instances of Python primitives +that Serenity does not model and treats as no-ops, such as +repr and FutureWarning. A small fraction was found to +be genuinely dead code, especially when Python files were +integral parts of a larger application, as they often are in +ETH150K. +To generate the slices, we started with leaf nodes, and +restricted ourselves to cases where the nodes had at least a +depth of 1 when the edges were reversed. We note that in a +majority of cases, leaf nodes were actually expressions, as +shown in the example code in Figure 7. We ignored these +in creating our dataset because we were focused on a prob- +lem that cannot be solved by a pure lexical analysis of code. +When we restricted ourselves to nodes that were potentially +function calls rather than expressions, we generated slices +from 65.35% of the programs where there existed at least one +slice where the leaf node was likely a function call. For the +train and test sets of programs, we generated 334,415 slices +and 162,847 slices respectively by iterating over all the leaf +nodes in dataflow graphs. Once we consider leaf nodes as +nodes for our prediction, there were a total of over 65K labels +that were generated across train and test sets for code com- +pletion. Figure 8 plots the cumulative frequency distribution +of labels against the number of labels. As shown in the Figure, +the distribution of labels follows the usual power law, but we +note that the most popular label appeared across train and +test only 1.7% of the time, and the top 10 labels cumulatively +appeared only 11.0% of the time. In other words, this is a +difficult classification problem5. We note that this method +of declaring code completion is more realistic compared to +other means for code completion (such as measuring next +token prediction), in the sense that this is often the case that +IDEs focus on. +4.2 +Language model selection +To decide on the best neural model to use as a basis for our +code completion experiments, we tested a number of code +related language models including CodeBERT [13], Graph- +CodeBERT [19], CuBERT [22] and CodeT5 [41]. CodeBERT[13] +5We will make the datasets and code for all the work reported in this section +available as open source. + 0 + 0.1 + 0.2 + 0.3 + 0.4 + 0.5 + 0.6 + 0.7 + 0.8 + 0.9 + 1 + 1 + 4 + 16 + 64 + 256 + 1024 + 4096 + 16384 +Cumulative Frequency +No. of Labels +Cumulative frequency of labels +Figure 8. Distribution of labels for the classification task +is a bimodal model trained on datasets with natural lan- +guage (NL) -programming language (PL) pairs (e.g. docu- +mentation/code pairs) across six programming languages +(Python, Java, JavaScript, PHP, Ruby, and Go). Similarly, +GraphCodeBERT uses NL-PL pairs for pretraining a code +language model, but based on local data flow graphs ex- +tracted from Abstract Syntax Trees. CuBERT [22] is another +BERT-based model fine-tuned on multiple classification tasks +such as checking the presence of certain bugs and predicting +exception types. CuBERT is trained only on Python code, +and furthermore uses language level tokens as inputs to the +model. CodeT5 [41] is an encoder-decoder model based on +T5 architecture [33] with code-specific knowledge trained +to distinguish which tokens are identifiers and recover them +when they are masked out. CodeT5 is fine-tuned using multi- +ple CodeXGLUE benchmarks including understanding tasks +such as code defect detection and clone detection, and gen- +eration tasks like code summarization and translation. +Figure 9 shows the performance of these different models +on the code completion task with no fine tuning for the top-1 +and top-5 cases. We modeled code completion as a mask pre- +diction task, with the function call to be predicted being the +masked token. As shown in the Figure 9, the best performing +model was CuBERT on this task, which is not surprising +because it was the only model trained exclusively on Python +and used language level tokens unlike the other models. We +note that the performance of CodeT5 was surprisingly poor, +but we think this may in part be due to the fact that it is +trained on NL-PL pairs and it is strictly a generative model, +7 + +, , +Wenting Zhao, Ibrahim Abdelaziz, Julian Dolby, Kavitha Srinivas, Mossad Helali, and Essam Mansour +21 +19 +13 +2.7 +33 +27 +23 +3.1 +0 +5 +10 +15 +20 +25 +30 +35 +40 +CuBERT +CodeBERT +GraphCodeBert +CodeT5 +Accuracy +Complete (Top1) +Complete (Top5) +Figure 9. Accuracy on top-5 and top-1 test data for base lan- +guage models. Performance on CuBERT for slices is based on +150,739 and 137,987 examples for complete and slice, respec- +tively because of tokenization issues. For all other systems +the number of testing examples was 167,816. +for which we needed to specify a length of generation. It +also needed the most tuning in terms of specifying different +search strategies for final token prediction, so it is possible +that we did not choose the optimal search strategy for it. For +our purposes though, we chose CuBERT as a base, primarily +because we expected to benefit most from fine tuning. We +point out that given our label distribution, for the language +model to provide even 21% performance on complete for +top-1 and 33% for top-5 is quite good. +We turn now to the problem of fine tuning CuBERT with +training inputs to see if analysis does in fact improve code +completion as we defined it. Note that CuBERT’s pretraining +was performed by feeding the model the logical lines of 5 +million programs - so at the minimal, some fine tuning for +the code context where the function call is to be predicted +is needed. As stated earlier, we contrasted three different +training conditions: +• complete: where we gave the model text starting from +the call, backwards, as shown in Figure 5 +• slice: where we used a backwards slice as shown in +Figure 6 +• combined where the text from complete and slice +were concatenated as input to the model using a sepa- +rator token. +The test was on complete text, or combined. We chose +these conditions because we observed from examples that +for the problem of code generation, data flow is not sufficient +by itself. Figure 10 shows such an example. In this code, lines +5 and 15 contain the clue needed to make the prediction +of id, but they are unrelated to the receiver for which the +call is being made on line 22. Yet, the local pattern of code +has the same variable names, and the same set of calls are +repeated across functions, suggesting that id may be a good +candidate label. By contrast, the corresponding slice con- +tains minimal information as shown in Figure 11, since the +1 +response.json.return_value = dict(response, +total_count=3, limit=0, offset=0) +↩→ +2 +projects = +self.redmine.project.all() +↩→ +3 +self.assertEqual(projects.limit, 0) +4 +self.assertEqual(projects.offset, +0) +↩→ +5 +self.assertEqual(projects[0].id, 1) +6 +self.assertEqual(projects[1].id, 2) +7 +self.assertEqual(projects[2].id, 3) +8 +9 +def test_offset_limit(self): +10 +response_with_limit_offset = +{'total_count': 2, 'limit': 3, +'offset': 1, 'projects': +response['projects'][1:3]} +↩→ +↩→ +↩→ +11 +self.response.json.return_value = +response_with_limit_offset +↩→ +12 +projects = +self.redmine.project.all()[1:3] +↩→ +13 +self.assertEqual(projects.limit, 3) +14 +self.assertEqual(projects.offset, +1) +↩→ +15 +self.assertEqual(projects[0].id, 2) +16 +self.assertEqual(projects[1].id, 3) +17 +18 +def test_offset_limit_mimic(self): +19 +projects = +self.redmine.project.all()[1:3] +↩→ +20 +self.assertEqual(projects.limit, 3) +21 +self.assertEqual(projects.offset, +1) +↩→ +22 +self.assertEqual(projects[0].? +Figure 10. Code snippet where local text can help prediction +1 +from tests import unittest, mock, Redmine, +URL +↩→ +2 +Redmine(self.url) +3 +projects = self.redmine.project.all()[1:3] +4 +self.assertEqual(projects[0].? +Figure 11. Code where data flow lacks sufficient context +receiver projects[0] was defined just within the function +test_offset_limit_mimic. +We note however that sometimes the slice can help even +when the truncation does not cut off key information for +prediction. Figure 12 shows one such example. The predicted +function is partial is imported in line 3, but the actual call +is on line 28. On the other hand, in Figure 13, the import is +the only call prior to the line, so the slice can make relevant +information proximal, such that the neural model can pay +greater attention to proximal elements of the code. +8 + +Serenity: Library Based Python Code Analysis for Code Completion and Automated Machine Learning +, , +1 +import sys +2 +import logging +3 +from functools import partial +4 +from datetime import datetime +5 +from abc import ABCMeta, abstractmethod +6 +import json +7 +from _config import AttrDict +8 +9 +__all__ = ['multikey_getter_gen', +'unescape_json', 'LogParser', +'JSONParser', 'LogLine', +↩→ +↩→ +10 +'AccessLog', 'CommonLogFormat', +'uWSGIParser'] +↩→ +11 +12 +def multikey_getter_gen(parser, keys, +is_indices=False, delimiter="\t"): +↩→ +13 +"""Generator meta-function to return a +function +↩→ +14 +parsing a logline and returning +multiple keys (tab-delimited)""" +↩→ +15 +if is_indices: +16 +keys = map(int, keys) +17 +18 +def multikey_getter(line, parser, +keyset): +↩→ +19 +data = parser(line.strip()) +20 +return +delimiter.join((unicode(data[k]) +for k in keyset)) +↩→ +↩→ +21 +22 +def multiindex_getter(line, parser, +keyset): +↩→ +23 +data = parser(line.strip()) +24 +return delimiter.join((unicode( +data.by_index( idx-1, +raw=True)) for idx in keys)) +↩→ +↩→ +25 +26 +if is_indices is True: +27 +# Field indices +28 +return ? +Figure 12. Example of where complete text may have text +relevant to the prediction, but distant from call site +1 +partial = #!/usr/bin/env python # +2 +from functools import partial +3 +keys = map(int, keys) +4 +return ? +Figure 13. Example of where dataflow is very focused +4.3 +Model details +We use the CuBERT model released by [22]6, which has 24 +layers with 16 attention heads and 1024 hidden units and +6The CuBERT model can be accessed at github.com/google-research/google- +research/tree/master/cubert +was pretrained on 4M unique Python files on Github. At +fine-tuning, we set the batch size to 10 and trained the model +using 8 Tesla V100 with 32GB memory. The learning rate +is 5e-5, and we gradually warmed up the learning rate for +the first 300 gradient updates, which are the default val- +ues provided by the HuggingFace library [43]. The training +stops after 20 epochs, or ends after the evaluation accuracy +hasn’t improved for three epochs. For the complete and +slice models we used the 512 tokens model, and when we +used combined, we used the 1024 tokens model such that +the exact same tokens present in complete and slice could +be used together along with the separator. +We apply CuBERT’s tokenization to Python programs in +ETH150K where the Python programs are first tokenized us- +ing the standard Python tokenizer (the tokenize module)7,8. +Then we further break down the program tokens into 49,558 +subwords using subword tokenization [39], as performed by +the cuBERT tokenizer. +4.4 +Results of fine tuning +Figure 14 shows the accuracy in predicting the function call +exactly across the different training and test conditions. As +shown in the Figure 14, training on slice was at 47% ac- +curacy when tested on the complete text (slice-complete), +which is significantly above the 21% of top-1 baseline from +cuBERT. Training on complete however was much better +at 62% on the same text (complete-complete), which is +not surprising given that inspection of examples (e.g., Fig- +ure 10) show that complete often contains the expressions +in slice when the dataflow is local, and furthermore, ben- +efits from repetition in coding patterns that might hint at +labels in the absence of any real connection. The key ques- +tion is whether slices provide any benefit over and above +what benefit is gained from complete. Training on combined +suggests that slices do provide a strong signal, with a 65% +accuracy on complete text (combined-complete), and 69% +accuracy on the combined text (combined-combined). We +also compared top-5 performance across conditions to allow +comparison to the baseline language models - not surpris- +ingly this result improved accuracy across all conditions, +with the combined-combined condition showing the best +performance at 78%. The results show that data flow analysis +can significantly augment code completion performance. +4.5 +How do slices help code completion? +We conducted an analysis of how slices might help code +completion performance; i.e., to understand if slices help +the model complete code better for rare labels compared +7github.com/python/cpython/blob/main/Lib/tokenize.py +8We note that tokenize only outputs tokens for the code snippet that is +free of any syntax errors; otherwise, it returns either IndentationError or +TokenError. To predict function calls we often feed code that is incomplete, +thus syntactically incorrect; therefore, we had to modify the original module +so that it always returns what has been already tokenized thus far. +9 + +, , +Wenting Zhao, Ibrahim Abdelaziz, Julian Dolby, Kavitha Srinivas, Mossad Helali, and Essam Mansour +21 +47 +62 +65 +69 +33 +66 +74 +75 +78 +0 +10 +20 +30 +40 +50 +60 +70 +80 +CuBERT baseline +slice-complete +complete-complete combined-complete combined-combined +Accuracy +Top-1 +Top-5 +Figure 14. Results of fine tuning on the different "training - +testing" conditions; i.e., training on {slice, complete or com- +bined} and tested on {complete or combined} + 0 + 0.1 + 0.2 + 0.3 + 0.4 + 0.5 + 0.6 + 0.7 + 0.8 + 0.9 + 1 + 1 + 4 + 16 + 64 + 256 + 1024 +Proportion of labels output correctly +Label Counts +Combined-Complete +Complete-Complete +Combined-Combined +Figure 15. Accuracy on labels with different counts +to more common ones, since statistical approaches likely +work better for common labels, but less well for rare ones. +Figure 15 shows the performance of the different models; the +presence of slices at training and test enhance code comple- +tion performance for rare labels more than common labels, +although the advantage does seem to be present for common +ones as well. The combined-combined model was 15% ac- +curate on labels with count 1, of about 18,000 labels, and that +number rapidly approaches 40% for labels with count 3. As +labels become more frequent, the differences between the +sklearn.datasets +sklearn.load_digits +sklearn.load_digits.data +sklearn.load_digits.target +sklearn.train_test_split +sklearn.LinearSVC +sklearn.svm +sklearn.cross_validation +Figure 16. KGpip’s training graph for our running example +after filtering out as input to the AutoML system. +models gets more noisy but the combined-combined case +still holds an advantage. +4.6 +Comparison to existing work +In this space, comparisons are tricky because there is no com- +mon or standard benchmark and because the exact problem +varies. We chose ETH150K, which is at least a well-known +code repository, but work that e.g. relies on its own sample +of GitHub makes results incomparable. The exact problem +varies too, with some tools, like us, predicting function calls, +others predicting only method calls and still others predict +the next token for all tokens. There is a real need for a bench- +mark in this space; as part of helping build such a benchmark, +we will release our own slice and complete dataset to the +community. [25] reports overall accuracy numbers around +0.7, which is almost the same as ours, but that paper is pre- +dicting the next token across all token types, so could benefit +from the fact that some predictions (e.g. ’)’ followed by ’:’ in +def) follow from the grammar. Pythia [37] uses a neural net- +work rather than a language model, but reports comparable +accuracy numbers for top-1. Their top-5 number is higher, +but their predictions seem to be for method calls, rather than +all functions as we do, for which the receiver may provide +context to aid prediction. +These approaches may be complementary, too. Our work +showed that adding slice data to local context greatly aided +the accuracy of our models. Other approaches also rely on +mostly local information, and could potentially benefit from +slices as well. We plan to investigate this further in our future +work. +5 +Automated ML Pipelines Application +The problem of automated machine learning pipelines (Au- +toML) focuses on automatically building pipelines by per- +forming a search over valid data transformations and learn- +ers, along with hyper-parameter optimization for each learner. +Our research question is whether we can perform learner and +transformation selection based on mining large repositories +10 + +Serenity: Library Based Python Code Analysis for Code Completion and Automated Machine Learning +, , +of abstracted ML python scripts obtained statically by Seren- +ity. Unlike dynamic analysis, Serenity’s static analysis of ML +pipelines has the advantage of scaling to millions of scripts +due to its low cost. Specifically, our question is whether the +extracted semantics of a set of pipelines by Serenity can help +in predicting a new pipeline when combined with neural +graph models. +We developed an AutoML system, called KGpip, described +in detail in a separate work [20], which builds a database +of datasets and their corresponding historically used ML +pipelines using Serenity analysis. KGpip formulates the au- +tomation of a ML pipeline as a graph generation problem. It +is based on two hypotheses that a neural graph generator +will: i) capture more succinctly multiple pipelines seen in +practice for a given dataset, and ii) capture statistical similar- +ities between different pipelines more effectively. In KGpip, +we filter out Serenity’s analysis to remove non-ML related +components such as calls to libraries other than target ML +libraries (Sklearn, XGBoost, and LGBM), and nodes indicat- +ing location of calls within a pipeline script, among others. +Figure 16 shows the filtered version of the graph for our +running example. We also show in Figure 17 an overview of +how KGpip works at training and inference phases. Using +code analysis graphs obtained from Serenity and dataset em- +beddings, KGpip trains a graph generation model optimized +to output a ML pipeline as close as possible to the target +pipelines of the training data. At inference time, KGpip iden- +tifies the closest dataset to the input dataset and uses its +embedding as input to the graph generation model which in +turn outputs a set of possible ML pipelines. These pipelines +are then validated and fed to a hyper-parameter optimizer to +get the best pipeline that results in the highest performance +on the input dataset. +KGpip is designed to work with AutoML systems, such +as AutoSklearn [15] and FLAML [40], to utilize their hyper- +parameter optimizers. With a collection of 2000 ML python +scripts, we trained a graph generation neural network that +learns to generate a ML pipeline graph for a given dataset. +We conducted a comprehensive evaluation using 77 datasets +from different benchmarks, such as AutoML and Penn Ma- +chine Learning Benchmark (PMLB), and different ML portals, +such as Kaggle and OpenML. Table 1 shows the overall KG- +pip performance which significantly improves the selection +of data transformation and learning algorithms of state-of- +the-art AutoML systems, namely, Auto-Sklearn and FLAML. +We note that AutoSklearn consults a database of pipelines +and datasets, and picks pipelines to start the search based +on a nearest neighbors to an unseen dataset, except that Au- +toSklearn’s dataset consists of effective pipelines based on +actual execution. We also compared KGpip to AL [6], which +uses dynamic code analysis on existing machine learning +pipelines to select optimized pipelines. AL was unable to +process 60 datasets because it ran out of time in searching +for pipelines. On a smaller set of 17/77 datasets on which +Average Performance +T-Test +AutoSklearn +0.71 (0.24) +- +KGpip + AutoSklearn +0.77 (0.22) +0.0002 +FLAML +0.71 (0.27) +- +KGpip + FLAML +0.77 (0.20) +0.0132 +Table 1. Performance (average and stdev) of KGpip com- +pared to FLAML and AutoSklearn. Both variations of KGpip +show significant improvements compared to existing sys- +tems, both with 2-tailed T-test 𝑝 < 0.05 +AL was able to work, AL achieved an average performance +of 0.36 compared to 0.745, 0.705, 0.79 and 0.765 by FLAML, +Auto-Sklearn, KGpip + FLAML, and KGpip + AutoSklearn, +respectively. This comparison with both AL and AutoSklearn +clearly illustrates the value provided by Serenity, compared +even to approaches that rely on dynamic runtime analysis. +6 +Related Work +Static Analysis for Python: Static analysis of Python has +attracted considerable interest lately, and there have been +a range of approaches. Type inference has been a focus of +much work, some using techniques such as abstract inter- +pretation and, more recently, there has been work using +machine learning. Likely the best-known work is MyPy [1] +and Pytype [2]. MyPy focuses on checking and inferring +types that conform to PEP 484 [38], which defines a syntax +for Python types. MyPy focuses on inference within a single +function, since types are expressed at function boundaries +in PEP 484. Pytype does type inference, and it can handle +cases where a variable has different types at different points. +It also does relatively little interprocedural analysis. +Moat et al. [28] present an abstract interpretation for type +inference of Python that models a variety of domains to +compute more accurate information, and it makes use of +the recency abstraction for aliasing. However, it is currently +limited in its support for interprocedural analysis, which is +enabled by inlining. Fritz and Hage [16] present a dataflow +analysis for type inference that provides a range of tradeoffs +for cost and precision. It does handle features like first-class +functions. +Machine learning is also used for type inference of Python. +TypeWriter [31] trained a neural model using a corpus of +code, with labeled data derived from user annotations. Type- +Writer considers comments in code as inputs to the neural +model, unlike program analysis based type inference. While +such systems are certainly performing analysis, their ap- +proach and mechanism are quite different from ours. +There have been other analyses of Python, often for spe- +cial purposes. Ariadne [11] makes use of WALA, as we do, +but focuses on inferring the shapes of tensors in machine +11 + +, , +Wenting Zhao, Ibrahim Abdelaziz, Julian Dolby, Kavitha Srinivas, Mossad Helali, and Essam Mansour +Figure 17. An overview of KGpip training and inference workflows +learning programs. Unlike our approach to libraries, Ariadne +models ML libraries as needed for tensor-related operations. +Code completion: Code completion has been a prominent +area of research towards achieving better productivity when +working within an IDE. One of the main challenges in this +domain is about code representation where the vast majority +of work has used either tokens or abstract syntax trees to +represent code (we refer the reader to [3] for a detailed survey +of this area). [25] represents Python and JavaScript codes +as ASTs and use a pointer network for better predicting +Out of Vocabulary words in code completion. Pythia [37] is +another approach that uses ASTs with an LSTM model for +code completion. +A number of approaches also tried to leverage representa- +tions based on data and control flow [4, 7, 14]. On JavaScript, +[21] utilized a program dependence graph to detect code du- +plication. [4] use AST based representation augmented with +local data and control flows for predicting variable names +and variables misuse. [14] combines token based represen- +tations of code with edges based on object uses, and AST +nodes to predict the documentation of a method. To perform +code completion over Java API calls, [29, 30] used a mostly +intraprocedural analysis for mining graphs augmented with +control and data flow. +With the rise of pre-trained language models such as BERT +[10] and GPT [5, 32], many recent approaches [13, 19, 22, +26, 41] started to leverage the already existing rich language +understanding in these models and fine tune it for various +code understanding tasks such as code summarization, trans- +lation, completion, bug detection, etc. CodeBERT[13] is a +BERT based model trained on pairs of natural language and +programming language samples across six programming +languages. GraphCodeBERT [19] uses BERT as well, but +represents the code using data flow graphs based on ASTs. +The dataflow in GraphCodeBERT is completely local and +not interprocedural, as in Serenity. For instance as an exam- +ple, it adds edges from all variables used in an expression +to their definitions. CuBERT [22] and CodeT5 [41] are an- +other two models based on BERT and T5 [33] architectures, +respectively. +Unlike our approach, all these methods represented code +either as a sequence of tokens [13, 22], ASTs [25, 27, 37, 41], +or data flows derived from ASTs [19]. +AutoML approaches: Several AutoML frameworks have +been proposed recently [6, 12, 15, 40]. In most AutoML sys- +tems, learner and pre-processing selection is driven by a +database of actual executions of pipelines and data. For in- +stance, [15, 36] compute a database of dataset meta-features +such as number of rows and columns, while [6] mines a +repository of run-time information of inputs to the learners +and preprocessors via dynamic code analysis of public ML +pipelines available e.g. on Kaggle. The predicted learners and +preprocessors are based on a similarity measure between the +target dataset and stored features. In KGpip we utilized dense +vector embeddings derived from raw contents of datasets to +measure this similarity and graph neural networks to select +the learners/preprocessors and generate the pipeline. +Some existing systems such as TPOT [24] or Recipe [9] +use evolutionary algorithms for pipeline generation. Others +approach it as a probabilistic matrix factorization [17], an +AI planning problem when combined with a user specified +grammar [23, 42], or a bayesian optimization problem com- +bined with Monte Carlo Tree Search [34]. None of these +approaches however use analysis to build up their database. +7 +Conclusion +In this paper, we introduced Serenity; a framework for Python +code static analysis. Serenity relies on two mechanisms (a) +dynamic dispatching at the core of language translation, +and (b) extreme abstraction of libraries. To demonstrate the +12 + +E +CSV +Deep graph +jupyter +generator +model +CsV +ML Pipelines +Hyperparameter +Skeleton +Optimization +Auto- +Sklearn +FLAMISerenity: Library Based Python Code Analysis for Code Completion and Automated Machine Learning +, , +utility of Serenity’s analysis, we used it in two important +code-related applications: code generation and automated +machine learning. Serenity’s analysis showed very promis- +ing performance in both applications, allowing us in some +cases to outperform approaches based on dynamic analysis, +and perform competitively for code completion. We also im- +plemented Serenity as an open-source implementation based +on WALA, a popular framework for program analysis. +References +[1] [n.d.]. MyPy. http://mypy-lang.org. Accessed: 2021-11-18. +[2] [n.d.]. Pytype. https://github.com/google/pytype. Accessed: 2021-11- +18. +[3] Miltiadis Allamanis, Earl T Barr, Premkumar Devanbu, and Charles +Sutton. 2018. 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Association for Computa- +tional Linguistics, Online, 38–45. https://www.aclweb.org/anthology/ +2020.emnlp-demos.6 +14 + diff --git a/E9E4T4oBgHgl3EQffg1w/content/tmp_files/load_file.txt b/E9E4T4oBgHgl3EQffg1w/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..19e2aed43a76ce064534f20571fb08ce8d64c4c2 --- /dev/null +++ b/E9E4T4oBgHgl3EQffg1w/content/tmp_files/load_file.txt @@ -0,0 +1,965 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf,len=964 +page_content='Serenity: Library Based Python Code Analysis for Code Completion and Automated Machine Learning Wenting Zhao Department of Computer Science Cornell University wzhao@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='cornell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='edu Ibrahim Abdelaziz Thomas J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Watson Research Center IBM Research Ibrahim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='abdelaziz1@ibm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='com Julian Dolby Thomas J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Watson Research Center IBM Research dolby@us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='ibm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='com Kavitha Srinivas Thomas J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Watson Research Center IBM Research Kavitha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='Srinivas@ibm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='com Mossad Helali Department of Computer Science Concordia University mossad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='helali@concordia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='ca Essam Mansour Department of Computer Science Concordia University essam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='mansour@concordia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='ca Abstract Dynamically typed languages such as Python have become very popular1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Among other strengths, Python’s dynamic na- ture and its straightforward linking to native code have made it the de-facto language for many research areas such as Ar- tificial Intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' This flexibility, however, makes static analysis very hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' While creating a sound, or a soundy, analysis for Python remains an open problem, we present in this work Serenity, a framework for static analysis of Python that turns out to be sufficient for some tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' The Serenity framework exploits two basic mechanisms: (a) reliance on dynamic dispatch at the core of language translation, and (b) extreme abstraction of libraries, to generate an abstraction of the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' We demonstrate the efficiency and usefulness of Serenity’s analysis in two applications: code completion and automated machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' In these two applications, we demonstrate that such analysis has a strong signal, and can be leveraged to establish state-of-the-art performance, com- parable to neural models and dynamic analysis respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Keywords: Python, Static Analysis, Code completion, Au- toML 1 Introduction Static analysis of Python is hard, due in part to features often regarded as strengths: its dynamic nature and its straightfor- ward linking to native code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Python is dynamically typed, so the aid static types provide to analysis of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Java is not available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Python has a dynamic object structure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' methods can be freely assigned and modified complicating resolving calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Even basic constructs such as method calls and ob- ject creations can be ambiguous in the basic syntax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Beyond the language itself, many of the rich collection of Python li- braries, especially the math-heavy libraries used in machine learning, are implemented in native code, which makes analy- sis require cross-language support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' For these reasons among 1https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='techrepublic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='com/article/programming-languages-pythons- growth-is-absolutely-explosive-says-anaconda-ceo-and-not-slowing- down/ others, to our knowledge, there is a lack of widely-used anal- ysis frameworks for Python, despite the value such analysis would have, for instance, for tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' However, while creating a sound, or a soundy, analysis for Python remains an open problem, we demonstrate Serenity2, a framework that turns out to be sufficient for some tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Beyond a relatively direct translation of Python Abstract Syntax Tree (AST) into a Control Flow Graph (CFG), Serenity exploits two basic mechanisms: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Reliance on dynamic dispatch at the core of language translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' It is not possible, always, even to tell whether a construct is an object creation or a function call, and this is just one example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Our approach to such situ- ations is to turn them into dynamic dispatches over types representing constituent constructs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' We detail how many subtleties of Python can be modeled in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Extreme abstraction of libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' User code often makes heavy use of APIs to create and operate upon domain objects, such as arrays in numpy, but these objects are often fairly opaque to the user code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' As such, we find it often suffices to treat libraries by just tracking the objects they create and methods called upon them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' This is not, nor is it designed to be, soundy, let alone sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' We show however that this enables useful modeling of user code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' We first discuss how Python is modeled and how the library abstraction still provides a useful analysis of user code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' We then demonstrate that this analysis is useful where we fo- cus on two applications that depend on the outputs of such analysis: Code Completion is a core functionality expected in all IDEs, where the goal is to suggest methods and functions to call given prior code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' We show how our dataflow analysis allows us to focus on relevant code at a point of completion, which when combined with 2With apologies to Reinhold Niebuhr, "give us courage to model what must be modeled, serenity to accept what cannot be modeled, and the insight to know the one from the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='" 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='05108v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='PL] 5 Jan 2023 , , Wenting Zhao, Ibrahim Abdelaziz, Julian Dolby, Kavitha Srinivas, Mossad Helali, and Essam Mansour local program context prior to the function call pro- duces much better code completion performance than the context alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Automated Machine Learning which takes a given dataset in the form of a structured table, and creates an effec- tive machine learning pipeline to learn to predict some columns based on other columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Prior approaches have been based on dynamic analysis, and we show static analysis does just as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Static analysis is more practical, as actually running these pipelines is an ardu- ous and expensive task;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' and one can mine large open repositories to populate such databases using analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' In the rest of the paper, we first describe Serenity’s tech- niques for modeling Python based on a running example (Sections 2 and 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' We then validate our analysis with two applications in code completion (Section 4) and automated machine learning (AutoML) (Section 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' We finally survey related work in Section 6 and conclude in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' 2 A Running Example Figure 1 shows the running example for this paper, a snip- pet of Python code adapted from the multi_class_svm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='py script of ETH150K [35] benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Our modification (lines 1 to 10) is to add some debugging options to illustrate com- plexities in Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' The variable fitit is assigned one of two functions depending on debug_level: either a closure or a function with an added assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' The code loads a digits dataset (line 20), reads specific fields of the dataset (line 21), manipulates the dataset (line 22), splits the data into train and test splits (line 23), and finally creates X_train_bias (line 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' The code then creates machine learning models FrankWolfeSSVM (line 42) and LinearSVC (line 72).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' The code then calls them: a fitit call on a LinearSVC (line 74) takes the model (line 26), and fitit is called on FrankWolfeSVC with X_train_bias (line 79).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Note that the call to the con- structor FrankWolfeSSVM is on line 42, and the fitit call on the object is on line 79, reflecting a property of most code it is non-local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' 3 Analysis 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='1 Background: call graph framework Grove et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' [18] provide a framework for expressing call graph algorithms for object-oriented languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' It encapsu- lates the bulk of the algorithm, parameterizing the algorithms with functions that determine how to add context sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' The details of the framework are beyond the scope of this paper, but we depend on two details: First, we will rely later on something called the Proce- dure Key Selection Function (PKS), which is essentially a way to specify when called functions should be ana- lyzed in a context-sensitive manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Second, the framework distinguishes between function call sites and object creation sites, which, as we shall 1 debug_level = 3 2 3 if debug_level > 5: 4 def fd(model, test, train): 5 assert len(test) < 500 6 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='fit(train, test) 7 fitit = fd 8 9 else: 10 fitit = lambda model, test, train: model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='fit(train, test) ↩→ 11 15 from sklearn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='svm import LinearSVC 16 17 from pystruct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='models import MultiClassClf 18 from pystruct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='learners import (NSlackSSVM, OneSlackSSVM, SubgradientSSVM, FrankWolfeSSVM) ↩→ ↩→ 19 20 digits = load_digits() 21 X, y = digits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='data, digits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='target 22 X = X / 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' 23 X_train, X_test, y_train, y_test = train_test_split(X, y) ↩→ 24 25 # we add a constant 1 feature for the bias 26 X_train_bias = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='hstack([X_train, np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='ones((X_train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='shape[0], 1))]) ↩→ 41 42 fw_bc_svm = FrankWolfeSSVM(model, C=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content="1, max_iter=50) ↩→ 71 72 libsvm = LinearSVC(multi_class='crammer_singer', C=." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='1) ↩→ ↩→ 73 start = time() 74 fitit(libsvm, X_train, y_train) 75 time_libsvm = time() - start 76 print("Score with sklearn and libsvm: %f (took %f seconds)" % (libsvm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='score(X_test, y_test), time_libsvm)) ↩→ ↩→ ↩→ 77 78 start = time() 79 fitit(fw_bc_svm, X_train_bias, y_train) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' A Running example see in Figure 3, is not possible in general in Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Hence, we combine the two sets into a single one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' The framework paper defines relevant program features at the top of page 694, which we excerpt here in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' We need two minor changes: InstVariables is taken to be the set of strings possibly used as field names, rather than a set of declared field names, which it is in the original framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' While 2 Serenity: Library Based Python Code Analysis for Code Completion and Automated Machine Learning , , field names can be defined in Python, this is entirely optional so we ignore such definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' NewSites becomes the same as the set CallSites to effect the second item above;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' that is, there is one set that combines all possible call sites and creation sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' This represents the fact that every site can potentially see both classes and functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Class all class declarations in the program InstVariable all instance variable declara- tions of the program Procedure all procedure declarations of the program Variable all variable names used in the pro- gram CallSite all call sites in the program NewSite all new sites in the program LoadSite all loads of instance variables in the program StoreSite all stores to instance variables in the program Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Program features from [18] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='2 Language modeling Figure 3 illustrates the kind of dynamism with which anal- ysis of Python must contend, in this case 5 different options for the meaning of X() on line 45 based on the value supplied as sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='argv[1] and sometimes sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='argv[2]: class1 class X (line 4) defines an ordinary class named X, of which line 45 creates an instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' class2 class X (line 11) defines a class named X that redefines the new operator, so line 45 just returns 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' def1 def X (line 16) defines a function named X, and calling it at line 45 returns 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' def2 X = lambda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' (line 20) creates a closure and assigns it to X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' calling the closure at line 45 returns 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' import The module X (line 23) overrides default module behavior to become callable and return 3 at line 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' method static X (line 32) is assigned the static method s of class X (line 28) which returns 5 at line 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' method instance X (line 41) gets a bound instance method (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' a closure over y) i of class X (list 35), returning 4 at line 45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Note that all of these definitions of X can flow to the same call at line 45, so there is literally no syntactic distinction between different kinds of allocations, calls, and even modules in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' And class and function names are all first class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Thus analysis must handle these basic operations in a dynamic manner, unlike e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Java, where calls, allocations and imports have clear syntactic distinctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Note further that even basic method calls require closures to handle line 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' 1 import sys 2 3 if sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='argv[1] == "class1" or sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='argv[1] == "inst": 4 class X: 5 pass 6 7 if sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='argv[1] == "inst": 8 X = X() 9 10 elif sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='argv[1] == "class2": 11 class X: 12 def __new__(*args): 13 return 0 14 15 elif sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='argv[1] == "def1": 16 def X(): 17 return 1 18 19 elif sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='argv[1] == "def2": 20 X = lambda: 2 21 22 elif sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='argv[1] == "import": 23 import X 24 25 elif sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='argv[1] == "method": 26 27 if sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='argv[2] == "static": 28 class X: 29 def s(): 30 return 5 31 32 X = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='s 33 34 elif sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='argv[2] == "instance": 35 class X: 36 v = 4 37 def i(self): 38 return self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='v 39 40 y = X() 41 X = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='i 42 43 44 print(str(X)) 45 print(str(X())) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Dynamic code examples The X() at line 45 is a call on X, and this allows us to use standard dynamic dispatch to model all of this behavior, using synthetic "methods" where needed to handle language semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' We will use a similar "dispatch" at field accesses to handle the difference between class and instance fields, which again can only be known from the object accessed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' We shall make use of these indirections to define our framework model in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' 3 , , Wenting Zhao, Ibrahim Abdelaziz, Julian Dolby, Kavitha Srinivas, Mossad Helali, and Essam Mansour We adopt the terminology of Grove et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' [18] to present our work as extensions to standard object-oriented call graph construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' To fit our dynamic Python context, we make a few changes to the core definitions of that work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' These changes reflect that Python does not require that fields be declared in order to be used, and it makes no syntactic dis- tinction between calls and allocations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Furthermore, as is standard for representing first-class entities in an object- oriented framework, we have one class for each first-class entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' As Figure 3 shows,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' classes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' functions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' methods and modules are all first-class,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' so our set of classes for analysis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='includes the following: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='𝐶𝑐𝑙𝑎𝑠𝑠 a class representing program class C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='𝐶𝑖𝑛𝑠𝑡 a class representing instances of class C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='𝑀𝑖𝑛𝑠𝑡 a class representing instances of module M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='𝐷𝑖𝑛𝑠𝑡 a class representing instances of function D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='𝑆𝑖𝑛𝑠𝑡 a class representing the instance of script S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='Now most of the irregularities of Python calls and creations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='are handled by treating every call site as a CallSite for each ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='receiver type ∗𝑖𝑛𝑠𝑡 and as a NewSite for every receiver type ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='∗𝑐𝑙𝑎𝑠𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' The site on line 45 in Figure 3 would have some types handled by each mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Fields are also handled seam- lessly: on line 32, X is a 𝐶𝑐𝑙𝑎𝑠𝑠, and on line 41 X is a 𝐶𝑖𝑛𝑠𝑡, so static and instance state are handled by making static fields be instance fields of the class object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Call graph construction starts with a root stub that creates an instance of the main script 𝑆𝑖𝑛𝑠𝑡 and calls it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='3 Framework modeling In many situations, it is difficult or impossible to find actual code for Python imports: there is no fixed relationship be- tween names in import statements and locations of actually source code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Even if there were, the structure of Python li- braries is such that large amounts of the code is native and hence a Python analysis framework is not applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Even if it were possible to find Python code, many libraries are large enough to make precise analysis challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' In our case, we are interested in the behavior of application code rather than library internals, so we minimize these issues by largely not analyzing framework code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Our model, called Turtles3, abstracts Python frameworks to capture how the framework interacts with user code and to ignore all of its internal details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Specifically, we model four aspects, all using the indirections of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='2: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' We model import statements as returning a new frame- work, denoted by the name of the imported module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' The framework is an opaque object with no function- ality beyond implementing the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Calls to framework functions and methods typically return something, which is then possibly used by the user code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' We model every call to the framework as 3from "turtles all the way down".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' This phrase is of unknown origin, see https://en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='org/wiki/Turtles_all_the_way_down returning a new object from it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' this model is transitive, so calls on those objects return further new objects from the framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' We label these objects with the path by which they are accessed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Accesses to fields of framework objects have little meaning in our model since we do not model the frame- work state at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' However, user code typically expects that a field access return something, so we model all such field accesses as returning the container object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Arguments to turtle methods are mostly ignored, since we do not model what the framework does to them;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' however, sometimes functions are passed as parame- ters, and we assume that the framework might call it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Since we do not model internal framework state, the model invokes callbacks from where they are passed as arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' The framework of Grove et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' [18] provides the customiza- tion support needed to implement this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' We start by introducing a new type of class, 𝑇𝑝𝑎𝑡ℎ, that represents a tur- tle, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' an opaque model object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Item 1 is implemented by modeling import M statements as a call to a synthetic import procedure with M as its argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' This call is modeled as returning a 𝑇𝑀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Item 2 is implemented as a Procedure Key Selection Function (PKS) which takes the receiver of a type 𝑇𝑝𝑎𝑡ℎ and the name 𝑛 of the called procedure and returns a new turtle of 𝑇𝑝𝑎𝑡ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Item 3 is implemented by simply re- turning self when reading any field of any 𝑇𝑝𝑎𝑡ℎ type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Item 4 is implemented as a PKS that generates calls for every argu- ment that is of a function type (this is not illustrated in our example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='4 Inheritance from Turtles One wrinkle in our data is that application classes often inherit from turtle classes, meaning that method calls on self should logically be turtle methods when the method read is never assigned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' That is, if a read of self is to a field or method that is never assigned and the class inherits from a turtle, the read should return a new turtle object to capture unknown superclass behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' However, this is tricky to do because, since methods and fields can be assigned anywhere in the code, it is not in general possible to know if one will not be assigned until analysis terminates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' What we need to do is record such reads and, when analysis terminates, process them as turtle reads and restart analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' This restarting itself may need to be repeated, since reading one turtle could make more code reachable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='5 Analysis of running example When this analysis is applied to the running example (Fig- ure 1), the result is the dataflow graph shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' To illustrate our framework model, observe the import call of LinearSVC on line 15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' as an import, this returns an object of type 𝐿𝑖𝑛𝑒𝑎𝑟𝑆𝑉𝐶𝑖𝑛𝑠𝑡, that is, an instance of the module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' When 4 Serenity: Library Based Python Code Analysis for Code Completion and Automated Machine Learning , , load_digits digits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='data digits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='target X y X/16 train_test_split X_train X_test y_train y_test hstack fit (fd) fit (fd) LinearSVC FrankWolfeSSVM Invocations arg 0, flow arg > 0 Reads fit (lambda) fit (lambda) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Dataflow graph for the running example this is called (line 72), it returns a turtle of type 𝑇𝐿𝑖𝑛𝑒𝑎𝑟𝑆𝑉𝐶, illustrated by the green node labeled LinearSVC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' When fit is called on this object in the fitit functions (line 74), item 2 means it returns a derived turtle of type𝑇𝐿𝑖𝑛𝑒𝑎𝑟𝑆𝑉𝐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='𝑓 𝑖𝑡, shown as a green node labeled fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Since fit is called on LinearSVC, a black data flow edge connects them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' On the other hand, the other non-self arguments to fit are shown with red arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Other turtle functions are shown similarly: load_digits, train_test_split, hstack, FrankWolfeSSVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Note that analysis has no idea what these functions do, just that they pass data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Note that fitit is a variable holding one of two first-class functions, and it is called for both of the ML models created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' To get the precise results shown in Figure 4 requires analysis infrastructure that handles first-class func- tions and also does context-sensitive analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' In particular, the model objects and the data flow to both the normal and debugging functions assigned to fit, since both potentially flow to fitit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' In the figure, the nodes are distinguished with labels of the function in which they occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Other nodes in Figure 4 represent local dataflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' The top- most two blue nodes represent reads of the data and target fields of data, so they have edges from the load_digits call and edges to their respective variables X and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' X is scaled by 16, shown by the nodes labeled X/16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' X/16 and y then flow to train_test_split with red edges since they are arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' This graph focuses on data flow, which captures patterns of how the various turtle APIs are used across programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' This allows us to learn patterns that enable our applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='6 Implementation Our analysis is implemented using WALA and its support for both Python 2 and Python 3 using the Jython system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' WALA is built to be extensible, and we used several features to ease our implementation work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' The main extension is for handling turtles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' For item 1, we override the model function that handles import to return a synthetic object with a turtle type named for the given mod- ule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' For item 2, we override the selection of called methods for turtle classes so that any call goes to a synthetic method that creates and returns a turtle with the appropriate ex- tended turtle name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' For item 2, this synthetic method mostly ignores its arguments, except generating a call to each one to handle callbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' For item 3, we override the code handling field reads to simply return the container if it is of turtle type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' The other configuration is to add aggressive context sen- sitivity for all turtle types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Since the synthetic methods are trivial anyway, it is cheap to ensure that every call site is analyzed separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' 4 Code Completion Application The core research question we ask is how useful Serenity’s analysis is and whether it can help other applications, de- spite the challenges in modeling dynamic languages such as Python accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' As a first application, we examine a code completion use case, which we cover below in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' By code completion, we refer to the problem where, when given a snippet of a program, the problem is to predict a function call, analogous to what an IDE does for method suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' We do not refer to code generation given natural language descriptions of code requirements, as in the Codex model that powers GitHub Co-Pilot [8] or even models that gener- ate entire functions in a generative style based on function signatures or snippets of code, such as CodeT5 [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Our observation is that for code completion, the analysis require- ment is that the methods be callable from a specific type, and so analysis for code completion is focused on detecting the types of objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' For languages such as Python, type infer- ence is hard, but our hypothesis is that code completion can benefit significantly from the data flow analysis that Serenity produces, simply because data flow can provide a focused context for code completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Recently, there have been a plethora of neural models of code such as [13], [19], [22], [41] trained with the objective of either predicting randomly masked tokens in code, or predicting the very next token, which one might assume is consistent with the task of code completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Our research question is whether one can leverage the extensive training of these models on millions of programs to perform code completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Specifically, we asked whether data flow analy- sis provided by Serenity can improve code completion when combined with these neural models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' If data flow analysis does provide any signal from Serenity, it should improve per- formance on code completion task even with the extensive training these language models already had.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' We therefore 5 , , Wenting Zhao, Ibrahim Abdelaziz, Julian Dolby, Kavitha Srinivas, Mossad Helali, and Essam Mansour 1 print("Score with pystruct subgradient ssvm: %f (took %f seconds)" % (np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content="mean(y_pred == y_test), time_subgradient_svm)) ↩→ ↩→ ↩→ 2 3 # the standard one-vs-rest multi-class 4 # would probably be as good and faster 5 # but solving a different model 6 libsvm = LinearSVC(multi_class='crammer_singer', C=." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='1) ↩→ ↩→ 7 start = time() 8 libsvm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='fit(X_train, y_train) 9 time_libsvm = time() - start 10 print("Score with sklearn and libsvm: %f (took %f seconds)" % (libsvm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='score(X_test, y_test), time_libsvm)) ↩→ ↩→ ↩→ 11 12 13 start = time() 14 fw_bc_svm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Code snippet used for prediction 1 from sklearn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='cross_validation import train_test_split ↩→ 2 from pystruct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='models import MultiClassClf 3 from pystruct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='learners import (NSlackSSVM, OneSlackSSVM, ↩→ 4 digits = load_digits() 5 digits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='data 6 digits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='target 7 X = X / 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' 8 train_test_split(X, y) 9 X_train_bias = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='hstack([X_train, np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='ones((X_train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='shape[0], 1))]) ↩→ 10 model = MultiClassClf(n_features=X_train_bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='shape[1], n_classes=10) ↩→ ↩→ 11 fw_bc_svm = FrankWolfeSSVM(model, C=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='1, max_iter=50) ↩→ 12 fw_bc_svm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Code snippet corresponding to a slice from the analysis graph modeled code completion as a fine tuning task, and varied the training inputs of fine tuning to be one of the three conditions shown below: All code as text prior to the function call A slice of the code restricted to source expressions that are relevant to a function call in data flow Both code as text, as well as the slice, separated by a token to distinguish the two inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' For all text code prior to a function call, there are limits on how many tokens modern language models can fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' That is, when the code goes beyond the limit, truncation is needed in order for the models to run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' A widely-used truncation strategy is to only keep 𝑛 tokens prior to the prediction point, where 𝑛 is the maximum sequence length, which can lead to fairly local information, as shown in Figure 5 for our running example shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' The key prediction in Figure 5 is to predict what method will be called on fw_bc_svm, but notice that the construction of fw_bc_svm is out of the scope of the truncation4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' For obtaining the slice restricted purely to dataflow, given a program and its corresponding dataflow graph, to predict the function call, we start at a node that we would like to predict, reverse all edges coming into the node, and find all reachable nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Each node in the reachability set corre- sponds to a source expression in the original program, and we only include the expressions that are not sub-expressions of any other expressions as features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Then, we order these expressions according to their positions in the source files, and add in variable names from the analysis artifacts so the code looks more or less like real code that the language mod- els have been trained on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Figure 6 shows an example of such a dataflow based slice looks for the code in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Here we start the fw_bc_svm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='fit call in Figure 4, reverse all edges coming into the node, and perform a reachability analysis, to gather the slice, adding variable names such as digits = load_digits().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' In this example, dataflow analysis does give important information useful for predicting the function call, because the slice brings in non-local but relevant code such as the definition of fw_bc_svm into the scope of text that can be fed to a neural model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='1 Dataset We used the popular benchmark of ETH150K [35], which comes with 100K programs used for training, and the re- maining 50K used as a testing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' ETH150K was analyzed using Serenity, and 147,288 of 150,000 files were successfully analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' For the analyzed files, we parsed each file with a Python AST parser, and gathered all function calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' For each function call identified by the AST, we examined whether we could find the function in the analysis output, and if it was found in the output, we checked if the source location of the call matched that in the AST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Our observation has been that the Jython source mappings can be wrong sometimes, so we used both metrics to measure the completeness of the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' The analysis found 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='77% of function calls in the AST with matching source locations, and 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='36% of function calls when the requirements to match source was relaxed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Manual inspection on a few cases where source locations did not match indicated that the problem was indeed mapping 4In this example, truncation was set to 1024 tokens, as per the requirements of one of the CuBERT models [22] 6 Serenity: Library Based Python Code Analysis for Code Completion and Automated Machine Learning , , 1 def f_Hp(self, pars, p, inpt, target): 2 eps = 1E-6 3 deriv = self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='fprime(pars, inpt, target) ↩→ 4 offseted = self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='fprime(pars + p * eps, inpt, target) ↩→ 5 return (offseted - deriv) / eps Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Example of code where a leaf node is an expression being incorrect in Jython.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Further investigation revealed that many of the missing calls are instances of Python primitives that Serenity does not model and treats as no-ops, such as repr and FutureWarning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' A small fraction was found to be genuinely dead code, especially when Python files were integral parts of a larger application, as they often are in ETH150K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' To generate the slices, we started with leaf nodes, and restricted ourselves to cases where the nodes had at least a depth of 1 when the edges were reversed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' We note that in a majority of cases, leaf nodes were actually expressions, as shown in the example code in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' We ignored these in creating our dataset because we were focused on a prob- lem that cannot be solved by a pure lexical analysis of code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' When we restricted ourselves to nodes that were potentially function calls rather than expressions, we generated slices from 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='35% of the programs where there existed at least one slice where the leaf node was likely a function call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' For the train and test sets of programs, we generated 334,415 slices and 162,847 slices respectively by iterating over all the leaf nodes in dataflow graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Once we consider leaf nodes as nodes for our prediction, there were a total of over 65K labels that were generated across train and test sets for code com- pletion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Figure 8 plots the cumulative frequency distribution of labels against the number of labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' As shown in the Figure, the distribution of labels follows the usual power law, but we note that the most popular label appeared across train and test only 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='7% of the time, and the top 10 labels cumulatively appeared only 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='0% of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' In other words, this is a difficult classification problem5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' We note that this method of declaring code completion is more realistic compared to other means for code completion (such as measuring next token prediction), in the sense that this is often the case that IDEs focus on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='2 Language model selection To decide on the best neural model to use as a basis for our code completion experiments, we tested a number of code related language models including CodeBERT [13], Graph- CodeBERT [19], CuBERT [22] and CodeT5 [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' CodeBERT[13] 5We will make the datasets and code for all the work reported in this section available as open source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='9 1 1 4 16 64 256 1024 4096 16384 Cumulative Frequency No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' of Labels Cumulative frequency of labels Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Distribution of labels for the classification task is a bimodal model trained on datasets with natural lan- guage (NL) -programming language (PL) pairs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' docu- mentation/code pairs) across six programming languages (Python, Java, JavaScript, PHP, Ruby, and Go).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Similarly, GraphCodeBERT uses NL-PL pairs for pretraining a code language model, but based on local data flow graphs ex- tracted from Abstract Syntax Trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' CuBERT [22] is another BERT-based model fine-tuned on multiple classification tasks such as checking the presence of certain bugs and predicting exception types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' CuBERT is trained only on Python code, and furthermore uses language level tokens as inputs to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' CodeT5 [41] is an encoder-decoder model based on T5 architecture [33] with code-specific knowledge trained to distinguish which tokens are identifiers and recover them when they are masked out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' CodeT5 is fine-tuned using multi- ple CodeXGLUE benchmarks including understanding tasks such as code defect detection and clone detection, and gen- eration tasks like code summarization and translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Figure 9 shows the performance of these different models on the code completion task with no fine tuning for the top-1 and top-5 cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' We modeled code completion as a mask pre- diction task, with the function call to be predicted being the masked token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' As shown in the Figure 9, the best performing model was CuBERT on this task, which is not surprising because it was the only model trained exclusively on Python and used language level tokens unlike the other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' We note that the performance of CodeT5 was surprisingly poor, but we think this may in part be due to the fact that it is trained on NL-PL pairs and it is strictly a generative model, 7 , , Wenting Zhao, Ibrahim Abdelaziz, Julian Dolby, Kavitha Srinivas, Mossad Helali, and Essam Mansour 21 19 13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='7 33 27 23 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='1 0 5 10 15 20 25 30 35 40 CuBERT CodeBERT GraphCodeBert CodeT5 Accuracy Complete (Top1) Complete (Top5) Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Accuracy on top-5 and top-1 test data for base lan- guage models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Performance on CuBERT for slices is based on 150,739 and 137,987 examples for complete and slice, respec- tively because of tokenization issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' For all other systems the number of testing examples was 167,816.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' for which we needed to specify a length of generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' It also needed the most tuning in terms of specifying different search strategies for final token prediction, so it is possible that we did not choose the optimal search strategy for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' For our purposes though, we chose CuBERT as a base, primarily because we expected to benefit most from fine tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' We point out that given our label distribution, for the language model to provide even 21% performance on complete for top-1 and 33% for top-5 is quite good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' We turn now to the problem of fine tuning CuBERT with training inputs to see if analysis does in fact improve code completion as we defined it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Note that CuBERT’s pretraining was performed by feeding the model the logical lines of 5 million programs - so at the minimal, some fine tuning for the code context where the function call is to be predicted is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' As stated earlier, we contrasted three different training conditions: complete: where we gave the model text starting from the call, backwards, as shown in Figure 5 slice: where we used a backwards slice as shown in Figure 6 combined where the text from complete and slice were concatenated as input to the model using a sepa- rator token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' The test was on complete text, or combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' We chose these conditions because we observed from examples that for the problem of code generation, data flow is not sufficient by itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Figure 10 shows such an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' In this code, lines 5 and 15 contain the clue needed to make the prediction of id, but they are unrelated to the receiver for which the call is being made on line 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Yet, the local pattern of code has the same variable names, and the same set of calls are repeated across functions, suggesting that id may be a good candidate label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' By contrast, the corresponding slice con- tains minimal information as shown in Figure 11, since the 1 response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='json.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='return_value = dict(response, total_count=3, limit=0, offset=0) ↩→ 2 projects = self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='redmine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='all() ↩→ 3 self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='assertEqual(projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='limit, 0) 4 self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='assertEqual(projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='offset, 0) ↩→ 5 self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='assertEqual(projects[0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='id, 1) 6 self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='assertEqual(projects[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='id, 2) 7 self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='assertEqual(projects[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content="id, 3) 8 9 def test_offset_limit(self): 10 response_with_limit_offset = {'total_count': 2, 'limit': 3, 'offset': 1, 'projects': response['projects'][1:3]} ↩→ ↩→ ↩→ 11 self." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='json.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='return_value = response_with_limit_offset ↩→ 12 projects = self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='redmine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='all()[1:3] ↩→ 13 self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='assertEqual(projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='limit, 3) 14 self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='assertEqual(projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='offset, 1) ↩→ 15 self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='assertEqual(projects[0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='id, 2) 16 self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='assertEqual(projects[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='id, 3) 17 18 def test_offset_limit_mimic(self): 19 projects = self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='redmine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='all()[1:3] ↩→ 20 self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='assertEqual(projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='limit, 3) 21 self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='assertEqual(projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='offset, 1) ↩→ 22 self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='assertEqual(projects[0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Code snippet where local text can help prediction 1 from tests import unittest, mock, Redmine, URL ↩→ 2 Redmine(self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='url) 3 projects = self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='redmine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='all()[1:3] 4 self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='assertEqual(projects[0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Code where data flow lacks sufficient context receiver projects[0] was defined just within the function test_offset_limit_mimic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' We note however that sometimes the slice can help even when the truncation does not cut off key information for prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Figure 12 shows one such example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' The predicted function is partial is imported in line 3, but the actual call is on line 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' On the other hand, in Figure 13, the import is the only call prior to the line, so the slice can make relevant information proximal, such that the neural model can pay greater attention to proximal elements of the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' 8 Serenity: Library Based Python Code Analysis for Code Completion and Automated Machine Learning ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' 1 import sys 2 import logging 3 from functools import partial 4 from datetime import datetime 5 from abc import ABCMeta,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=" abstractmethod 6 import json 7 from _config import AttrDict 8 9 __all__ = ['multikey_getter_gen'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=" 'unescape_json'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=" 'LogParser'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=" 'JSONParser'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=" 'LogLine'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=" ↩→ ↩→ 10 'AccessLog'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=" 'CommonLogFormat'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=" 'uWSGIParser'] ↩→ 11 12 def multikey_getter_gen(parser," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' keys,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' is_indices=False,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' delimiter="\\t"): ↩→ 13 """Generator meta-function to return a function ↩→ 14 parsing a logline and returning multiple keys (tab-delimited)""" ↩→ 15 if is_indices: 16 keys = map(int,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' keys) 17 18 def multikey_getter(line,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' parser,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' keyset): ↩→ 19 data = parser(line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='strip()) 20 return delimiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='join((unicode(data[k]) for k in keyset)) ↩→ ↩→ 21 22 def multiindex_getter(line, parser, keyset): ↩→ 23 data = parser(line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='strip()) 24 return delimiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='join((unicode( data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='by_index( idx-1, raw=True)) for idx in keys)) ↩→ ↩→ 25 26 if is_indices is True: 27 # Field indices 28 return ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Example of where complete text may have text relevant to the prediction, but distant from call site 1 partial = #!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='/usr/bin/env python # 2 from functools import partial 3 keys = map(int, keys) 4 return ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Example of where dataflow is very focused 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='3 Model details We use the CuBERT model released by [22]6, which has 24 layers with 16 attention heads and 1024 hidden units and 6The CuBERT model can be accessed at github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='com/google-research/google- research/tree/master/cubert was pretrained on 4M unique Python files on Github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' At fine-tuning, we set the batch size to 10 and trained the model using 8 Tesla V100 with 32GB memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' The learning rate is 5e-5, and we gradually warmed up the learning rate for the first 300 gradient updates, which are the default val- ues provided by the HuggingFace library [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' The training stops after 20 epochs, or ends after the evaluation accuracy hasn’t improved for three epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' For the complete and slice models we used the 512 tokens model, and when we used combined, we used the 1024 tokens model such that the exact same tokens present in complete and slice could be used together along with the separator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' We apply CuBERT’s tokenization to Python programs in ETH150K where the Python programs are first tokenized us- ing the standard Python tokenizer (the tokenize module)7,8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Then we further break down the program tokens into 49,558 subwords using subword tokenization [39], as performed by the cuBERT tokenizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='4 Results of fine tuning Figure 14 shows the accuracy in predicting the function call exactly across the different training and test conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' As shown in the Figure 14, training on slice was at 47% ac- curacy when tested on the complete text (slice-complete), which is significantly above the 21% of top-1 baseline from cuBERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Training on complete however was much better at 62% on the same text (complete-complete), which is not surprising given that inspection of examples (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=', Fig- ure 10) show that complete often contains the expressions in slice when the dataflow is local, and furthermore, ben- efits from repetition in coding patterns that might hint at labels in the absence of any real connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' The key ques- tion is whether slices provide any benefit over and above what benefit is gained from complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Training on combined suggests that slices do provide a strong signal, with a 65% accuracy on complete text (combined-complete), and 69% accuracy on the combined text (combined-combined).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' We also compared top-5 performance across conditions to allow comparison to the baseline language models - not surpris- ingly this result improved accuracy across all conditions, with the combined-combined condition showing the best performance at 78%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' The results show that data flow analysis can significantly augment code completion performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='5 How do slices help code completion?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' We conducted an analysis of how slices might help code completion performance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=', to understand if slices help the model complete code better for rare labels compared 7github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='com/python/cpython/blob/main/Lib/tokenize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='py 8We note that tokenize only outputs tokens for the code snippet that is free of any syntax errors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' otherwise, it returns either IndentationError or TokenError.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' To predict function calls we often feed code that is incomplete, thus syntactically incorrect;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' therefore, we had to modify the original module so that it always returns what has been already tokenized thus far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' 9 , , Wenting Zhao, Ibrahim Abdelaziz, Julian Dolby, Kavitha Srinivas, Mossad Helali, and Essam Mansour 21 47 62 65 69 33 66 74 75 78 0 10 20 30 40 50 60 70 80 CuBERT baseline slice-complete complete-complete combined-complete combined-combined Accuracy Top-1 Top-5 Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Results of fine tuning on the different "training - testing" conditions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=', training on {slice, complete or com- bined} and tested on {complete or combined} 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='9 1 1 4 16 64 256 1024 Proportion of labels output correctly Label Counts Combined-Complete Complete-Complete Combined-Combined Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Accuracy on labels with different counts to more common ones, since statistical approaches likely work better for common labels, but less well for rare ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Figure 15 shows the performance of the different models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' the presence of slices at training and test enhance code comple- tion performance for rare labels more than common labels, although the advantage does seem to be present for common ones as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' The combined-combined model was 15% ac- curate on labels with count 1, of about 18,000 labels, and that number rapidly approaches 40% for labels with count 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' As labels become more frequent, the differences between the sklearn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='datasets sklearn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='load_digits sklearn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='load_digits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='data sklearn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='load_digits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='target sklearn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='train_test_split sklearn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='LinearSVC sklearn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='svm sklearn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='cross_validation Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' KGpip’s training graph for our running example after filtering out as input to the AutoML system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' models gets more noisy but the combined-combined case still holds an advantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='6 Comparison to existing work In this space, comparisons are tricky because there is no com- mon or standard benchmark and because the exact problem varies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' We chose ETH150K, which is at least a well-known code repository, but work that e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' relies on its own sample of GitHub makes results incomparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' The exact problem varies too, with some tools, like us, predicting function calls, others predicting only method calls and still others predict the next token for all tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' There is a real need for a bench- mark in this space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' as part of helping build such a benchmark, we will release our own slice and complete dataset to the community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' [25] reports overall accuracy numbers around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='7, which is almost the same as ours, but that paper is pre- dicting the next token across all token types, so could benefit from the fact that some predictions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' ’)’ followed by ’:’ in def) follow from the grammar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Pythia [37] uses a neural net- work rather than a language model, but reports comparable accuracy numbers for top-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Their top-5 number is higher, but their predictions seem to be for method calls, rather than all functions as we do, for which the receiver may provide context to aid prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' These approaches may be complementary, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Our work showed that adding slice data to local context greatly aided the accuracy of our models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Other approaches also rely on mostly local information, and could potentially benefit from slices as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' We plan to investigate this further in our future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' 5 Automated ML Pipelines Application The problem of automated machine learning pipelines (Au- toML) focuses on automatically building pipelines by per- forming a search over valid data transformations and learn- ers, along with hyper-parameter optimization for each learner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Our research question is whether we can perform learner and transformation selection based on mining large repositories 10 Serenity: Library Based Python Code Analysis for Code Completion and Automated Machine Learning , , of abstracted ML python scripts obtained statically by Seren- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Unlike dynamic analysis, Serenity’s static analysis of ML pipelines has the advantage of scaling to millions of scripts due to its low cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Specifically, our question is whether the extracted semantics of a set of pipelines by Serenity can help in predicting a new pipeline when combined with neural graph models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' We developed an AutoML system, called KGpip, described in detail in a separate work [20], which builds a database of datasets and their corresponding historically used ML pipelines using Serenity analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' KGpip formulates the au- tomation of a ML pipeline as a graph generation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' It is based on two hypotheses that a neural graph generator will: i) capture more succinctly multiple pipelines seen in practice for a given dataset, and ii) capture statistical similar- ities between different pipelines more effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' In KGpip, we filter out Serenity’s analysis to remove non-ML related components such as calls to libraries other than target ML libraries (Sklearn, XGBoost, and LGBM), and nodes indicat- ing location of calls within a pipeline script, among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Figure 16 shows the filtered version of the graph for our running example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' We also show in Figure 17 an overview of how KGpip works at training and inference phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Using code analysis graphs obtained from Serenity and dataset em- beddings, KGpip trains a graph generation model optimized to output a ML pipeline as close as possible to the target pipelines of the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' At inference time, KGpip iden- tifies the closest dataset to the input dataset and uses its embedding as input to the graph generation model which in turn outputs a set of possible ML pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' These pipelines are then validated and fed to a hyper-parameter optimizer to get the best pipeline that results in the highest performance on the input dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' KGpip is designed to work with AutoML systems, such as AutoSklearn [15] and FLAML [40], to utilize their hyper- parameter optimizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' With a collection of 2000 ML python scripts, we trained a graph generation neural network that learns to generate a ML pipeline graph for a given dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' We conducted a comprehensive evaluation using 77 datasets from different benchmarks, such as AutoML and Penn Ma- chine Learning Benchmark (PMLB), and different ML portals, such as Kaggle and OpenML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Table 1 shows the overall KG- pip performance which significantly improves the selection of data transformation and learning algorithms of state-of- the-art AutoML systems, namely, Auto-Sklearn and FLAML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' We note that AutoSklearn consults a database of pipelines and datasets, and picks pipelines to start the search based on a nearest neighbors to an unseen dataset, except that Au- toSklearn’s dataset consists of effective pipelines based on actual execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' We also compared KGpip to AL [6], which uses dynamic code analysis on existing machine learning pipelines to select optimized pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' AL was unable to process 60 datasets because it ran out of time in searching for pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' On a smaller set of 17/77 datasets on which Average Performance T-Test AutoSklearn 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='71 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='24) KGpip + AutoSklearn 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='77 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='22) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='0002 FLAML 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='71 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='27) KGpip + FLAML 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='77 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='20) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='0132 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Performance (average and stdev) of KGpip com- pared to FLAML and AutoSklearn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Both variations of KGpip show significant improvements compared to existing sys- tems, both with 2-tailed T-test 𝑝 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='05 AL was able to work, AL achieved an average performance of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='36 compared to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='745, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='705, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='79 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='765 by FLAML, Auto-Sklearn, KGpip + FLAML, and KGpip + AutoSklearn, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' This comparison with both AL and AutoSklearn clearly illustrates the value provided by Serenity, compared even to approaches that rely on dynamic runtime analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' 6 Related Work Static Analysis for Python: Static analysis of Python has attracted considerable interest lately, and there have been a range of approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Type inference has been a focus of much work, some using techniques such as abstract inter- pretation and, more recently, there has been work using machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Likely the best-known work is MyPy [1] and Pytype [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' MyPy focuses on checking and inferring types that conform to PEP 484 [38], which defines a syntax for Python types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' MyPy focuses on inference within a single function, since types are expressed at function boundaries in PEP 484.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Pytype does type inference, and it can handle cases where a variable has different types at different points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' It also does relatively little interprocedural analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Moat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' [28] present an abstract interpretation for type inference of Python that models a variety of domains to compute more accurate information, and it makes use of the recency abstraction for aliasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' However, it is currently limited in its support for interprocedural analysis, which is enabled by inlining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Fritz and Hage [16] present a dataflow analysis for type inference that provides a range of tradeoffs for cost and precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' It does handle features like first-class functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Machine learning is also used for type inference of Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' TypeWriter [31] trained a neural model using a corpus of code, with labeled data derived from user annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Type- Writer considers comments in code as inputs to the neural model, unlike program analysis based type inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' While such systems are certainly performing analysis, their ap- proach and mechanism are quite different from ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' There have been other analyses of Python, often for spe- cial purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Ariadne [11] makes use of WALA, as we do, but focuses on inferring the shapes of tensors in machine 11 , , Wenting Zhao, Ibrahim Abdelaziz, Julian Dolby, Kavitha Srinivas, Mossad Helali, and Essam Mansour Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' An overview of KGpip training and inference workflows learning programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Unlike our approach to libraries, Ariadne models ML libraries as needed for tensor-related operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Code completion: Code completion has been a prominent area of research towards achieving better productivity when working within an IDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' One of the main challenges in this domain is about code representation where the vast majority of work has used either tokens or abstract syntax trees to represent code (we refer the reader to [3] for a detailed survey of this area).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' [25] represents Python and JavaScript codes as ASTs and use a pointer network for better predicting Out of Vocabulary words in code completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Pythia [37] is another approach that uses ASTs with an LSTM model for code completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' A number of approaches also tried to leverage representa- tions based on data and control flow [4, 7, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' On JavaScript, [21] utilized a program dependence graph to detect code du- plication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' [4] use AST based representation augmented with local data and control flows for predicting variable names and variables misuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' [14] combines token based represen- tations of code with edges based on object uses, and AST nodes to predict the documentation of a method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' To perform code completion over Java API calls, [29, 30] used a mostly intraprocedural analysis for mining graphs augmented with control and data flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' With the rise of pre-trained language models such as BERT [10] and GPT [5, 32], many recent approaches [13, 19, 22, 26, 41] started to leverage the already existing rich language understanding in these models and fine tune it for various code understanding tasks such as code summarization, trans- lation, completion, bug detection, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' CodeBERT[13] is a BERT based model trained on pairs of natural language and programming language samples across six programming languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' GraphCodeBERT [19] uses BERT as well, but represents the code using data flow graphs based on ASTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' The dataflow in GraphCodeBERT is completely local and not interprocedural, as in Serenity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' For instance as an exam- ple, it adds edges from all variables used in an expression to their definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' CuBERT [22] and CodeT5 [41] are an- other two models based on BERT and T5 [33] architectures, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Unlike our approach, all these methods represented code either as a sequence of tokens [13, 22], ASTs [25, 27, 37, 41], or data flows derived from ASTs [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' AutoML approaches: Several AutoML frameworks have been proposed recently [6, 12, 15, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' In most AutoML sys- tems, learner and pre-processing selection is driven by a database of actual executions of pipelines and data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' For in- stance, [15, 36] compute a database of dataset meta-features such as number of rows and columns, while [6] mines a repository of run-time information of inputs to the learners and preprocessors via dynamic code analysis of public ML pipelines available e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' on Kaggle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' The predicted learners and preprocessors are based on a similarity measure between the target dataset and stored features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' In KGpip we utilized dense vector embeddings derived from raw contents of datasets to measure this similarity and graph neural networks to select the learners/preprocessors and generate the pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Some existing systems such as TPOT [24] or Recipe [9] use evolutionary algorithms for pipeline generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Others approach it as a probabilistic matrix factorization [17], an AI planning problem when combined with a user specified grammar [23, 42], or a bayesian optimization problem com- bined with Monte Carlo Tree Search [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' None of these approaches however use analysis to build up their database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' 7 Conclusion In this paper, we introduced Serenity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' a framework for Python code static analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Serenity relies on two mechanisms (a) dynamic dispatching at the core of language translation, and (b) extreme abstraction of libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' To demonstrate the 12 E CSV Deep graph jupyter generator model CsV ML Pipelines Hyperparameter Skeleton Optimization Auto- Sklearn FLAMISerenity: Library Based Python Code Analysis for Code Completion and Automated Machine Learning , , utility of Serenity’s analysis, we used it in two important code-related applications: code generation and automated machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Serenity’s analysis showed very promis- ing performance in both applications, allowing us in some cases to outperform approaches based on dynamic analysis, and perform competitively for code completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' We also im- plemented Serenity as an open-source implementation based on WALA, a popular framework for program analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' References [1] [n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' MyPy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' http://mypy-lang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Accessed: 2021-11-18.' 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Processing: System Demonstrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' Association for Computa- tional Linguistics, Online, 38–45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='aclweb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='org/anthology/ 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='emnlp-demos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} +page_content='6 14' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E4T4oBgHgl3EQffg1w/content/2301.05108v1.pdf'} diff --git a/FNE5T4oBgHgl3EQfVQ8Q/content/tmp_files/2301.05549v1.pdf.txt b/FNE5T4oBgHgl3EQfVQ8Q/content/tmp_files/2301.05549v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6bb4bb9b8d8b746a5349765411cf934e7c9a6baf --- /dev/null +++ b/FNE5T4oBgHgl3EQfVQ8Q/content/tmp_files/2301.05549v1.pdf.txt @@ -0,0 +1,471 @@ +arXiv:2301.05549v1 [quant-ph] 12 Jan 2023 +1 +On the explainability of quantum neural networks +based on variational quantum circuits +Ammar Daskin +Abstract—Ridge functions are used to describe and study the +lower bound of the approximation done by the neural networks +which can be written as a linear combination of activation +functions. If the activation functions are also ridge functions, +these networks are called explainable neural networks. +In this paper, we first show that quantum neural networks +which are based on variational quantum circuits can be written +as a linear combination of ridge functions. Consequently, we +show that the interpretability and explainability of such quantum +neural networks can be directly considered and studied as an +approximation with the linear combination of ridge functions. +Index Terms—Quantum neural networks, explainability, inter- +pretability, ridge functions +I. INTRODUCTION +Neural networks have applications almost in every field +of the science ranging from health to banking. The ability +to interpret the result of a model and explain the learning +behavior may be deemed important especially in critical in- +dustries such as medicine and health care[1–3]. Limitations +of the approximation rates of the classical neural networks +can be understood better by using linear combination of ridge +functions as an approximation to neural networks. +The power and the limitations of quantum neural networks +are yet to be fully understood. In this paper, we show that +quantum neural networks can be written as a sum of ridge +functions. Therefore, the math and methodologies that are used +to understand classical neural networks can be used to study +quantum ones. +A. Approximation with ridge functions +For a random variable y, if we have the observations +y1, . . . , yn at points x1, . . . , xn, a standard regression model +can be described by yi = ˆyi + ri, where ˆyi defines the +dependence of yi on xi When xi are univariate real values, +the assumption is that the dependence is smooth. This leads +the following estimation for the regression model[4]: +E(y | x) = f(x), +(1) +where f is a smoothing function. In linear smoothing, ˆy = +(ˆy1, . . . , ˆyn)T can be written in the form of matrix vector +transformation: ˆy = Sy, where S is the smoother matrix +that does not depend on y. When there are more than one +predictors, the estimating regression surface is hard because +of the curse of dimensionality (the data sparseness in high +A. +Daskin +was +with +the +Department +of +Computer +Engineering, +Istanbul +Medeniyet +University, +Istanbul, +Turkey +e-mail: +(see +https://sites.google.com/view/adaskin). +dimensions)[5]. The general approach is to use the one- +dimensional smoother as the building block in an additive +model [4]. Given predictors xijs for each yi outcome, i.e. +{yi, xi1, . . . , xip}, the additive model can be described as: +E(yi|xi1, . . . , xip) = α + +p +� +j=1 +fj(xij) + ǫ. +(2) +where ǫ is the inherent error, α is a constant parameter, and fjs +represent unspecified smooth functions. Fitting can be done +by using the backfitting algorithm [5, 6] which is in matrix +form equivalent to Gauss-Seidel method in numerical linear +algebra[7]. +Generalizing this model leads to the projection pursuit +model proposed in [5] where the regression surface is pre- +dicted by a linear combination of ridge functions as in the +following form: +f(x) = +K +� +k=1 +fk(wk · x). +(3) +Here, wks represent weight vectors (projection indices) and +fks are ridge functions [8–11]: Any multivariate function +fk : Rd → R. The vector wk is called the direction and fk +gives a constant on certain hyper-planes whose normal vectors +are parallel to this direction. Ridge functions are used in +approximation theory, partial differential equations, and neural +networks. For instance, a feed forward neural network can be +defined as [8]: +� +k +γkσ (wk · x + bk) , +(4) +where bk, αk, and wk represents parameters that describe the +neural network. σ is a univariate function (activation function). +The degree of approximation by σ functions can be bounded +by the degree of approximation by ridge functions (we refer +the reader to Ref.[8] for the properties and other uses of the +ridge functions). The lower bound of the approximation of the +neural networks can be also studied through the relations of +the activation functions with ridge functions (e.g., [12–14]). +If σ is chosen as a ridge function these networks are recently +called explainable neural networks [15]: e.g. an example +architecture which has three important structures, a projection +layer, sub-network, and a combination layer is described to +learn the following: +f(x) = µ + +K +� +k=1 +γkfk(wk · x). +(5) +Here, µ and γks are shift and scaling parameters, respectively. +In comparison to standard neural networks, the learning in + +2 +this model can be understood by the ”explainable” features: +linear projections and uni-variate functions (in other words, the +mechanisms used to learn the model can be clearly explained +by studying the constitutes that are ridge functions.). +II. EXPLAINABILITY OF QUANTUM NEURAL NETWORKS +Quantum neural networks[16–19] are generally based on +variational quantum circuits[20] and can be described by +⟨x| W(θ) ˆOW(θ) |x⟩ , +(6) +where ˆO represents the measurement operator, |x⟩ is the input +vector formatted as a quantum state and W(θ) is a unitary +matrix generated by the quantum gates with the angle values +defined by θ. Here, by abuse of the notation, we can consider +ˆO as a selector set on the parts of W(θ) |x⟩: e.g., to obtain +the measurement output of the first qubit in |0⟩ state, in vector +forms, we select the first half of the output and combine their +squared absolute values. Then, the output quantum state of the +quantum circuit applied to |x⟩ can be rewritten as: +� +i∈ ˆ +O +| +� +wi|x +� +|2 = +� +i∈ ˆ +O +fi( +� +wi|x +� +), +(7) +where ⟨wi| represents a row of the unitary matrix W(θ). +In variational quantum circuits, generally any change of +the vector element of θ may affect multiple row of W(θ). +This can affect the studies that try to understand the quantum +neural network model. Therefore, to make any fi independent +from each other, we can use the linear combination of unitary +matrices[21, 22]. By following Ref.[22], we can write the rows +of any matrix W(θ) as the first rows of matrices and combine +them on a block diagonal matrix: +V = + + + + + + + + + + + + + + + + + + + + + + + + + + +⟨w1| +• +... +• + + + + + + +N×N +... + + + + + + +⟨wN| +• +... +• + + + + + + +N×N + + + + + + + + + + + + + + + + + + + + +N 2×N 2 +(8) +where N is the dimension of W(θ). Using the direct sum, +we can write V = +N +� +i +Vi with Vi representing the unitary +matrix for wi. Note that any N-dimensional vector can be +formed with O(N) quantum gates as the leading row of a +unitary matrix by using its Schmidt decomposition. Therefore, +the construction V for a generic W(θ) requires at most O(N 2) +gates (See Ref.[22] for complexity analysis of the circuit.). +The equivalent quantum state to the output of W(θ) |x⟩ +can be generated as a part of the outcome of the following +transformation: +|ψ⟩ = V + + + + +|x⟩ +... +|x⟩ + + + + +N 2×1 +. +(9) +In a simplified form let |ψi⟩ represents the ith element of |ψ⟩ +with 0 ≥ i < N 2, we can define a new selector operator that +selects every |ψi⟩, where i mod N = 0. That means we can +still use the definition similar to Eq.(7): +N 2−1 +� +i,i mod N=0 +| +� +wi|x +� +|2 +(10) +Note that by writing a quantum operator in this way, we simply +make the weight vectors independent from each other. That +means the impact of an angle-change in one quantum gate +can be limited to affect only one vector. Therefore, this model +is at least as much powerful as using a single unitary matrix +where a change may affect multiple rows. In addition, studying +the approximations in this model may be easier since we can +easily see how many independent weight vectors are needed +and how each weight vector affect the result and training. +A. Explainability of networks that are represented by expo- +nentials +Ridge functions are also used to approximate the integral +forms of functions[23]: One of the most commonly used one +is given by the following Fourier form: +Ψwk := eix·wk, +(11) +Here, the complex plane can be rewritten using the only real +valued functions, therefore Ψwk can be considered as a ridge +function for each wk. Therefore, quantum neural networks +such as [24] can be also explained as an approximation through +the linear combination of ridge functions. +This can be also used to understand quantum circuits that +are defined through the Ising type Hamiltonian[25] or machine +learning tasks that are solved through the adiabatic quantum +computation [26, 27] +III. DISCUSSION AND CONCLUSION +For a given set of weight vectors, in literature there are many +works on the conditions to approximate a given function with +unknown ridge functions. For instance[28], consider C(X) is +the space of continuous functions on X, then for a given +{w1, w2}, for an appropriate choice of some continuous +functions h1 and h2, one can write g1(h1(x)) + g2(h2(x)) = +C(X) if and only if the lengths of h1 − h2 paths in X are +bounded by some positive integer. From the Kolmogorov- +Arnold representation theorem [29], we know that any multi- +variate continuous function can be approximated as a sum of +univariate functions. By this theorem, it can be also explained +how the hidden layers in the classical neural networks help +in approximations[29–31]. However, a continuous function + +3 +may not be represented exactly by an approximation with +ridge functions [10, 29]. Therefore, the approximation rates +in quantum neural networks that reduces to Eq.(7) and (10) +can be well understood. On the other hand, this indicates that +the limitations of the approximations with the ridge functions +may be also limitations for these types of quantum neural +networks. +In this brief paper, we show that quantum neural networks +can be understood as a linear combination of ridge functions, +which is used to understand the interpretability and explain- +ability of the classical neural networks. In particular, Eq.(7) +and Eq.(10) can be used to describe quantum neural network +models. In addition, since it can be written in the form of a +linear combination of ridge functions, the approximation errors +and upper and lower bounds on the errors of quantum neural +networks can be studied through this formulation. +In quantum neural networks which can be reduced to +Eq.(7) and (10), the approximation is done by using N ridge +functions. Here, N is in general exponential in the number of +qubits and can be considered a very large number. For general +function approximations, using these equations may be helpful +to make a decision on the size of N and required number of +operations and qubits before any application. However, the +problems solved by the neural networks are in general not +easy to define by functions, therefore it is not easy to decide +how many functions (and quantum operations and qubits) are +required to make the model more trainable. +REFERENCES +[1] E. 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K˚urkov´a, “Kolmogorov’s theorem and multilayer neu- +ral networks,” Neural networks, vol. 5, no. 3, pp. 501– +506, 1992. + diff --git a/FNE5T4oBgHgl3EQfVQ8Q/content/tmp_files/load_file.txt b/FNE5T4oBgHgl3EQfVQ8Q/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..38db6ab194d4aebb211143c1bc2d9510381f6ca8 --- /dev/null +++ b/FNE5T4oBgHgl3EQfVQ8Q/content/tmp_files/load_file.txt @@ -0,0 +1,322 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf,len=321 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content='05549v1 [quant-ph] 12 Jan 2023 1 On the explainability of quantum neural networks based on variational quantum circuits Ammar Daskin Abstract—Ridge functions are used to describe and study the lower bound of the approximation done by the neural networks which can be written as a linear combination of activation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' If the activation functions are also ridge functions, these networks are called explainable neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' In this paper, we first show that quantum neural networks which are based on variational quantum circuits can be written as a linear combination of ridge functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' Consequently, we show that the interpretability and explainability of such quantum neural networks can be directly considered and studied as an approximation with the linear combination of ridge functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' Index Terms—Quantum neural networks, explainability, inter- pretability, ridge functions I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' INTRODUCTION Neural networks have applications almost in every field of the science ranging from health to banking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' The ability to interpret the result of a model and explain the learning behavior may be deemed important especially in critical in- dustries such as medicine and health care[1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' Limitations of the approximation rates of the classical neural networks can be understood better by using linear combination of ridge functions as an approximation to neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' The power and the limitations of quantum neural networks are yet to be fully understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' In this paper, we show that quantum neural networks can be written as a sum of ridge functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' Therefore, the math and methodologies that are used to understand classical neural networks can be used to study quantum ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' Approximation with ridge functions For a random variable y, if we have the observations y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' , yn at points x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' , xn, a standard regression model can be described by yi = ˆyi + ri, where ˆyi defines the dependence of yi on xi When xi are univariate real values, the assumption is that the dependence is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' This leads the following estimation for the regression model[4]: E(y | x) = f(x), (1) where f is a smoothing function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' In linear smoothing, ˆy = (ˆy1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' , ˆyn)T can be written in the form of matrix vector transformation: ˆy = Sy, where S is the smoother matrix that does not depend on y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' When there are more than one predictors, the estimating regression surface is hard because of the curse of dimensionality (the data sparseness in high A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' Daskin was with the Department of Computer Engineering, Istanbul Medeniyet University, Istanbul, Turkey e-mail: (see https://sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content='com/view/adaskin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' dimensions)[5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' The general approach is to use the one- dimensional smoother as the building block in an additive model [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' Given predictors xijs for each yi outcome, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' {yi, xi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' , xip}, the additive model can be described as: E(yi|xi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' , xip) = α + p � j=1 fj(xij) + ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' (2) where ǫ is the inherent error, α is a constant parameter, and fjs represent unspecified smooth functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' Fitting can be done by using the backfitting algorithm [5, 6] which is in matrix form equivalent to Gauss-Seidel method in numerical linear algebra[7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' Generalizing this model leads to the projection pursuit model proposed in [5] where the regression surface is pre- dicted by a linear combination of ridge functions as in the following form: f(x) = K � k=1 fk(wk · x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' (3) Here, wks represent weight vectors (projection indices) and fks are ridge functions [8–11]: Any multivariate function fk : Rd → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' The vector wk is called the direction and fk gives a constant on certain hyper-planes whose normal vectors are parallel to this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' Ridge functions are used in approximation theory, partial differential equations, and neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' For instance, a feed forward neural network can be defined as [8]: � k γkσ (wk · x + bk) , (4) where bk, αk, and wk represents parameters that describe the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' σ is a univariate function (activation function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' The degree of approximation by σ functions can be bounded by the degree of approximation by ridge functions (we refer the reader to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' [8] for the properties and other uses of the ridge functions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' The lower bound of the approximation of the neural networks can be also studied through the relations of the activation functions with ridge functions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=', [12–14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' If σ is chosen as a ridge function these networks are recently called explainable neural networks [15]: e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' an example architecture which has three important structures, a projection layer, sub-network, and a combination layer is described to learn the following: f(x) = µ + K � k=1 γkfk(wk · x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' (5) Here, µ and γks are shift and scaling parameters, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' In comparison to standard neural networks, the learning in 2 this model can be understood by the ”explainable” features: linear projections and uni-variate functions (in other words, the mechanisms used to learn the model can be clearly explained by studying the constitutes that are ridge functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' EXPLAINABILITY OF QUANTUM NEURAL NETWORKS Quantum neural networks[16–19] are generally based on variational quantum circuits[20] and can be described by ⟨x| W(θ) ˆOW(θ) |x⟩ , (6) where ˆO represents the measurement operator, |x⟩ is the input vector formatted as a quantum state and W(θ) is a unitary matrix generated by the quantum gates with the angle values defined by θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' Here, by abuse of the notation, we can consider ˆO as a selector set on the parts of W(θ) |x⟩: e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=', to obtain the measurement output of the first qubit in |0⟩ state, in vector forms, we select the first half of the output and combine their squared absolute values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' Then, the output quantum state of the quantum circuit applied to |x⟩ can be rewritten as: � i∈ ˆ O | � wi|x � |2 = � i∈ ˆ O fi( � wi|x � ), (7) where ⟨wi| represents a row of the unitary matrix W(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' In variational quantum circuits, generally any change of the vector element of θ may affect multiple row of W(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' This can affect the studies that try to understand the quantum neural network model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' Therefore, to make any fi independent from each other, we can use the linear combination of unitary matrices[21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' By following Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' [22], we can write the rows of any matrix W(θ) as the first rows of matrices and combine them on a block diagonal matrix: V = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed ⟨w1| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 N×N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed ⟨wN| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 N×N \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 N 2×N 2 (8) where N is the dimension of W(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' Using the direct sum, we can write V = N � i Vi with Vi representing the unitary matrix for wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' Note that any N-dimensional vector can be formed with O(N) quantum gates as the leading row of a unitary matrix by using its Schmidt decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' Therefore, the construction V for a generic W(θ) requires at most O(N 2) gates (See Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' [22] for complexity analysis of the circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' The equivalent quantum state to the output of W(θ) |x⟩ can be generated as a part of the outcome of the following transformation: |ψ⟩ = V \uf8eb \uf8ec \uf8ec \uf8ed |x⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' |x⟩ \uf8f6 \uf8f7 \uf8f7 \uf8f8 N 2×1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' (9) In a simplified form let |ψi⟩ represents the ith element of |ψ⟩ with 0 ≥ i < N 2, we can define a new selector operator that selects every |ψi⟩, where i mod N = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' That means we can still use the definition similar to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' (7): N 2−1 � i,i mod N=0 | � wi|x � |2 (10) Note that by writing a quantum operator in this way, we simply make the weight vectors independent from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' That means the impact of an angle-change in one quantum gate can be limited to affect only one vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' Therefore, this model is at least as much powerful as using a single unitary matrix where a change may affect multiple rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' In addition, studying the approximations in this model may be easier since we can easily see how many independent weight vectors are needed and how each weight vector affect the result and training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' Explainability of networks that are represented by expo- nentials Ridge functions are also used to approximate the integral forms of functions[23]: One of the most commonly used one is given by the following Fourier form: Ψwk := eix·wk, (11) Here, the complex plane can be rewritten using the only real valued functions, therefore Ψwk can be considered as a ridge function for each wk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' Therefore, quantum neural networks such as [24] can be also explained as an approximation through the linear combination of ridge functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' This can be also used to understand quantum circuits that are defined through the Ising type Hamiltonian[25] or machine learning tasks that are solved through the adiabatic quantum computation [26, 27] III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' DISCUSSION AND CONCLUSION For a given set of weight vectors, in literature there are many works on the conditions to approximate a given function with unknown ridge functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' For instance[28], consider C(X) is the space of continuous functions on X, then for a given {w1, w2}, for an appropriate choice of some continuous functions h1 and h2, one can write g1(h1(x)) + g2(h2(x)) = C(X) if and only if the lengths of h1 − h2 paths in X are bounded by some positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' From the Kolmogorov- Arnold representation theorem [29], we know that any multi- variate continuous function can be approximated as a sum of univariate functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' By this theorem, it can be also explained how the hidden layers in the classical neural networks help in approximations[29–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' However, a continuous function 3 may not be represented exactly by an approximation with ridge functions [10, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' Therefore, the approximation rates in quantum neural networks that reduces to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' (7) and (10) can be well understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' On the other hand, this indicates that the limitations of the approximations with the ridge functions may be also limitations for these types of quantum neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' In this brief paper, we show that quantum neural networks can be understood as a linear combination of ridge functions, which is used to understand the interpretability and explain- ability of the classical neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' In particular, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' (7) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' (10) can be used to describe quantum neural network models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' In addition, since it can be written in the form of a linear combination of ridge functions, the approximation errors and upper and lower bounds on the errors of quantum neural networks can be studied through this formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' In quantum neural networks which can be reduced to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' (7) and (10), the approximation is done by using N ridge functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' Here, N is in general exponential in the number of qubits and can be considered a very large number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'} +page_content=' For general function approximations, using these 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+Université Paris Cité, Institut de Mathématiques de Jussieu-Paris Rive Gauche, F-75013, +Paris, France +E-mail: antoine.chambert-loir@u-paris.fr +Maxim Kontsevich +Institut des Hautes Études Scientifiques, 35 route de Chartres, 91440 Bures-sur-Yvette, +France +E-mail: maxim@ihes.fr +Yuri Tschinkel +Courant Institute, NYU, 251 Mercer St. New York, NY 10012, USA +Simons Foundation, 160 5th Av., New York, NY 10010, USA +E-mail: tschinkel@cims.nyu.edu +Abstract. — We develop a theory of Burnside rings in the context of birational equivalences +of algebraic varieties equipped with logarithmic volume forms. +We introduce a residue +homomorphism and construct an additive invariant of birational morphisms. We also define +a specialization homomorphism. +Résumé. — +Nous proposons une théorie d’anneaux de Burnside dans le contexte de la +géométrie birationnelle des variétés algébriques munies d’une forme volume à pôles logarith- +miques. Nous introduisons un homomorphisme « résidu », construisons un invariant additif +des morphismes birationnels. Nous définissons aussi un homomorphisme de spécialisation. +2000 Mathematics Subject Classification. — 14E08, 14E07, 14D06. +Key words and phrases. — Birational geometry, Burnside rings, logarithmic volume +forms, specialization. +Contents +1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +2 +2. Logarithmic differential forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +4 +3. Burnside rings for logarithmic forms. .. . . . . . . . . . . . . . . . . . . . . . . . . . . +6 +4. Residues. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +9 +5. A complex of Burnside rings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 +6. Algebraic structure of Burn(k) after localization at 2. . . . . . . . . . . 17 +7. Birational morphisms preserving volume forms. . . . . . . . . . . . . . . . . . 20 +8. Specialization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 +References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 + +2 +ANTOINE CHAMBERT-LOIR, MAXIM KONTSEVICH & YURI TSCHINKEL +1. Introduction +The study of birationality of algebraic varieties is a classical and well studied +subject, with many open problems. In some cases, it is interesting to study birational +maps preserving additional structure, for example group actions, symplectic forms, +or volume forms. Such a study is already implicit in many questions of birational +geometry, eg, in the notion of crepant resolution of singularities. +In this paper, we consider the case of varieties endowed with volume forms with +logarithmic poles and develop a formalism of Burnside rings along the lines of their +counterpart introduced by Kontsevich & Tschinkel (2019) to establish the spe- +cialization of rationality, and its equivariant version by Kresch & Tschinkel +(2022b). +Let k be a field of characteristic zero. For each integer n, we define +Burnn(k) +as the free abelian group on birational equivalence classses of pairs (X, ω) consisting +of an integral smooth proper k-variety X of dimension n equipped with an n-form ω +with at most logarithmic poles. +The graded abelian group +Burn(k) = +� +n∈N +Burnn(k) +carries a ring structure, induced by taking products of varieties, decomposed into ir- +reducible components, and equipped with the external product of the volume forms. +In section 4, we define morphisms of abelian groups +∂ : Burnn(k) → Burnn−1(k). +When X is smooth and the polar divisor of ω has strict normal crossings, the image +of the class [X, ω] is given by the following formula. Let (Dα)α∈A be the family +of irreducible components of the polar divisor of ω. For each subset A of A , the +intersection DA = � +α∈A Dα is a union of integral smooth varieties of codimension |A|; +taking iterated residues, we may equip it with a volume form with logarithmic +poles ωA. Then +∂([X, ω]) = +� +∅̸=A⊆A +(−1)|A|−1[DA, ωA] · T|A|−1, +where +T = [P1, dt/t]. +In particular, the existence of the map ∂ relies on the birational invariance of this +expression, see theorem 4.7. +This construction is reminiscent of the boundary map in polar homology +(Khesin & Rosly, 2003; Khesin et al, 2004; Gorchinskiy & Rosly, 2015). +However, apart from the obvious difference that we only record birational types of +strata, rather than the strata themselves, our formula takes into account strata of +all codimensions, rather than those of codimension one. + +BURNSIDE RINGS AND VOLUME FORMS WITH LOGARITHMIC POLES +3 +The map ∂ is additive. Furthermore, we prove in theorem 4.10 that +∂(a · b) = εn · ∂(a) · b + a · ∂(b) − T · ∂(a) · ∂(b), +when a ∈ Burnm(k) and b ∈ Burnn(k). Here ε is the class of the point Spec(k) +equipped with the volume form equal to −1. +In theorem 5.1, we show that +∂ ◦ ∂ = 0. +These formulas may look complicated. However, as we explain in §6, they simplify +significantly after inverting 2. +Inspired by the constructions of Lin et al (2020); Lin & Shinder (2022); +Kresch & Tschinkel (2022a), we define in §7 a homomorphism +c: Bir(X, ω) → Burnn−1(k), +from the group of birational automorphisms of the pair (X, ω), where X is an n- +dimensional integral proper smooth variety equipped with a logarithmic volume +form ω. As in the above references, our map c is defined at the groupoid level of +birational maps preserving logarithmic volume forms. +When the birational isomorphism ϕ: (X, ω) ��� (Y, η) is described by a diagram +W +X +Y +← +→ +p +← +→ +q +← +→ +ϕ +of smooth proper integral k-varieties, with birational morphisms p and q, the two +logarithmic volume forms p∗ω and q∗η on W are equal, and the element c(ϕ) ∈ +Burnn−1(k) is given by +c(ϕ) = +� +E∈Exc(q) +[E, p∗ωE] − +� +D∈Exc(p) +[D, q∗ηD] +where Exc(p) is the set of irreducible components of the exceptional divisor of p, +and where, for each such component D, the logarithmic volume form p∗ωD on D is +obtained by taking the residue of p∗ω along D (and similarly for q). +Finally, consider a discrete valuation ring with residue field k and field of frac- +tions K and let t be a uniformizing element. In this context, we define a specialization +map +ρt : Burnn(K) → Burnn(k). +The image of the class [X, ω] involves the combinatorics of a good model (X , ω) +over the valuation ring, and a certain subcomplex of the Clemens complex of the +special fiber. In the particular case where X is smooth, the polar divisor of ω is +a relative divisor with normal crossings, and denoting by ωo the restriction of ω to +the special fiber Xo, one has +ρt([X, ω]) = [Xo, ωo]. +Note that the existence of such a specialization map implies, as in theorem 1 +of Kontsevich & Tschinkel (2019), or as in (Nicaise & Shinder, 2019), that + +4 +ANTOINE CHAMBERT-LOIR, MAXIM KONTSEVICH & YURI TSCHINKEL +birational equivalence of varieties with logarithmic volume forms is preserved under +“good specializations”. +In the geometric case, where the valuation is the local ring of a curve C at point o, +the construction of the specialization map can be viewed as a restriction to the +special fiber of a normalization of a global residue map ∂ that takes place on a proper +model whose special fiber is a divisor with normal crossings. The normalization +procedure extracts a subcomplex of the Clemens complex of the special fiber. A +similar situation appeared in the study of Tamagawa measures on analytic manifolds +(Chambert-Loir & Tschinkel, 2010). +Related constructions emerged from the seminal work of Kontsevich & Soibelman +(2006) inspired by mirror symmetry, and its subsequent developments, eg, by +Mustaţă & Nicaise (2015); Nicaise & Xu (2016); Boucksom & Jonsson +(2017); Jonsson & Nicaise (2020). +Our constructions use essentially only formal properties of the residue maps. Con- +sequently, one can envision analogous theories for logarithmic forms of smaller de- +gree, Milnor K-theory, or even for the cycle modules of Rost (1996). +Acknowledgments. — The third author was partially supported by NSF grant +2000099. +2. Logarithmic differential forms +2.1. Kähler differentials. — Let k be a field of characteristic zero and let K be +a finitely generated extension of k; let n be its transcendence degree. The space of +Kähler differentials ΩK/k is the K-vector space generated by symbols da, for a ∈ K, +subject to the relations: +(1) For a ∈ k, one has da = 0; +(2) For a, b ∈ K, one has d(a + b) = da + db and d(ab) = adb + b(da). +For any integer m ⩾ 0, we may consider its mth exterior power Ωm +K/k, which is +a K-vector space of dimension +�n +m +� +; in particular, it vanishes if m > n, Ω1 +K/k has +dimension n, and Ωn +K/k has dimension one. One has Ω0 +K/k = k, canonically. +Elements of Ωn +K/k, for n = tr. degk(K), are also called volume forms. +For a ∈ K×, we also write dlog a = da/a ∈ ΩK/k. +2.2. Models. — Let m be an integer and let ω ∈ Ωm +K/k. +A model of K is an +integral k-scheme X together with a k-isomorphism K ≃ k(X); we say that this +model is proper, resp. smooth if X is proper, resp. smooth over k. Given such a +model, ω induces a meromorphic global section ωX of Ωm +X/k. The polar ideal of ωX +is the subsheaf of OX whose local sections are the a ∈ OX such that aωX is induced +by a regular m-form. Let D be the zero-locus of this ideal. Its complement U is +the largest open subscheme of X such that ωX is induced by a regular m-form on U. +If X is smooth, then ωX is locally free, hence the scheme D is an effective divisor +(Hartogs’s principle); we call it the polar divisor of ω on X. + +BURNSIDE RINGS AND VOLUME FORMS WITH LOGARITHMIC POLES +5 +2.3. Logarithmic forms. — By Hironaka’s theorem on embedded resolution of +singularities, there exist smooth projective models (X, ωX) of (K, ω) such that the +polar divisor D of ωX has normal crossings. +Following (Deligne, 1970, chap. 2, §3), we then say that ωX has at most loga- +rithmic poles, or that ω has at most logarithmic poles on X, if both ωX and dωX +have at most simple poles along D. +The following lemma implies that this condition is essentially independent of the +choice of X such that the polar divisor of ωX has normal crossings. +Lemma 2.4. — Let g : X′ → X be a morphism of smooth k-varieties, let D be a +divisor with normal crossings in X and let D′ be a divisor with normal crossings +in X′ such that D′ = g−1(D). Let ω be a regular m-form on X +D and let ω′ = g∗ω. +(1) If ω has at most logarithmic poles along D, then ω′ has at most logarithmic +poles along D′. +(2) The converse holds if g is proper and surjective. +Proof. — The first assertion is (Deligne, 1970, chap. II, prop. 3.2, (iv)). Let us +prove the second one. +Consider the generic point η of X and a point η′ ∈ X′ +D′ which is algebraic +over k(η). The Zariski closure X′ +1 of η′ is proper and generically finite over X, and +D′ +1 = D′ ∩ X′ +1 is a divisor. There is a proper modification h: X′ +2 → X′ +1 such that +D′ +2 = h−1(D′ +1) has normal crossings. By the first part, the form h∗ω′|X′ +1 has at most +logarithmic poles along D′ +2. Replacing g by g◦h, we may assume that g is generically +finite. +Since the sheaf of forms with at most logarithmic poles along D is locally free and +X is smooth, we can delete from X a subset of codimension at least 2. Thus, we +may assume that g is flat, D is smooth and irreducible, and g is étale outside of D. +It suffices to argue étale locally at the generic point of D. By the local description +of ramified morphisms, there are étale local coordinates (z1, . . . , zn) on X such that +Dred = V(z1), local coordinates (z′ +1, . . . , z′ +n) on X′ such that g∗z1 = (z′ +1)e, g∗z2 = z′ +2, +etc., where e is the ramification index of g along D. Let d be the order of the pole +of ω along D; write ω = α/zd +1 + β ∧ dz1/zd +1, where α, β are regular forms which do +not involve dz1. Then +ω′ = g∗ω = g∗α/(z′ +1)de + e g∗β ∧ dz′ +1/(z′ +1)1+(d−1)e. +Assume, by contradiction, that d ⩾ 2, so that de ⩾ 2 and 1 + (d − 1)e ⩾ 2. Since ω′ +has at most logarithmic poles along D, we get g∗α = 0 and g∗β = 0. This implies +that both α and β are multiples of z1, contradicting the hypothesis that d was the +order of the pole of ω along D. Therefore, d ⩽ 1. This concludes the proof. +2.5. — We say that an m-form ω ∈ Ωm +K/k is logarithmic if for all proper smooth +models X of K such that the polar divisor of ωX has normal crossings, the meromor- +phic differential form ωX has at most logarithmic poles. By resolution of singularities + +6 +ANTOINE CHAMBERT-LOIR, MAXIM KONTSEVICH & YURI TSCHINKEL +(Hironaka, 1964), two models are dominated by a third one, hence lemma 2.4, im- +plies that it suffices that this condition is satisfied on some proper smooth model +for which the polar divisor of ωX has normal crossings. +Analogously, if X is a reduced k-variety, then we say that a meromorphic m-form ω +on X is logarithmic “everywhere” if for all proper birational models (X′, ω′) of (X, ω), +the meromorphic m-form ω′ on X′ has at most logarithmic poles. It suffices that +this holds on one such model. +3. Burnside rings for logarithmic forms +3.1. Burnside rings. — Let k be a field of characteristic zero and n an integer +such that n ⩾ 0. Kontsevich & Tschinkel (2019) defined the Burnside group +Burnn(k) as the free abelian group on isomorphism classes of finitely generated +extensions of k of transcendence degree n. +Any integral k-variety X of dimension n has a class [X] in Burnn(k). This gives rise +to alternative useful presentations of Burnn(k), for example involving only classes +of integral projective smooth varieties. +The group +Burn(k) = +� +n≥0 +Burnn(k) +carries a natural commutative ring structure, with multiplication defined by taking +products of (smooth projective) k-varieties: +[X] · [X′] = [X × X′]. +3.2. Definition of a Burnside group for volume forms. — Let k be a field +of characteristic zero and let n be an integer ⩾ 0. We define Burnn(k) to be the +free abelian group on isomorphisms classes of pairs (K, ω), where +– K is a finitely generated extension of k of transcendence degree n and +– ω ∈ Ωn +K/k is a logarithmic volume form. +We write +[K, ω] ∈ Burnn(k) +for the class of a pair (K, ω). +Remark 3.3. — This definition has obvious more geometric formulations. +For +example, we can take for generators equivalence classes of pairs (X, ω), where +– X is a smooth integral k-scheme of dimension n, and +– ω a regular volume form on X which is logarithmic “everywhere”, +modulo the smallest equivalence relation that identifies (X, ω) and (X′, ω′) if there +exists an open immersion f : X′ → X such that ω′ = f ∗ω. +Alternatively, we can assume that X is proper, smooth and integral, the form ω is +a logarithmic volume form on X, and consider the smallest equivalence relation that +identifies (X, ω) and (X′, ω′) if there exists a proper birational morphism f : X′ → X +such that ω′ = f ∗ω. By the weak factorization theorem of (Abramovich et al, + +BURNSIDE RINGS AND VOLUME FORMS WITH LOGARITHMIC POLES +7 +2002), this equivalence relation is generated by such morphisms f which are blowing- +ups along smooth centers in good position with respect to the polar divisor of X. +In both contexts, if X is an n-dimensional k-variety and ω is a meromorphic n-form +on X which is logarithmic “everywhere”, then we define [X, ω] to be the sum, over +all irreducible components Y of X which have dimension n, of the classes [Y, ω|Y]. +Example 3.4. — Finitely generated extensions of k of transcendence degree 0 are +finite extensions of k. Let K be such an extension. Since k has characteristic zero, +one has Ω1 +K/k = 0. However, Ω0 +K/k, which is its 0th exterior power, is canonically +isomorphic to K. Consequently, Burn0(k) is the free abelian group on isomorphism +classes of pairs (K, λ), where K is a finite extension of k and λ ∈ K. +We will let 1 = [Spec(k), 1] and ε = [Spec(k), −1]. +Example 3.5. — Let K = k(t). The differential form dt/t is a logarithmic volume +form; indeed X = P1 +k is a model of K and this form has poles of order 1 at 0 and ∞, +and no other poles. We write T for the class of (k(t), dt/t). +Note that the k-isomorphism of K that maps t to 1/t maps dt/t to its opposite; +consequently, we also have T = [k(t), −dt/t] = ε · T. +In the context of birational geometry in presence of logarithmic volume forms, +“rational varieties” would have class in Tn, and similarly for stable birationality. +3.6. Multiplicative structure. — We view the direct sum +Burn(k) = +� +n∈N +Burnn(k) +as a graded abelian group. It is endowed with a multiplication such that +[X, ω] · [X′, ω′] = [X × X′, ω ∧ ω′] +when X, X′ are proper, smooth and integral and ω, resp. ω′ are logarithmic volume +forms on X, resp. X′, and Y ranges over the set of irreducible components of X×X′. +Let s: X′ × X → X × X′ be the isomorphism exchanging the two factors. One has +s∗(ω ∧ ω′) = (−1)nn′ω′ ∧ ω, +if n = dim(X), n′ = dim(X′), ω is a volume form on X and ω′ is a volume form +on X′. Consequently, +a · b = εnn′ · b · a +for a ∈ Burnn(k) and b ∈ Burnn′(k). In particular, classes in Burnn(k), for even n, +are central in Burn(k). +We remark that the element T ∈ Burn1(k) is central as well. Let indeed a ∈ +Burnn(k). If n is even, then a · T = T · a. Otherwise, we have a · T = ε · T · a, but +we have seen in example 3.5 that T = ε · T. As a consequence, a · T = T · a. +However, the ring Burn(k) is not commutative. Indeed, consider curves X, X′ +without automorphisms and no nonconstant morphism between them. Then the +switch is the only isomorphism from X′ × X to X × X′. Take nonzero logarithmic + +8 +ANTOINE CHAMBERT-LOIR, MAXIM KONTSEVICH & YURI TSCHINKEL +1-forms ω, ω′ on X, X′ respectively. The classes [X × X′, ω ∧ ω′] and [X′ × X, ω′ ∧ ω] +are then distinct. +3.7. Functoriality. — Let k′ be an extension of k. Then there is a natural ring +homomorphism +Burn(k) → Burn(k′) +described as follows. Let (X, ω) be an integral k-variety of dimension n equiped with +a logarithmic q-form. Let X′ = X ⊗k k′ be its base change to k′, and let ω′ be the +volume form on X′ deduced from ω by base change. Then the class of (X, ω) maps to +the sum of classes (Y, ω′|Y), where Y runs the (finite) set of irreducible components +of X′. +If k′ is a finite extension of k, we also have a trace map +Trk′/k : Burn(k′) → Burn(k) +obtained by averaging over a set of representatives of automorphisms of the Galois +closure of k′ over k modulo those preserving k′. +3.8. Relation with the classical Burnside group. — Forgetting the form ω +gives a ring morphism +π: Burn(k) → Burn(k). +On the other hand, if K is a finitely generated extension of k of transcendence +degree n, we can endow it with the zero n-form. The resulting map +̟: Burn(k) → Burn(k) +identifies Burn(k) with an ideal of Burn(k). One has π ◦ ̟ = id. +3.9. Variations on the theme. — The construction of the Burnside ring Burn(k) +admits several natural variants that are relevant in more specific contexts. Some of +them will be used in later sections. +3.9.1. A relative ring. — Let n be an integer. +For any k-scheme S, we define +Burnn(S/k) as the free abelian group on triples (X, ω, u) where X is an integral +smooth n-dimensional k-scheme, ω ∈ Ωn +X/k is a regular volume form which is loga- +rithmic “everywhere”, and u: X → S is a morphism, modulo the smallest equivalence +relation that identifies (X, ω, u) and (X′, ω′, u′) if there exists an open immersion +f : X′ → X such that ω′ = f ∗ω and u′ = u ◦ f. +Let h: S → T be a morphism of k-schemes. It induces a morphism of abelian +groups +h∗ : Burnn(S/k) → Burnn(T/k) +such that h∗([X, ω, u]) = [X, ω, h ◦ u] for any triple (X, ω, u) as above. +3.9.2. Pluriforms. — One can replace volume forms with volume r-pluriforms, that +is, elements of (Ωn +K/k)⊗r, for some given integer r. The corresponding logarithmic +condition requires that the pluriform has poles of order at most r on an adequate +model. Note that when r is even, the obtained ring is commmutative. + +BURNSIDE RINGS AND VOLUME FORMS WITH LOGARITHMIC POLES +9 +3.9.3. Forms up to scalars. — In the construction, we may wish to identify (K, ω) +and (K′, ω′) if there exists λ ∈ k×, resp. λ ∈ {±1}, and a k-isomorphism f : K → K′ +such that f ∗ω′ = λω. These variants also give rise to a commutative ring. +3.9.4. Group actions. — Let G be a profinite group scheme over k. One can also +consider pairs (K, ω), where the field K is endowed with an action of G leaving the +form ω invariant. The obtained ring will be denoted by BurnG(k). +4. Residues +4.1. Residue of a volume form. — Let X be an equidimensional smooth k- +variety of dimension n. +Let D be a smooth divisor on X. We denote by Ωm +X/k(log D) the sheaf of m-forms +on X with logarithmic poles along D, locally of the form η ∧d log f +η′, where η and +η′ are regular and f is a local equation of D. The residue map is the homomorphism +of OX-modules +ρD : Ωm +X/k(log D) → Ωm−1 +D/k , +characterized by the relation +ρD(η ∧ d log f + η′) = η|D +for every local sections η ∈ Ωm−1 +X/k and η′ ∈ Ωm +X/k, and any local generator f of the +ideal of D. +If ω is a logarithmic m-form on X, there is an open subset U of X such that +U ∩ D ̸= ∅ and such that ω|U belongs to Ωm +X/k(log D). Its residue ρD(ω|U) is then a +meromorphic section of Ωm−1 +D/k . +Lemma 4.2. — Let ω be a logarithmic differential form of degree m on X. Then +ρD(ω) is a logarithmic (m − 1)-form on D. +Proof. — We may assume that the sum of D and of the polar divisor of ω has strict +normal crossings. The assertion is then evident in local coordinates. +4.3. Blowing-ups and normal bundles. — Let Y be a smooth closed sub- +scheme of X. The blow-up BlY(X) of X along Y is a smooth k-variety. The blowing- +up morphism bY : BlY(X) → X is an isomorphism over the complement of Y. If Y +is nowhere dense and nonempty, then EY = b−1 +Y (Y) is a smooth divisor in BlY(X). +In general, EY = b−1 +Y (Y) identifies, as an Y-scheme, with the projectivization of +the normal bundle NY(X) of Y in X. +Let W be a closed smooth subscheme of X. Assume that W and Y are transversal. +Then the Zariski closure of b−1 +Y (W +(Y ∩ W)) is called the strict transform of W +in BlY(X). It identifies with BlY∩W(W). +Let now ω be a logarithmic m-form on X. +Then the form b∗ +Yω on BlY(X) is +logarithmic; assuming that Y is nonempty and nowhere dense, we can consider the +residue ρY(ω) of b∗ +Yω along EY. It is a logarithmic (n − 1)-form on P(NY(X )). + +10 +ANTOINE CHAMBERT-LOIR, MAXIM KONTSEVICH & YURI TSCHINKEL +Definition 4.4. — Let X be an irreducible proper smooth k-variety, let n be its +dimension and let ω ∈ Ωn +X/k be logarithmic volume form whose polar divisor D has +strict normal crossings. Let (Dα)α∈A be the family of its irreducible components; for +A ⊆ A , we let DA = � +α∈A Dα. We then define an element ρ(X, ω) in Burnn−1(X/k) +by the formula: +ρ(X, ω) = +� +∅̸=A⊆A +(−1)|A|−1ρDA(X, ω). +(In this formula and all similar ones below, it is always implicit that the terms +where DA = ∅ are omitted.) +4.5. Iterated residues. — We retain the notation of definition 4.4 +Fix a logarithmic volume form ω on X and a nonempty subset A of A such +that DA ̸= ∅. It will be useful to compute inductively the logarithmic volume form +ρDA(ω) that appears in definition 4.4. +Let bA : ˜X → X be the blowing-up of X along DA and let E be its exceptional +divisor. +When A = {α} has a single element, DA is the divisor Dα, the blowing-up mor- +phism bA is an isomorphism and the exceptional divisor identifies with DA. Then +ρDA(X, ω) = [Dα, ρDα(ω), jα], +where jα is the immersion of Dα into X. +This construction can be pursued in higher codimension, using iterated residues. +Fix a total order on A . There is a unique, strictly increasing sequence (α1, . . . , αm) +in A such that A = {α1, . . . , αm}. Given the chosen order on A , we may apply the +iterated residues construction and obtain a logarithmic form of degree n − m +ρDA(ω) = ρDα1 ◦ · · · ◦ ρDαm(ω). +On a nonempty open subset U of X that meets DA, we may write +ω = η ∧ dlog(fα1) ∧ . . . dlog(fαm), +for a regular form η, and then one has ρDA(ω) = η|U∩DA. +Denote by bA the blowing-up of X along DA and by EA its exceptional divisor; +recall that EA identifies with the projectivized normal bundle NDA(X) of DA in X. +Using local equations for the divisors Dα, for α ∈ A, we trivialize NDA(X) on a dense +open subscheme of DA; this gives a birational isomorphism of EA with DA × Pm−1 +(with m = |A|), and a local computation gives the formula +ρDA(X, ω) = [DA, ρDA(ω)] · Tm−1 +in Burnn−1(DA/k). +When m ⩾ 2, the definition of ρDA actually depends on the chosen order of A , but +only up to a sign, so that the class [DA, ρDA(ω)] is well defined up to multiplication +by the class ε ∈ Burn0(k). On the other hand, it is multiplied by Tm−1 and we +have observed that ε · T = T. + +BURNSIDE RINGS AND VOLUME FORMS WITH LOGARITHMIC POLES +11 +Proposition 4.6. — Let (X, ω), D, and (Dα)α∈A be as in definition 4.4. Let Y be +a strict irreducible subvariety of X; let AY be the set of all α ∈ A such that Y ̸⊆ Dα; +we assume that � +α∈AY Dα meets Y transversally. +Let g : X′ → X be the blowing-up of X along Y and let ω′ = g∗ω; it is a logarithmic +form, its polar divisor has strict normal crossings, and we have +g∗ρ(X′, ω′) = ρ(X, ω) +in Burn(X/k). +Proof. — Let E = g−1(Y) be the exceptional divisor; for each α ∈ A , let D′ +α be the +strict transform of Dα. The blow-up X′ is smooth; the divisor E + � +α∈A D′ +α has +strict normal crossings and contains the polar divisor of ω′. +Let B be the set of all β ∈ A such that Y ⊆ Dβ, so that DB is the minimal +stratum containing Y. +We now split the discussion into two cases. +(1) Assume that dim(Y) < dim(DB). Since g is ramified along E, its Jacobian +vanishes along E. Since ω has poles of order at most one, the form ω′ = g∗ω is +regular at the generic point of E. Consequently, the polar divisor of ω′ does not +contain E and we have to compare +� +∅̸=A⊆A +(−1)|A|−1ρD′ +A(ω′) +with +� +∅̸=A⊆A +(−1)|A|−1ρDA(ω). +Since g is a local isomorphism around the generic points of Dα, for α ∈ A , we +see that the polar divisor of ω′ is equal to � +α∈A D′ +α. For every nonempty subset A +of A , one has +g∗ρDA(X′, ω′) = ρDA(X, ω) +for every nonempty subset A of A , which implies the desired formula in this case. +(2) Assume that dim(Y) = dim(DB). In this case, Y is an irreducible component +of DB. Since D∅ = X and Y ̸= X, we have B ̸= ∅. We have to compare the expression +� +∅̸=A⊆A +(−1)|A|−1ρD′ +A(ω′) + +� +A⊆A +(−1)|A|ρE∩D′ +A(ω′) +with +� +∅̸=A⊆A +(−1)|A|−1ρDA(ω). +The argument takes place in a neighborhood of Y, which allows us to assume that +Y = DB. +Let A be a nonempty subset of A . One has D′ +A = ∅ whenever B ⊆ A, and the +corresponding terms are absent from the second expression. On the other hand, +if B ̸⊆ A, the morphism g identifies D′ +A with the blow-up of DA along DA ∩ Y = +DA∪B. In particular, g induces a birational isomorphism from D′ +A to DA, so that + +12 +ANTOINE CHAMBERT-LOIR, MAXIM KONTSEVICH & YURI TSCHINKEL +g∗ρD′ +A(X′, ω′) = ρDA(X, ω). Moreover, E ∩ D′ +A is the projectivized normal bundle +PNDA∪B(DA), and +g∗ρE∩D′ +A(X′, ω′) = ρDA∪B(X, ω). +Similarly, one has +g∗ρE(X′, ω′) = ρDB(X, ω). +This gives a formula of the form +g∗ρ(X′, ω′) = +� +∅̸=A⊆A +B̸⊆A +(−1)|A|−1ρDA(X, ω) + +� +A⊆A +B̸⊆A +(−1)|A|ρDA∪B(X, ω) += +� +∅̸=A⊆A +n′ +AρDA(X, ω), +where +n′ +A = + + + +(−1)|A|−1 +if B ̸⊆ A, +� +C⊆A +B̸⊆C +C∪B=A +(−1)|C| +if B ⊆ A. +It suffices to prove that n′ +A = nA for any nonempty subset A of A . This is obvious +when B ̸⊆ A, so let us assume that B ⊆ A. In the sum that defines n′ +A, we write +C = (C +B)∪C′, where C′ = C∩B is a subset of B; the condition C∪B = A means +C +B = A +B; the condition B ̸⊆ C means C′ ̸= B. Consequently, we have +n′ +A = (−1)|A B| � +C′⊆B +C′̸=B +(−1)|C′| += (−1)|A B| +� � +C′⊆B +(−1)|C′| − (−1)|B| +� += (−1)|A B| � +(1 − 1)|B| − (−1)|B|� += (−1)|A|−1, +since |B| ⩾ 1. This concludes the proof of the proposition. +Theorem 4.7. — Let (X, ω) be as in definition 4.4. If X is proper, then the image +of ρ(X, ω) in Burnn−1(k) only depends on the class [X, ω] ∈ Burnn(k). It gives rise +to a morphism of abelian groups +∂n : Burnn(k) → Burnn−1(k). +Proof. — By the definition of Burnn(k) involving pairs (X, ω) where X is proper, +it suffices to consider two pairs (X, ω) and (X′, ω′) as in definition 4.4 which are +related by a proper birational morphism g : X′ → X such that g∗ω = ω′. By the +weak factorization theorem of Abramovich et al (2002), in order to prove the +theorem, we may assume that g is a blowing-up of X along a smooth subvariety +which is transversal to the polar divisor of ω. In this case, proposition 4.6 asserts + +BURNSIDE RINGS AND VOLUME FORMS WITH LOGARITHMIC POLES +13 +that g∗ρ(X′, ω′) = ρ(X, ω) in Burn(X/k). In particular, the images in Burn(k) of +ρ(X′, ω′) and ρ(X, ω) are equal. +Example 4.8. — The meromorphic differential form dt/t on P1 +k has residues 1 +and −1 at 0 and ∞ respectively. By construction, we thus have +∂1(T) = [Spec(k), 1] + [Spec(k), −1] = 1 + ε. +Let n be an integer such that n ⩾ 2 and let us compute ∂n(Tn). We view Tn as +the class of Pn, with homogeneous coordinates [1 : x1 : . . . : xn], and with the toric +differential form +ωn = (dx1/x1) ∧ . . . (dxn/xn). +Its divisor is the sum of the toric hyperplanes D0, . . . , Dn. Each of these hyperplanes +identifies with Pn−1, and ρDj(ωn) is (−1)n−jωn−1. Let A = {0, . . . , n}. If A = A , +then DA = ∅. Otherwise, we see by induction that DA is isomorphic to Pn−|A| and +ρDA(ωn) identifies with ±ωn−|A|, so that +[DA, ρDA(ωn)] · T|A|−1 = [Gm +n−1, ±ωn−1] = Tn−1, +since n − 1 ⩾ 1. Then, +∂n(Tn) = +� +∅̸=A⊆A +(−1)|A|−1[DA, ρDA(ωn)] · T|A|−1 += +� +∅̸=A⊊A +(−1)|A|−1Tn−1. +Now, +� +∅̸=A⊊A +(−1)|A|−1 = 1 − (1 − 1)n+1 + (−1)n+1 = +� +2 +if n is odd; +0 +if n is even. +We get ∂n(Tn) = 2Tn−1 if n is odd and ∂n(Tn) = 0 if n is even. (Remind that +n ⩾ 2.) Since T = ε · T, the following formula unifies the various cases: for n ⩾ 1, +we have +∂n(Tn) = (1 + (−1)n−1ε) · Tn−1. +Proposition 4.9. — For every class b ∈ Burnn(k), we have +∂n+1(b · T) = −∂n(b) · T + b · ∂1T. +Proof. — We may assume that b = [X, ω], where X is a proper integral smooth +variety of dimension n, and ω is a logarithmic volume form on X whose polar divisor +has strict normal crossings. Let (Dα)α∈A be the family of its irreducible components. +We view b·T as the class of [X×P1, ω ∧dt/t]. The polar divisor of ω ∧dt/t is equal +to +� +α∈A +Dα × P1 + X × {0} + X × {∞}. + +14 +ANTOINE CHAMBERT-LOIR, MAXIM KONTSEVICH & YURI TSCHINKEL +It has strict normal crossings, and its strata are of the form DA × P1, for nonempty +A ⊆ A , or DA × {0}, or DA × {∞}, for A ⊆ A . This decomposes ∂n+1(b × T) as +the sum of three terms. +The first one is +� +∅̸=A⊆A +[DA × P1, ρDA×P1(ω ∧ dt/t)] · T|A|−1. +For any nonempty subset A of A , one has +ρDA×P1(ω ∧ dt/t) = ±ρDA(ω) ∧ dt/t, +so that +[DA × P1, ρDA×P1(ω ∧ dt/t)] · T|A|−1 = [DA, ρDA(ω)] · T · T|A|−1. +Consequently, the first term equals ∂n(b) × T. +Write D0 = X × {0} and D∞ = X × {∞}, and identify both divisors to X. For a +subset A of A , we have +ρDA∪{0}(ω ∧ dt/t) = ρDA ◦ ρD0(ω ∧ dt/t) = ρDA(ω). +Consequently, the second term is equal to +� +A⊆A +(−1)|A|[DA, ρDA(ω)] · T|A| = [X, ω] − ∂n(b) · T. +Similarly, the third term is equal to +[X, −ω] − ∂n(b) · T. +Summing up these three terms, we get +∂n+1(b × T) = −∂n(b) · T + [X, ω] + [X, −ω]. +We now recall that ∂1(T) = [Spec(k), 1] + [Spec(k), −1], so that +[X, ω] + [X, −ω] = [X, ω] · ∂1(T) = b · ∂1(T). +This concludes the proof. +Theorem 4.10. — Let a ∈ Burnm(k) and b ∈ Burnn(k); we have +∂m+n(a · b) = εn · ∂m(a) · b + a · ∂n(b) − T · ∂m(a) · ∂n(b) +in Burnm+n−1(k). +Proof. — It suffices to treat the case where a and b are classes of proper integral +smooth varieties (X, ω), (Y, η), endowed with meromorphic volume forms whose +polar divisors have strict normal crossings and no multiplicities. Let (Dα)α∈A be +the irreducible components of the polar divisor of ω, let (Eβ)β∈B be the irreducible +components of the polar divisor of η. Then [X, ω]·[Y, η] is the class of [X×Y, ω ∧η]; +the polar divisor of ω ∧ η is equal to +� +α∈A +Dα × Y + +� +β∈B +X × Eβ. + +BURNSIDE RINGS AND VOLUME FORMS WITH LOGARITHMIC POLES +15 +We fix a total order on the disjoint union of A and B such that the elements of A +are smaller than those of B. For any subsets A, B of A and B, observe that we +have +ρDA∪B(ω ∧ η) = ±ρDA(ω) ∧ ρEB(η), +where ρDA has to be understood as the identity when A is empty, and similarly +for ρEB. The sign is 1 when A = ∅; when B = ∅, it is equal to (−1)|A|n; we won’t +need to use its explicit value in the other cases. Then we can write ∂([X, ω] · [Y, η]) +as +� +A⊆A +B⊆B +A∪B̸=∅ +(−1)|A|+|B|−1[DA × EB, ±ρA(ω) ∧ ρEB(η)] · T|A∪B|−1 +and we split it into the sum of three terms, according to which B = ∅, or A = ∅, or +none of them is empty. The first two terms are respectively equal to +� +∅̸=A⊆A +(−1)|A|−1[DA × Y, (−1)n|A|ρDA(ω) ∧ η] · T|A|−1 = ∂([X, (−1)nω]) · [Y, η] +and +� +∅̸=B⊆B +(−1)|B|−1[X × EB, ω ∧ ρEB(η)] · T|B|−1 = [X, ω] · ∂([Y, η]), +since T belongs to the center of Burn(k). As for the third one, we obtain +− +� +∅̸=B⊆B +(−1)|B|−1 +� +∅̸=A⊆A +(−1)|A|−1[DA, ρDA(ω)] · [EB, ρEB(η)] · T|A|+|B|−2 +which equals +−∂([X, ω]) · ∂([Y, η]) · T. +Finally, we get +∂m+n(a · b) = ∂m+n([X, ω] · [Y, η]) += ∂m([X, (−1)nω) · [Y, η] + [X, ω] · ∂n([Y, η]) +− T · ∂m([X, ω]) · ∂n([Y, η]) += εn · ∂m(a) · b + a · ∂n(b) − T · ∂m(a) · ∂n(b) +as was to be shown. +In particular, using the computation of example 4.8, we obtain the following +generalization of proposition 4.9. +Corollary 4.11. — For any a ∈ Burnm(k) and any integer n, we have +∂m+n(a · Tn) = +� +∂m(a) · Tn +if n is even; +−∂m(a) · Tn + a · ∂n(Tn) +if n is odd. +Remark 4.12. — For the variant of Burn(k) where we consider forms up to sign, +the formula of theorem 4.10 simplifies to +∂m+n(a · b) = ∂m(a) · b + a · ∂n(b) − T · ∂m(a) · ∂n(b). + +16 +ANTOINE CHAMBERT-LOIR, MAXIM KONTSEVICH & YURI TSCHINKEL +5. A complex of Burnside rings +Theorem 5.1. — For any integer n ⩾ 2, we have +∂n−1 ◦ ∂n = 0. +In other words, the residue morphisms of Burnside groups give rise to a complex +· · · → Burnn(k) → Burnn−1(k) → · · · → Burn1(k) → Burn0(k) +Proof. — It suffices to prove the following result: +Let (X, ω) be an integral +proper smooth variety of dimension n equipped with a meromorphic volume +form ω whose polar divisor has strict normal crossings and no multiplicities; then +∂n−1(∂n([X, ω])) = 0. +Let (Dα)α∈A be the family of irreducible components of the polar divisor of ω +in X. By definition, one has +∂n([X, ω]) = +� +∅̸=A⊆A +(−1)|A|−1ρDA(X, ω). +Fix a total order on A . Let (α1, . . . , αm) be a strictly increasing sequence in A +and let A = {α1, . . . , αm}. We have seen in §4.5 that ρDA(X, ω) can be defined via +iterated residue maps: +ρDA([X, ω]) = [DA, ρDα1 ◦ · · · ◦ ρDαm(ω)] · T|A|−1 = [DA, ωA] · T|A|−1 +where we wrote ωA for the composition ρDα1 ◦ · · · ◦ ρDαm(ω). When |A| is odd, we +have +∂(ρDA([X, ω])) = ∂([DA, ωA]) · T|A|−1, +while when |A| is even, we have +∂(ρDA([X, ω])) = −∂([DA, ωA]) · Ta−1 + [DA, ωA] · ∂(T|A|−1). +Consequently, we have +∂ ◦ ∂([X, ω]) = +� +∅̸=A⊆A +∂([DA, ωA]) · T|A|−1 − +� +∅̸=A⊆A +|A| even +[DA, ωA] · ∂(T|A|−1). +The polar divisor of the form ωA on DA is equal to � +β̸∈A Dβ ∩ DA, so that, by +definition (and computation of ∂ via iterated residues), +∂([DA, ωA]) = +� +∅̸=B⊆∁A +(−1)|B|−1[DAB, ωA∪B] · T|B|−1. +Also, when A is nonempty and of even cardinality, ∂(T|A|−1) = 2T|A|−2. When we +put these two formulas into the antepenultimate one and collect the various terms, +we obtain +∂ ◦ ∂([X, ω]) = +� +C⊆A +2⩽|C| +nC[DC, ωC] · T|C|−2, + +BURNSIDE RINGS AND VOLUME FORMS WITH LOGARITHMIC POLES +17 +where +nC = − +� +∅̸=A,B +A∪B=C,A∩B=∅ +(−1)|B| − 2δ|C| is even. +In the first sum, the terms A = ∅ or B = ∅ are omitted, while if we put them in, we +obtain +� +A∪B=C +A∩B=∅ +(−1)|B| = +|C| +� +b=0 +�|C| +b +� +(−1)b = (1 − 1)|C| = 0 +since |C| ⩾ 1. Consequently, +nC = 1 + (−1)|C| − 2δ|C| is even = 0. +This concludes the proof. +6. Algebraic structure of Burn(k) after localization at 2 +In this section, we study the algebraic structure of the Burnside ring Burn(k), +endowed with its elements ε, T and the operator ∂. +6.1. — By construction, Burn(k) = � +n⩾0 Burnn(k) is an associative unital Z⩾0- +graded ring, ε ∈ Burn0(k), T ∈ Burn1(k) and ∂ is a homogeneous additive map +of degree −1. They satisfy the following relations, for homogeneous elements a, b ∈ +Burn(k): +b · a = ε|a||b| · a · b +(§3.6); +(1) +ε2 = 1 +(example 3.4); +(2) +T = ε · T +(example 3.5); +(3) +∂(T) = 1 + ε +(example 4.8); +(4) +∂(a · b) = ε|b| · ∂(a) · b + a · ∂(b) − T · ∂(a) · ∂(b) +(theorem 4.10); +(5) +∂(∂(a)) = 0 +(theorem 5.1). +(6) +By (1), the element ε is central, and by (2), we may view Burn(k) as an algebra over +Z[ε]/(ε2 − 1). After inverting 2, the algebra Burn(k) splits into two components +Burnε=1(k) and Burnε=−1(k), one over which ε = 1, and the other over which +ε = −1. +In the rest of this section, we implicitly assume that 2 is inverted, without changing +the notation. + +18 +ANTOINE CHAMBERT-LOIR, MAXIM KONTSEVICH & YURI TSCHINKEL +6.2. Sector ε = −1. — Here, we have T = −T, hence T = 0 since 2 is invertible. +As a consequence, after replacing ∂ with ∂′ : a �→ (−1)1+|a|∂(a), one gets from (5) +the usual graded Leibniz rule +∂′(a · b) = ∂′(a) · b + (−1)|a|a · ∂′(b) +and therefore Burnε=−1(k) is a classical differential graded (super-)commutative +algebra, similar to, eg, the de Rham complex. +6.3. Sector ε = 1. — The algebra Burnε=1 is now commutative (and not graded +commutative). This reflects the intuition in our constructions that they speak about +volume forms (as opposed to top-degree differential forms) for which we have com- +mutativity (as reflected by the change of order of integration in multiple integrals). +Lemma 6.4. — The map F: a �→ a − T · ∂(a) is a ring endomorphism of +Burnε=1(k), and F2 = id. Moreover, one has F ◦ ∂ = ∂ = −∂ ◦ F. +Proof. — This map is additive. One has F(1) = 1 − T · ∂(1) = 1. Let us show +multiplicativity. Indeed, for a, b ∈ Burnε=1(k), one has +F(a) · F(b) = (a − T · ∂(a)) · (b − T · ∂(b)) += a · b − T · ∂(a) · b − T · a · ∂(b) + T2 · ∂(a) · ∂(b) += a · b − T · (∂(a) · b + a · ∂(b) − T · ∂(a) · ∂(b)) += a · b − T · ∂(a · b) +(using (5)) += F(a · b). +Since ∂2 = 0, one has +F(∂(a)) = ∂(a) − T · ∂(∂(a)) = ∂(a). +On the other hand, +∂(F(a)) = ∂(a − T · ∂(a)) += ∂(a) − ∂(T · ∂(a)) += ∂(a) − ∂(T) · ∂(a) − T · ∂(∂(a)) + T · ∂(T) · ∂(∂(a)) += −∂(a) +using that ∂(T) = 2 and ∂2 = 0. +Consequently, for a ∈ Burnε=1(k), we have +F2(a) = F(a) − T · ∂(F(a)) = a − T · ∂(a) + T · ∂(a) = a +since ∂ ◦ F = −∂. + +BURNSIDE RINGS AND VOLUME FORMS WITH LOGARITHMIC POLES +19 +6.5. — To simplify the notation, write B = Burnε=1(k). Since F2 = id and 2 is +invertible, the algebra B splits as a direct sum +B = B+ ⊕ B−, +such that F acts as id on B+ and as − id on B−. Moreover, B+ is a subalgebra. +Since the operator ∂ anticommutes with F, it induces maps +∂± : B+ → B−, +∂∓ : B− → B+. +Note that +F(T) = T − T · ∂(T) = −T, +so that T ∈ B−. Consequently, the multiplication by T map induces two maps +t± : B+ → B−, +t∓ : B− → B+. +Lemma 6.6. — The map ∂ vanishes on B+. Equivalently, ∂± = 0. +The maps 1 +2∂∓ and t± are inverses the one of the other. +Proof. — For a ∈ B+, one has ∂(a) = −∂(F(a)) = −∂(a), since ∂ ◦ F = −∂, hence +∂(a) = 0. +On the other hand, for a ∈ B+, one has +∂(T · a) = 2 · a + T · ∂(a) − 2T · ∂(a) = 2a − T · ∂(a) = a + F(a) = 2a +while for a ∈ B−, we have +T · ∂(a) = a − F(a) = 2a. +This concludes the proof of the lemma. +In particular, we see that the cohomology of the differential ∂ vanishes in the +sector Burnε=1(k) = B. +6.7. — It follows from the lemma that we have a ring isomorphism +B = B+[t](t2 − T2), +from which we see that all the algebraic structure of B+ (namely δ, T, F) can be +canonically reconstructed from a unital commutative associative Z⩾0-graded ring B+ +endowed with an element in degree +2 (namely, the element T2). +Remark 6.8. — The situation clarifies even more if we invert the class T. Then +we can write ∂(a) = (a − F(a))/T, and all relations happen to follow from the fact +that F is an involution such that F(T) = −1. Indeed, +∂2(a) = ∂(a) − F(∂(a)) +T += 1 +T +�a − ∂(a) +T +− F(a − ∂(a) +T +� += 0 + +20 +ANTOINE CHAMBERT-LOIR, MAXIM KONTSEVICH & YURI TSCHINKEL +explains that ∂2 = 0. Moreover, for a, b ∈ B, we have +∂(a · b) = a · b − F(a · b) +T += a · b − F(a) · F(b) +T += a − F(a) +T +· b + a · b − F(b) +T +− T · a − F(a) +T +· b − F(b) +T += ∂(a) · b + a · ∂(b) − T · ∂(a) · ∂(b). +7. Birational morphisms preserving volume forms +7.1. — Let (X, ωX) be a smooth integral k-variety of dimension n equipped with +a meromorphic form with poles of order at most one on X and let f : Y → X be a +proper birational morphism. +Let E be an exceptional divisor in Y, that is, such that dim(f(E)) < dim(E). +Then the Jacobian of p vanishes along E. Since ω has poles of order at most one +on X, the meromorphic form f ∗ω on Y is regular at the generic point of E; its +restriction to E is a meromorphic form with poles of order at most one and we may +consider the class [E, f ∗ωX|E] in Burnn−1(k). +We define c(f; X, ω) to be the sum of all such classes [E, f ∗ω|E] in the free abelian +group Burnn−1(k). +Lemma 7.2. — Let g : Z → Y be a proper birational morphism of smooth integral +varieties of dimension n. Then g ◦ f is a proper birational morphism and one has +c(g ◦ f; X, ω) = c(g; Y, f ∗ω) + c(f; X, ω) +in Burnn−1(k). +Proof. — An integral divisor F in Z is exceptional for g ◦ f if and only if one of the +two mutually excluding situations happen: +– The divisor F is exceptional for g; +– Or g(F) is a divisor in Y which is exceptional for f. +Moreover, any divisor E in Y which is exceptional for f appears once and only as +a divisor of the form g(F). In the first case, F contributes to c(g ◦ f; X, ω) by a +term [F, (g ◦ f)∗ω|F], and it contributes to c(g; Y, f ∗ω) by precisely the same term. +In the second case, F contributes to c(g ◦ f; X, ω) by a term [F, g∗(f ∗ω|E)], while E +contributes to c(f; X, ω) by the term [E, f ∗ω|E]. Since g is birational around the +generic point of E, they coincide, and this concludes the proof. +7.3. — Let (X, ωX) and (Y, ωY) be proper smooth k-varieties equipped with loga- +rithmic volume forms and let +ϕ: (X, ωX) ��� (Y, ωY) + +BURNSIDE RINGS AND VOLUME FORMS WITH LOGARITHMIC POLES +21 +be a birational map preserving the volume forms. By definition, this means that +there exists a diagram +W +X +Y +← +→ +p +← +→ +q +← +→ +ϕ +of integral k-varieties such that that p and q are proper and birational, and such +that p∗ω = q∗ω′ on W. In this situation, we may assume that W is smooth. +Lemma 7.4. — With this notation, the element +c(ϕ) = c(q) − c(p) ∈ Burnn−1(k) +only depends on the birational map ϕ, and not on the choice of the triple (W, p, q). +Proof. — Consider two possible diagrams X +p←− V +q−→ Y and X +r←− W +s−→ Y describ- +ing ϕ. Considering for example a resolution of singularities U of V ×X W, we can fit +these two diagrams in a common commutative diagram of the following form: +U +V +W +X +Y +← +→ +u +← +→ +v +← +→ +p +← +→ +q +← +→ +r +← +→ +s +← +→ +ϕ +The equalities p∗ωX = q∗ωY and r∗ωX = s∗ωY imply that +(p ◦ u)∗ωX = u∗p∗ωX = u∗q∗ωY = (q ◦ u)∗ωY = (s ◦ v)∗ωY. +By lemma 7.2, we then have +c(p) − c(q) = c(p ◦ u) − c(q ◦ u) = c(r ◦ v) − c(s ◦ v) = c(r) − c(s). +This concludes the proof. +Theorem 7.5. — If ψ: (Y, ωY) ��� (Z, ωZ) is another birational map preserving +volume forms, then one has +c(ψ ◦ ϕ) = c(ψ) + c(ψ). +Proof. — Consider two diagrams X +p←− V +q−→ Y and Y +r←− W +s−→ Y describing ϕ +and ϕ. Considering for example a resolution of singularities U of V ×Y W, we can + +22 +ANTOINE CHAMBERT-LOIR, MAXIM KONTSEVICH & YURI TSCHINKEL +fit these two diagrams in a common commutative diagram of the following form: +U +V +W +X +Y +Z +← +→ +u +← +→ +v +← +→ +p +← +→ +q +← +→ +r +← +→ +s +← +→ +ϕ +← +→ +ψ +and the diagram X +p◦u +←−− U +s◦v +−−→ describes the birational map ψ◦ϕ. Since q◦u = r◦v, +we then have +c(ψ ◦ ϕ) = c(p ◦ u) − c(s ◦ v) += c(p ◦ u) − c(q ◦ u) + c(r ◦ v) − c(s ◦ v) += c(p) − c(q) + +c(r) − c(s) += c(ϕ) + c(ψ), +as was to be shown. +Corollary 7.6. — Let Bir(X, ω) be the set of birational automorphisms of X pre- +serving ω. The map c induces a homomorphism of abelian groups +Bir(X, ω) → Burnn−1(k). +Its kernel contains the group of automorphisms of X that preserve ω. +8. Specialization +Let K be the field of fractions of a discrete valuation ring R with residue field k. +Fix a uniformizer t ∈ R. +In this context, Kontsevich & Tschinkel (2019) have defined two (distinct) +specialization morphisms +ρt : Burnn(K) → Burnn(k), +relating the Burnside groups of K and k (see 3.1), one of which is a ring homo- +morphism. +(The latter homomorphism actually depends on the choice of t, see +example 6.2 of (Kresch & Tschinkel, 2022b).) +The goal of this section is to define a similar homomorphism +ρt : Burn(K) → Burn(k) +for varieties with logarithmic volume forms. + +BURNSIDE RINGS AND VOLUME FORMS WITH LOGARITHMIC POLES +23 +8.1. — Let X be an integral proper scheme over R, of relative dimension n, whose +special fiber ∆ is a divisor with strict normal crossings. +Let (∆α)α∈A be the family of irreducible components of the special fiber ∆; for +α ∈ A , let eα be the multiplicity of ∆α in ∆. For every nonempty subset A of A , +let ∆A be the intersection of all divisors ∆α, for α ∈ A and eA be the greatest +common divisor of the eα, for α ∈ A; let also ∆◦ +A be the complement ∆A +� +α̸∈A ∆α. +The first specialization morphism of (Kontsevich & Tschinkel, 2019) is de- +fined by +(8.2) +ρt([XK]) = +� +∅̸=A⊆A +(−1)|A|−1[∆A]L|A|−1, +where L ∈ Burn(k) is the class of the affine line. +Although this map is not multiplicative, it proved sufficient for many applications +to rationality problems. +To ensure multiplicativity, a more delicate construction was necessary, valued in +the Burnside ring +Burn�µ(k) +of varieties endowed with an action of the profinite group �µ, limit of finite groups of +roots of unity. +Fix a nonempty subset A of A . We identify the normal bundle of ∆A in X as a +direct sum of line bundles: +N∆A(X ) ≃ +� +α∈A +N∆α(X )|∆A. +Let us consider its open subscheme N ◦ +∆A(X ) obtained by restricting to ∆◦ +A and +taking out all “coordinate” hyperplanes. This furnishes a morphism +νA : N ◦ +∆A(X ) → +� +α∈A +N∆α(X )⊗eα|∆◦ +A. +Since the uniformizer t has divisor − � +α∈A eα∆α on X , it trivializes the line bundle +on the target of νA. We set ∆′ +A = ν−1 +A (t). By construction, the projection ∆′ +A → ∆A +is a torsor with group µeA. +With this notation, the correct, multiplicative, specialization map of (Kontsevich & Tschinkel, +2019) is given by the formula +�ρt(X) = +� +∅̸=A⊆A +(−1)|A|−1[∆′ +A]L|A|−1 +in Burn�µ(k). +Remark 8.3. — The relation between the two specialization morphisms is as fol- +lows. Fix a nonempty subset A of A . The group Gm acts diagonally on N ◦ +∆A(X ) +(the factors of index α /∈ A don’t act), and this induces an action of the finite group +of roots of unity of order eA on ∆′ +A, hence an action of �µ, so that �ρt(X) naturally +lives in the equivariant Burnside ring Burn�µ(k). Moreover, taking the �µ-invariants + +24 +ANTOINE CHAMBERT-LOIR, MAXIM KONTSEVICH & YURI TSCHINKEL +of ∆′ +A, we get ∆◦ +A, so that the specialization map ρt is the composition of �ρt with +the map +Burn�µ(k) → Burn(k) +obtained by taking �µ-invariants. +Taking invariants does not commute with taking products, in general, so that ρt +is not multiplicative. +8.4. — Let us explain how to define analogous specialization homomorphisms in +our context of Burnside groups with volume forms. +For simplicity, we only consider the case where K has transcendence degree 1 +over k, in which case the idea can be explained geometrically as follows. We assume +that there exists an smooth integral curve C together with a k-point o ∈ C(k) such +that K = k(C) and R = OC,o. We fix a local parameter t ∈ R such that V(t) = o. +Let us consider a pair (X, ω) consisting of an integral proper K-variety X of di- +mension n and a logarithmic n-form ω on X. +8.5. — Consider a regular flat proper model X is of X over C, let ∆ = (Xo)red +be its reduced special fiber, and consider a divisor D with relative normal cross- +ings on X . +We assume that the divisor D + ∆ has normal crossings. +In this +situation, Deligne (1970, §3.3.2) says that a meromorphic relative differential m- +form on X /C is logarithmic with respect to D + ∆ if it is (locally) the image of +a logarithmic m-form ˜ω in Ωm +X /k with poles D + ∆ under the natural morphism +Ωm +X /k → Ωm +X /C. +Consider a logarithmic relative n-form ω on X /C. We consider an associated +volume form ω′ on X , defined locally by +ω′ = ˜ω ∧ dt/t, +where ˜ω is any local lift of ω. This form ω′ is logarithmic and we can compute its +“residue along ∆” as in §4, only taking into account the strata of the polar divisor +of ω′ which are contained in the special fiber ∆. +There exists a subset Ao of A and a subset Bo of B such that the polar divisor +of ω′ is given by +� +α∈Ao +∆α + +� +β∈Bo +Dβ. +We thus set +ρt(X , ω) = +� +∅̸=A⊆Ao +B⊆Bo +(−1)|A|+|B|−1ρ∆A∩DB(X , ω). +This is an element of Burnn(Xo/k). +Proposition 8.6. — Let Y be an irreducible closed subscheme of X which is +transverse to D + ∆ and let g : X ′ → X be the blowing-up of X along Y . The +form g∗ω on X ′ is logarithmic and we have +g∗ρt(X ′, g∗ω) = ρt(X , ω) + +BURNSIDE RINGS AND VOLUME FORMS WITH LOGARITHMIC POLES +25 +in Burnn(Xo/k). +Proof. — With the notation of §4, the difference +ρ(X , ˜ω) − ρt(X , ω) +is exactly the part of ρ(X , ˜ω) which lies over the complement of the special fiber Xo +in X . We have seen in theorem 4.7 that +g∗ρ(X ′, ˜ω′) = ρ(X , ω), +and a similar formula holds over X +Xo. This implies the proposition. +8.7. — Starting from a smooth proper K-variety X and a logarithmic volume +form ω on X, we can define a model X /C, with D and ∆ as above, but the form ω +will not necessarily extend to a logarithmic relative form with respect to D +∆, nor +does the volume form ˜ω on X . However, this can be achieved by multiplying ω by +a suitable power of the uniformizing element. +Let us write the polar divisor of ˜ω on X as +divX (˜ω) = D + ∆ = +� +α∈A +dα∆α + +� +β∈B +dβDβ. +With this notation, the condition for ˜ω to be logarithmic on X is just that +dα ⩾ −1, +dβ ⩾ −1. +In particular, while the conditions at the horizontal components follow from their +counterparts on the generic fiber, those for the vertical components are not auto- +matic. On the other hand, for any κ ∈ Z, the form tκ˜ω is logarithmic if and only +if +κeα + dα ⩾ −1 +for all α ∈ A , that is, if and only if κ ⩾ κ(ω), where κ(ω) is defined by +κ(ω) = inf +α∈A +1 − dα +eα +. +Since the rational number κ(ω) is defined in terms of logarithmic forms, it only +depends on the class of (X, ω) in Burnn(K), and not on the actual model which is +chosen to compute it. +8.8. — We assume for the moment that κ(ω) ∈ Z. This holds in particular if the +special fiber Xo is reduced. Let then Ao be the subset of A consisting of all α such +that +κ(ω)eα + dα = −1, +and let Bo be the subset of B consisting of all β such that dβ = −1. The polar +divisor of tκ˜ω is equal to +� +α∈Ao +∆α + +� +β∈Bo +Dβ, + +26 +ANTOINE CHAMBERT-LOIR, MAXIM KONTSEVICH & YURI TSCHINKEL +and we set +ρt(X , ω) = ρt(X , tκ(ω)ω) +in Burnn(Xo/k). +In the particular case where D is empty, the strata of the Clemens complex of +the special fiber that actually appear in the definition of this class are those defined +by Kontsevich & Soibelman (2006), more precisely, by the adjustment provided +by Mustaţă & Nicaise (2015). +8.9. — In the general case, the rational number κ(ω) is not an integer. Let us +consider the finite ramified extension Kd = K(t1/d) of K, whose ramification index d +is a multiple of the denominator of κ(ω), but which induces an isomorphism on +the residue field. Geometrically, this furnishes a morphism π: Cd → C which is +ramified at the point o, together with a lift of o in Cd(k) (still denoted by o), and a +distinguished uniformizing element t1/d. +We consider the extension of (X, ω) to Kd and introduce a model (Xd, ωd) as +above, over Cd. Now, the corresponding κ-parameter is integral, so that any choice +of a uniformizing element t1/d in Rd induces a class ρt1/d(Xd, ωd) in Burn(k). In +fact, we can assume that the scheme Xd carries an action of the group scheme µd of +dth roots of unity induced by its action on Spec(Rd), leaving the logarithmic form ωd +invariant. In other words, we obtain a class in the group Burn�µ(k). +Combinining these classes, we obtain the desired group homomorphism +�ρt : Burn(K) → Burn�µ(k). +In fact, as explained in (Nicaise, 2013, §2.3), especially proposition 2.3.2, one +can compute the normalisation of X ⊗ Rd in terms of the given model X . This +gives an explicit decomposition of �ρt(X, ω) as a sum +� +∅̸=A⊆Ao +(−1)|A|−1[D′ +A, ν∗ +AωA] · T|A|−1, +where νA : D′ +A → DA is the µdA-torsor introduced in §8.1 for the definition of the +classical specialization map. +Remark 8.10. — In the case of specialization of rationality, it has proved fruitful +to consider models with singularities on the special fiber, mild enough so that the +special fiber computes the specialization of the birational type of the generic fiber. +This is in particular the case for rational double points. +A parallel study can be developped in the context of varieties with logarithmic +forms. +Following (Kontsevich & Tschinkel, 2019) and keeping track of the various +logarithmic volume forms on the strata, we have: +Theorem 8.11. — The morphism �ρt is a ring homomorphism. + +BURNSIDE RINGS AND VOLUME FORMS WITH LOGARITHMIC POLES +27 +References +D. Abramovich, K. Karu, K. Matsuki & J. 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Nicaise (2020), “Convergence of p-adic pluricanonical measures +to Lebesgue measures on skeleta in Berkovich spaces”. Journal de l’École poly- +technique — Mathématiques, 7, pp. 287–336. +B. Khesin & A. Rosly (2003), “Polar Homology”. Canadian Journal of Mathe- +matics, 55 (5), pp. 1100–1120. +B. Khesin, A. Rosly & R. Thomas (2004), “A polar de Rham theorem”. Topology, +43 (5), pp. 1231–1246. +M. Kontsevich & Y. Soibelman (2006), “Affine structures and non-Archimedean +analytic spaces”. The Unity of Mathematics, Progr. Math. 244, pp. 321–385, +Birkhäuser Boston, Boston, MA. +M. Kontsevich & Y. Tschinkel (2019), “Specialization of birational types”. +Inventiones mathematicae, 217 (2), pp. 415–432. +A. Kresch & Y. Tschinkel (2022a), “Burnside groups and orbifold invariants of +birational maps”. arXiv:2208.05835. +A. Kresch & Y. Tschinkel (2022b), “Equivariant birational types and Burnside +volume”. Annali Scuola Normale Superiore - Classe Di Scienze, 23 (2), pp. 1013– +1052. +H.-Y. Lin & E. Shinder (2022), “Motivic invariants of birational maps”. +arXiv:2207.07389. +H.-Y. Lin, E. Shinder & S. Zimmermann (2020), “Factorization centers in di- +mension two and the Grothendieck ring of varieties”. arXiv:2012.04806. +M. Mustaţă & J. Nicaise (2015), “Weight functions on non-Archimedean analytic +spaces and the Kontsevich–Soibelman skeleton”. Algebraic Geometry, 2 (3), pp. +365–404. +J. Nicaise (2013), “Geometric criteria for tame ramification”. Mathematische +Zeitschrift, 273 (3-4), pp. 839–868. + +28 +ANTOINE CHAMBERT-LOIR, MAXIM KONTSEVICH & YURI TSCHINKEL +J. Nicaise & E. Shinder (2019), “The motivic nearby fiber and degeneration of +stable rationality”. Inventiones mathematicae, 217 (2), pp. 377–413. +J. Nicaise & C. Xu (2016), “The essential skeleton of a degeneration of algebraic +varieties”. American Journal of Mathematics, 138 (6), pp. 1645–1667. +M. Rost (1996), “Chow Groups with Coefficients”. Documenta Mathematica, +1 (16), pp. 319–393. + diff --git a/GtE1T4oBgHgl3EQfFQMH/content/tmp_files/load_file.txt b/GtE1T4oBgHgl3EQfFQMH/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c0d3e8801e8b40a55eb23bcbca454c05a182f3c0 --- /dev/null +++ b/GtE1T4oBgHgl3EQfFQMH/content/tmp_files/load_file.txt @@ -0,0 +1,1183 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf,len=1182 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='02899v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='AG] 7 Jan 2023 Burnside rings and volume forms with logarithmic poles Antoine Chambert-Loir Université Paris Cité, Institut de Mathématiques de Jussieu-Paris Rive Gauche, F-75013, Paris, France E-mail: antoine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='chambert-loir@u-paris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='fr Maxim Kontsevich Institut des Hautes Études Scientifiques, 35 route de Chartres, 91440 Bures-sur-Yvette, France E-mail: maxim@ihes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='fr Yuri Tschinkel Courant Institute, NYU, 251 Mercer St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' New York, NY 10012, USA Simons Foundation, 160 5th Av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=', New York, NY 10010, USA E-mail: tschinkel@cims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='nyu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='edu Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — We develop a theory of Burnside rings in the context of birational equivalences of algebraic varieties equipped with logarithmic volume forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' We introduce a residue homomorphism and construct an additive invariant of birational morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' We also define a specialization homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Résumé.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — Nous proposons une théorie d’anneaux de Burnside dans le contexte de la géométrie birationnelle des variétés algébriques munies d’une forme volume à pôles logarith- miques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Nous introduisons un homomorphisme « résidu », construisons un invariant additif des morphismes birationnels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Nous définissons aussi un homomorphisme de spécialisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' 2000 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — 14E08, 14E07, 14D06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — Birational geometry, Burnside rings, logarithmic volume forms, specialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Logarithmic differential forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' 27 2 ANTOINE CHAMBERT-LOIR, MAXIM KONTSEVICH & YURI TSCHINKEL 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Introduction The study of birationality of algebraic varieties is a classical and well studied subject, with many open problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' In some cases, it is interesting to study birational maps preserving additional structure, for example group actions, symplectic forms, or volume forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Such a study is already implicit in many questions of birational geometry, eg, in the notion of crepant resolution of singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' In this paper, we consider the case of varieties endowed with volume forms with logarithmic poles and develop a formalism of Burnside rings along the lines of their counterpart introduced by Kontsevich & Tschinkel (2019) to establish the spe- cialization of rationality, and its equivariant version by Kresch & Tschinkel (2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Let k be a field of characteristic zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' For each integer n, we define Burnn(k) as the free abelian group on birational equivalence classses of pairs (X, ω) consisting of an integral smooth proper k-variety X of dimension n equipped with an n-form ω with at most logarithmic poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' The graded abelian group Burn(k) = � n∈N Burnn(k) carries a ring structure, induced by taking products of varieties, decomposed into ir- reducible components, and equipped with the external product of the volume forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' In section 4, we define morphisms of abelian groups ∂ : Burnn(k) → Burnn−1(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' When X is smooth and the polar divisor of ω has strict normal crossings, the image of the class [X, ω] is given by the following formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Let (Dα)α∈A be the family of irreducible components of the polar divisor of ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' For each subset A of A , the intersection DA = � α∈A Dα is a union of integral smooth varieties of codimension |A|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' taking iterated residues, we may equip it with a volume form with logarithmic poles ωA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Then ∂([X, ω]) = � ∅̸=A⊆A (−1)|A|−1[DA, ωA] · T|A|−1, where T = [P1, dt/t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' In particular, the existence of the map ∂ relies on the birational invariance of this expression, see theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' This construction is reminiscent of the boundary map in polar homology (Khesin & Rosly, 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Khesin et al, 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Gorchinskiy & Rosly, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' However, apart from the obvious difference that we only record birational types of strata, rather than the strata themselves, our formula takes into account strata of all codimensions, rather than those of codimension one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' BURNSIDE RINGS AND VOLUME FORMS WITH LOGARITHMIC POLES 3 The map ∂ is additive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Furthermore, we prove in theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='10 that ∂(a · b) = εn · ∂(a) · b + a · ∂(b) − T · ∂(a) · ∂(b), when a ∈ Burnm(k) and b ∈ Burnn(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Here ε is the class of the point Spec(k) equipped with the volume form equal to −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' In theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='1, we show that ∂ ◦ ∂ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' These formulas may look complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' However, as we explain in §6, they simplify significantly after inverting 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Inspired by the constructions of Lin et al (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Lin & Shinder (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Kresch & Tschinkel (2022a), we define in §7 a homomorphism c: Bir(X, ω) → Burnn−1(k), from the group of birational automorphisms of the pair (X, ω), where X is an n- dimensional integral proper smooth variety equipped with a logarithmic volume form ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' As in the above references, our map c is defined at the groupoid level of birational maps preserving logarithmic volume forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' When the birational isomorphism ϕ: (X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' ω) ��� (Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' η) is described by a diagram W X Y ← → p ← → q ← → ϕ of smooth proper integral k-varieties,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' with birational morphisms p and q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' the two logarithmic volume forms p∗ω and q∗η on W are equal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' and the element c(ϕ) ∈ Burnn−1(k) is given by c(ϕ) = � E∈Exc(q) [E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' p∗ωE] − � D∈Exc(p) [D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' q∗ηD] where Exc(p) is the set of irreducible components of the exceptional divisor of p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' and where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' for each such component D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' the logarithmic volume form p∗ωD on D is obtained by taking the residue of p∗ω along D (and similarly for q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Finally, consider a discrete valuation ring with residue field k and field of frac- tions K and let t be a uniformizing element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' In this context, we define a specialization map ρt : Burnn(K) → Burnn(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' The image of the class [X, ω] involves the combinatorics of a good model (X , ω) over the valuation ring, and a certain subcomplex of the Clemens complex of the special fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' In the particular case where X is smooth, the polar divisor of ω is a relative divisor with normal crossings, and denoting by ωo the restriction of ω to the special fiber Xo, one has ρt([X, ω]) = [Xo, ωo].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Note that the existence of such a specialization map implies, as in theorem 1 of Kontsevich & Tschinkel (2019), or as in (Nicaise & Shinder, 2019), that 4 ANTOINE CHAMBERT-LOIR, MAXIM KONTSEVICH & YURI TSCHINKEL birational equivalence of varieties with logarithmic volume forms is preserved under “good specializations”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' In the geometric case, where the valuation is the local ring of a curve C at point o, the construction of the specialization map can be viewed as a restriction to the special fiber of a normalization of a global residue map ∂ that takes place on a proper model whose special fiber is a divisor with normal crossings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' The normalization procedure extracts a subcomplex of the Clemens complex of the special fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' A similar situation appeared in the study of Tamagawa measures on analytic manifolds (Chambert-Loir & Tschinkel, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Related constructions emerged from the seminal work of Kontsevich & Soibelman (2006) inspired by mirror symmetry, and its subsequent developments, eg, by Mustaţă & Nicaise (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Nicaise & Xu (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Boucksom & Jonsson (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Jonsson & Nicaise (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Our constructions use essentially only formal properties of the residue maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Con- sequently, one can envision analogous theories for logarithmic forms of smaller de- gree, Milnor K-theory, or even for the cycle modules of Rost (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — The third author was partially supported by NSF grant 2000099.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Logarithmic differential forms 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Kähler differentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — Let k be a field of characteristic zero and let K be a finitely generated extension of k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' let n be its transcendence degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' The space of Kähler differentials ΩK/k is the K-vector space generated by symbols da, for a ∈ K, subject to the relations: (1) For a ∈ k, one has da = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' (2) For a, b ∈ K, one has d(a + b) = da + db and d(ab) = adb + b(da).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' For any integer m ⩾ 0, we may consider its mth exterior power Ωm K/k, which is a K-vector space of dimension �n m � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' in particular, it vanishes if m > n, Ω1 K/k has dimension n, and Ωn K/k has dimension one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' One has Ω0 K/k = k, canonically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Elements of Ωn K/k, for n = tr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' degk(K), are also called volume forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' For a ∈ K×, we also write dlog a = da/a ∈ ΩK/k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — Let m be an integer and let ω ∈ Ωm K/k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' A model of K is an integral k-scheme X together with a k-isomorphism K ≃ k(X);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' we say that this model is proper, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' smooth if X is proper, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' smooth over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Given such a model, ω induces a meromorphic global section ωX of Ωm X/k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' The polar ideal of ωX is the subsheaf of OX whose local sections are the a ∈ OX such that aωX is induced by a regular m-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Let D be the zero-locus of this ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Its complement U is the largest open subscheme of X such that ωX is induced by a regular m-form on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' If X is smooth, then ωX is locally free, hence the scheme D is an effective divisor (Hartogs’s principle);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' we call it the polar divisor of ω on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' BURNSIDE RINGS AND VOLUME FORMS WITH LOGARITHMIC POLES 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Logarithmic forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — By Hironaka’s theorem on embedded resolution of singularities, there exist smooth projective models (X, ωX) of (K, ω) such that the polar divisor D of ωX has normal crossings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Following (Deligne, 1970, chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' 2, §3), we then say that ωX has at most loga- rithmic poles, or that ω has at most logarithmic poles on X, if both ωX and dωX have at most simple poles along D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' The following lemma implies that this condition is essentially independent of the choice of X such that the polar divisor of ωX has normal crossings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — Let g : X′ → X be a morphism of smooth k-varieties, let D be a divisor with normal crossings in X and let D′ be a divisor with normal crossings in X′ such that D′ = g−1(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Let ω be a regular m-form on X D and let ω′ = g∗ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' (1) If ω has at most logarithmic poles along D, then ω′ has at most logarithmic poles along D′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' (2) The converse holds if g is proper and surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — The first assertion is (Deligne, 1970, chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' II, prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='2, (iv)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Let us prove the second one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Consider the generic point η of X and a point η′ ∈ X′ D′ which is algebraic over k(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' The Zariski closure X′ 1 of η′ is proper and generically finite over X, and D′ 1 = D′ ∩ X′ 1 is a divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' There is a proper modification h: X′ 2 → X′ 1 such that D′ 2 = h−1(D′ 1) has normal crossings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' By the first part, the form h∗ω′|X′ 1 has at most logarithmic poles along D′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Replacing g by g◦h, we may assume that g is generically finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Since the sheaf of forms with at most logarithmic poles along D is locally free and X is smooth, we can delete from X a subset of codimension at least 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Thus, we may assume that g is flat, D is smooth and irreducible, and g is étale outside of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' It suffices to argue étale locally at the generic point of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' By the local description of ramified morphisms, there are étale local coordinates (z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' , zn) on X such that Dred = V(z1), local coordinates (z′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' , z′ n) on X′ such that g∗z1 = (z′ 1)e, g∗z2 = z′ 2, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=', where e is the ramification index of g along D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Let d be the order of the pole of ω along D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' write ω = α/zd 1 + β ∧ dz1/zd 1, where α, β are regular forms which do not involve dz1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Then ω′ = g∗ω = g∗α/(z′ 1)de + e g∗β ∧ dz′ 1/(z′ 1)1+(d−1)e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Assume, by contradiction, that d ⩾ 2, so that de ⩾ 2 and 1 + (d − 1)e ⩾ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Since ω′ has at most logarithmic poles along D, we get g∗α = 0 and g∗β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' This implies that both α and β are multiples of z1, contradicting the hypothesis that d was the order of the pole of ω along D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Therefore, d ⩽ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — We say that an m-form ω ∈ Ωm K/k is logarithmic if for all proper smooth models X of K such that the polar divisor of ωX has normal crossings, the meromor- phic differential form ωX has at most logarithmic poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' By resolution of singularities 6 ANTOINE CHAMBERT-LOIR, MAXIM KONTSEVICH & YURI TSCHINKEL (Hironaka, 1964), two models are dominated by a third one, hence lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='4, im- plies that it suffices that this condition is satisfied on some proper smooth model for which the polar divisor of ωX has normal crossings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Analogously, if X is a reduced k-variety, then we say that a meromorphic m-form ω on X is logarithmic “everywhere” if for all proper birational models (X′, ω′) of (X, ω), the meromorphic m-form ω′ on X′ has at most logarithmic poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' It suffices that this holds on one such model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Burnside rings for logarithmic forms 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Burnside rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — Let k be a field of characteristic zero and n an integer such that n ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Kontsevich & Tschinkel (2019) defined the Burnside group Burnn(k) as the free abelian group on isomorphism classes of finitely generated extensions of k of transcendence degree n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Any integral k-variety X of dimension n has a class [X] in Burnn(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' This gives rise to alternative useful presentations of Burnn(k), for example involving only classes of integral projective smooth varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' The group Burn(k) = � n≥0 Burnn(k) carries a natural commutative ring structure, with multiplication defined by taking products of (smooth projective) k-varieties: [X] · [X′] = [X × X′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Definition of a Burnside group for volume forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — Let k be a field of characteristic zero and let n be an integer ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' We define Burnn(k) to be the free abelian group on isomorphisms classes of pairs (K, ω), where – K is a finitely generated extension of k of transcendence degree n and – ω ∈ Ωn K/k is a logarithmic volume form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' We write [K, ω] ∈ Burnn(k) for the class of a pair (K, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — This definition has obvious more geometric formulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' For example, we can take for generators equivalence classes of pairs (X, ω), where – X is a smooth integral k-scheme of dimension n, and – ω a regular volume form on X which is logarithmic “everywhere”, modulo the smallest equivalence relation that identifies (X, ω) and (X′, ω′) if there exists an open immersion f : X′ → X such that ω′ = f ∗ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Alternatively, we can assume that X is proper, smooth and integral, the form ω is a logarithmic volume form on X, and consider the smallest equivalence relation that identifies (X, ω) and (X′, ω′) if there exists a proper birational morphism f : X′ → X such that ω′ = f ∗ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' By the weak factorization theorem of (Abramovich et al, BURNSIDE RINGS AND VOLUME FORMS WITH LOGARITHMIC POLES 7 2002), this equivalence relation is generated by such morphisms f which are blowing- ups along smooth centers in good position with respect to the polar divisor of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' In both contexts, if X is an n-dimensional k-variety and ω is a meromorphic n-form on X which is logarithmic “everywhere”, then we define [X, ω] to be the sum, over all irreducible components Y of X which have dimension n, of the classes [Y, ω|Y].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — Finitely generated extensions of k of transcendence degree 0 are finite extensions of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Let K be such an extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Since k has characteristic zero, one has Ω1 K/k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' However, Ω0 K/k, which is its 0th exterior power, is canonically isomorphic to K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Consequently, Burn0(k) is the free abelian group on isomorphism classes of pairs (K, λ), where K is a finite extension of k and λ ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' We will let 1 = [Spec(k), 1] and ε = [Spec(k), −1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — Let K = k(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' The differential form dt/t is a logarithmic volume form;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' indeed X = P1 k is a model of K and this form has poles of order 1 at 0 and ∞, and no other poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' We write T for the class of (k(t), dt/t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Note that the k-isomorphism of K that maps t to 1/t maps dt/t to its opposite;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' consequently, we also have T = [k(t), −dt/t] = ε · T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' In the context of birational geometry in presence of logarithmic volume forms, “rational varieties” would have class in Tn, and similarly for stable birationality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Multiplicative structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — We view the direct sum Burn(k) = � n∈N Burnn(k) as a graded abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' It is endowed with a multiplication such that [X, ω] · [X′, ω′] = [X × X′, ω ∧ ω′] when X, X′ are proper, smooth and integral and ω, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' ω′ are logarithmic volume forms on X, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' X′, and Y ranges over the set of irreducible components of X×X′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Let s: X′ × X → X × X′ be the isomorphism exchanging the two factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' One has s∗(ω ∧ ω′) = (−1)nn′ω′ ∧ ω, if n = dim(X), n′ = dim(X′), ω is a volume form on X and ω′ is a volume form on X′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Consequently, a · b = εnn′ · b · a for a ∈ Burnn(k) and b ∈ Burnn′(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' In particular, classes in Burnn(k), for even n, are central in Burn(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' We remark that the element T ∈ Burn1(k) is central as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Let indeed a ∈ Burnn(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' If n is even, then a · T = T · a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Otherwise, we have a · T = ε · T · a, but we have seen in example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='5 that T = ε · T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' As a consequence, a · T = T · a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' However, the ring Burn(k) is not commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Indeed, consider curves X, X′ without automorphisms and no nonconstant morphism between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Then the switch is the only isomorphism from X′ × X to X × X′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Take nonzero logarithmic 8 ANTOINE CHAMBERT-LOIR, MAXIM KONTSEVICH & YURI TSCHINKEL 1-forms ω, ω′ on X, X′ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' The classes [X × X′, ω ∧ ω′] and [X′ × X, ω′ ∧ ω] are then distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Functoriality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — Let k′ be an extension of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Then there is a natural ring homomorphism Burn(k) → Burn(k′) described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Let (X, ω) be an integral k-variety of dimension n equiped with a logarithmic q-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Let X′ = X ⊗k k′ be its base change to k′, and let ω′ be the volume form on X′ deduced from ω by base change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Then the class of (X, ω) maps to the sum of classes (Y, ω′|Y), where Y runs the (finite) set of irreducible components of X′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' If k′ is a finite extension of k, we also have a trace map Trk′/k : Burn(k′) → Burn(k) obtained by averaging over a set of representatives of automorphisms of the Galois closure of k′ over k modulo those preserving k′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Relation with the classical Burnside group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — Forgetting the form ω gives a ring morphism π: Burn(k) → Burn(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' On the other hand, if K is a finitely generated extension of k of transcendence degree n, we can endow it with the zero n-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' The resulting map ̟: Burn(k) → Burn(k) identifies Burn(k) with an ideal of Burn(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' One has π ◦ ̟ = id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Variations on the theme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — The construction of the Burnside ring Burn(k) admits several natural variants that are relevant in more specific contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Some of them will be used in later sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' A relative ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — Let n be an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' For any k-scheme S, we define Burnn(S/k) as the free abelian group on triples (X, ω, u) where X is an integral smooth n-dimensional k-scheme, ω ∈ Ωn X/k is a regular volume form which is loga- rithmic “everywhere”, and u: X → S is a morphism, modulo the smallest equivalence relation that identifies (X, ω, u) and (X′, ω′, u′) if there exists an open immersion f : X′ → X such that ω′ = f ∗ω and u′ = u ◦ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Let h: S → T be a morphism of k-schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' It induces a morphism of abelian groups h∗ : Burnn(S/k) → Burnn(T/k) such that h∗([X, ω, u]) = [X, ω, h ◦ u] for any triple (X, ω, u) as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Pluriforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — One can replace volume forms with volume r-pluriforms, that is, elements of (Ωn K/k)⊗r, for some given integer r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' The corresponding logarithmic condition requires that the pluriform has poles of order at most r on an adequate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Note that when r is even, the obtained ring is commmutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' BURNSIDE RINGS AND VOLUME FORMS WITH LOGARITHMIC POLES 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Forms up to scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — In the construction, we may wish to identify (K, ω) and (K′, ω′) if there exists λ ∈ k×, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' λ ∈ {±1}, and a k-isomorphism f : K → K′ such that f ∗ω′ = λω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' These variants also give rise to a commutative ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Group actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — Let G be a profinite group scheme over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' One can also consider pairs (K, ω), where the field K is endowed with an action of G leaving the form ω invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' The obtained ring will be denoted by BurnG(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Residues 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Residue of a volume form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — Let X be an equidimensional smooth k- variety of dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Let D be a smooth divisor on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' We denote by Ωm X/k(log D) the sheaf of m-forms on X with logarithmic poles along D, locally of the form η ∧d log f +η′, where η and η′ are regular and f is a local equation of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' The residue map is the homomorphism of OX-modules ρD : Ωm X/k(log D) → Ωm−1 D/k , characterized by the relation ρD(η ∧ d log f + η′) = η|D for every local sections η ∈ Ωm−1 X/k and η′ ∈ Ωm X/k, and any local generator f of the ideal of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' If ω is a logarithmic m-form on X, there is an open subset U of X such that U ∩ D ̸= ∅ and such that ω|U belongs to Ωm X/k(log D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Its residue ρD(ω|U) is then a meromorphic section of Ωm−1 D/k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — Let ω be a logarithmic differential form of degree m on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Then ρD(ω) is a logarithmic (m − 1)-form on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — We may assume that the sum of D and of the polar divisor of ω has strict normal crossings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' The assertion is then evident in local coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Blowing-ups and normal bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — Let Y be a smooth closed sub- scheme of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' The blow-up BlY(X) of X along Y is a smooth k-variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' The blowing- up morphism bY : BlY(X) → X is an isomorphism over the complement of Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' If Y is nowhere dense and nonempty, then EY = b−1 Y (Y) is a smooth divisor in BlY(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' In general, EY = b−1 Y (Y) identifies, as an Y-scheme, with the projectivization of the normal bundle NY(X) of Y in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Let W be a closed smooth subscheme of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Assume that W and Y are transversal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Then the Zariski closure of b−1 Y (W (Y ∩ W)) is called the strict transform of W in BlY(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' It identifies with BlY∩W(W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Let now ω be a logarithmic m-form on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Then the form b∗ Yω on BlY(X) is logarithmic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' assuming that Y is nonempty and nowhere dense, we can consider the residue ρY(ω) of b∗ Yω along EY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' It is a logarithmic (n − 1)-form on P(NY(X )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' 10 ANTOINE CHAMBERT-LOIR, MAXIM KONTSEVICH & YURI TSCHINKEL Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — Let X be an irreducible proper smooth k-variety, let n be its dimension and let ω ∈ Ωn X/k be logarithmic volume form whose polar divisor D has strict normal crossings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Let (Dα)α∈A be the family of its irreducible components;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' for A ⊆ A , we let DA = � α∈A Dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' We then define an element ρ(X, ω) in Burnn−1(X/k) by the formula: ρ(X, ω) = � ∅̸=A⊆A (−1)|A|−1ρDA(X, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' (In this formula and all similar ones below, it is always implicit that the terms where DA = ∅ are omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=') 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Iterated residues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — We retain the notation of definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='4 Fix a logarithmic volume form ω on X and a nonempty subset A of A such that DA ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' It will be useful to compute inductively the logarithmic volume form ρDA(ω) that appears in definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Let bA : ˜X → X be the blowing-up of X along DA and let E be its exceptional divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' When A = {α} has a single element, DA is the divisor Dα, the blowing-up mor- phism bA is an isomorphism and the exceptional divisor identifies with DA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Then ρDA(X, ω) = [Dα, ρDα(ω), jα], where jα is the immersion of Dα into X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' This construction can be pursued in higher codimension, using iterated residues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Fix a total order on A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' There is a unique, strictly increasing sequence (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' , αm) in A such that A = {α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' , αm}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Given the chosen order on A , we may apply the iterated residues construction and obtain a logarithmic form of degree n − m ρDA(ω) = ρDα1 ◦ · · · ◦ ρDαm(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' On a nonempty open subset U of X that meets DA, we may write ω = η ∧ dlog(fα1) ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' dlog(fαm), for a regular form η, and then one has ρDA(ω) = η|U∩DA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Denote by bA the blowing-up of X along DA and by EA its exceptional divisor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' recall that EA identifies with the projectivized normal bundle NDA(X) of DA in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Using local equations for the divisors Dα, for α ∈ A, we trivialize NDA(X) on a dense open subscheme of DA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' this gives a birational isomorphism of EA with DA × Pm−1 (with m = |A|), and a local computation gives the formula ρDA(X, ω) = [DA, ρDA(ω)] · Tm−1 in Burnn−1(DA/k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' When m ⩾ 2, the definition of ρDA actually depends on the chosen order of A , but only up to a sign, so that the class [DA, ρDA(ω)] is well defined up to multiplication by the class ε ∈ Burn0(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' On the other hand, it is multiplied by Tm−1 and we have observed that ε · T = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' BURNSIDE RINGS AND VOLUME FORMS WITH LOGARITHMIC POLES 11 Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — Let (X, ω), D, and (Dα)α∈A be as in definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Let Y be a strict irreducible subvariety of X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' let AY be the set of all α ∈ A such that Y ̸⊆ Dα;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' we assume that � α∈AY Dα meets Y transversally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Let g : X′ → X be the blowing-up of X along Y and let ω′ = g∗ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' it is a logarithmic form, its polar divisor has strict normal crossings, and we have g∗ρ(X′, ω′) = ρ(X, ω) in Burn(X/k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — Let E = g−1(Y) be the exceptional divisor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' for each α ∈ A , let D′ α be the strict transform of Dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' The blow-up X′ is smooth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' the divisor E + � α∈A D′ α has strict normal crossings and contains the polar divisor of ω′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Let B be the set of all β ∈ A such that Y ⊆ Dβ, so that DB is the minimal stratum containing Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' We now split the discussion into two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' (1) Assume that dim(Y) < dim(DB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Since g is ramified along E, its Jacobian vanishes along E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Since ω has poles of order at most one, the form ω′ = g∗ω is regular at the generic point of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Consequently, the polar divisor of ω′ does not contain E and we have to compare � ∅̸=A⊆A (−1)|A|−1ρD′ A(ω′) with � ∅̸=A⊆A (−1)|A|−1ρDA(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Since g is a local isomorphism around the generic points of Dα, for α ∈ A , we see that the polar divisor of ω′ is equal to � α∈A D′ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' For every nonempty subset A of A , one has g∗ρDA(X′, ω′) = ρDA(X, ω) for every nonempty subset A of A , which implies the desired formula in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' (2) Assume that dim(Y) = dim(DB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' In this case, Y is an irreducible component of DB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Since D∅ = X and Y ̸= X, we have B ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' We have to compare the expression � ∅̸=A⊆A (−1)|A|−1ρD′ A(ω′) + � A⊆A (−1)|A|ρE∩D′ A(ω′) with � ∅̸=A⊆A (−1)|A|−1ρDA(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' The argument takes place in a neighborhood of Y, which allows us to assume that Y = DB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Let A be a nonempty subset of A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' One has D′ A = ∅ whenever B ⊆ A, and the corresponding terms are absent from the second expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' On the other hand, if B ̸⊆ A, the morphism g identifies D′ A with the blow-up of DA along DA ∩ Y = DA∪B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' In particular, g induces a birational isomorphism from D′ A to DA, so that 12 ANTOINE CHAMBERT-LOIR, MAXIM KONTSEVICH & YURI TSCHINKEL g∗ρD′ A(X′, ω′) = ρDA(X, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Moreover, E ∩ D′ A is the projectivized normal bundle PNDA∪B(DA), and g∗ρE∩D′ A(X′, ω′) = ρDA∪B(X, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Similarly, one has g∗ρE(X′, ω′) = ρDB(X, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' This gives a formula of the form g∗ρ(X′, ω′) = � ∅̸=A⊆A B̸⊆A (−1)|A|−1ρDA(X, ω) + � A⊆A B̸⊆A (−1)|A|ρDA∪B(X, ω) = � ∅̸=A⊆A n′ AρDA(X, ω), where n′ A = \uf8f1 \uf8f2 \uf8f3 (−1)|A|−1 if B ̸⊆ A, � C⊆A B̸⊆C C∪B=A (−1)|C| if B ⊆ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' It suffices to prove that n′ A = nA for any nonempty subset A of A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' This is obvious when B ̸⊆ A, so let us assume that B ⊆ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' In the sum that defines n′ A, we write C = (C B)∪C′, where C′ = C∩B is a subset of B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' the condition C∪B = A means C B = A B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' the condition B ̸⊆ C means C′ ̸= B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Consequently, we have n′ A = (−1)|A B| � C′⊆B C′̸=B (−1)|C′| = (−1)|A B| � � C′⊆B (−1)|C′| − (−1)|B| � = (−1)|A B| � (1 − 1)|B| − (−1)|B|� = (−1)|A|−1, since |B| ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' This concludes the proof of the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — Let (X, ω) be as in definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' If X is proper, then the image of ρ(X, ω) in Burnn−1(k) only depends on the class [X, ω] ∈ Burnn(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' It gives rise to a morphism of abelian groups ∂n : Burnn(k) → Burnn−1(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — By the definition of Burnn(k) involving pairs (X, ω) where X is proper, it suffices to consider two pairs (X, ω) and (X′, ω′) as in definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='4 which are related by a proper birational morphism g : X′ → X such that g∗ω = ω′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' By the weak factorization theorem of Abramovich et al (2002), in order to prove the theorem, we may assume that g is a blowing-up of X along a smooth subvariety which is transversal to the polar divisor of ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' In this case, proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='6 asserts BURNSIDE RINGS AND VOLUME FORMS WITH LOGARITHMIC POLES 13 that g∗ρ(X′, ω′) = ρ(X, ω) in Burn(X/k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' In particular, the images in Burn(k) of ρ(X′, ω′) and ρ(X, ω) are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — The meromorphic differential form dt/t on P1 k has residues 1 and −1 at 0 and ∞ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' By construction, we thus have ∂1(T) = [Spec(k), 1] + [Spec(k), −1] = 1 + ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Let n be an integer such that n ⩾ 2 and let us compute ∂n(Tn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' We view Tn as the class of Pn, with homogeneous coordinates [1 : x1 : .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' : xn], and with the toric differential form ωn = (dx1/x1) ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' (dxn/xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Its divisor is the sum of the toric hyperplanes D0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' , Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Each of these hyperplanes identifies with Pn−1, and ρDj(ωn) is (−1)n−jωn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Let A = {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' If A = A , then DA = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Otherwise, we see by induction that DA is isomorphic to Pn−|A| and ρDA(ωn) identifies with ±ωn−|A|, so that [DA, ρDA(ωn)] · T|A|−1 = [Gm n−1, ±ωn−1] = Tn−1, since n − 1 ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Then, ∂n(Tn) = � ∅̸=A⊆A (−1)|A|−1[DA, ρDA(ωn)] · T|A|−1 = � ∅̸=A⊊A (−1)|A|−1Tn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Now, � ∅̸=A⊊A (−1)|A|−1 = 1 − (1 − 1)n+1 + (−1)n+1 = � 2 if n is odd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' 0 if n is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' We get ∂n(Tn) = 2Tn−1 if n is odd and ∂n(Tn) = 0 if n is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' (Remind that n ⩾ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=') Since T = ε · T, the following formula unifies the various cases: for n ⩾ 1, we have ∂n(Tn) = (1 + (−1)n−1ε) · Tn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — For every class b ∈ Burnn(k), we have ∂n+1(b · T) = −∂n(b) · T + b · ∂1T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — We may assume that b = [X, ω], where X is a proper integral smooth variety of dimension n, and ω is a logarithmic volume form on X whose polar divisor has strict normal crossings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Let (Dα)α∈A be the family of its irreducible components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' We view b·T as the class of [X×P1, ω ∧dt/t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' The polar divisor of ω ∧dt/t is equal to � α∈A Dα × P1 + X × {0} + X × {∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' 14 ANTOINE CHAMBERT-LOIR, MAXIM KONTSEVICH & YURI TSCHINKEL It has strict normal crossings, and its strata are of the form DA × P1, for nonempty A ⊆ A , or DA × {0}, or DA × {∞}, for A ⊆ A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' This decomposes ∂n+1(b × T) as the sum of three terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' The first one is � ∅̸=A⊆A [DA × P1, ρDA×P1(ω ∧ dt/t)] · T|A|−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' For any nonempty subset A of A , one has ρDA×P1(ω ∧ dt/t) = ±ρDA(ω) ∧ dt/t, so that [DA × P1, ρDA×P1(ω ∧ dt/t)] · T|A|−1 = [DA, ρDA(ω)] · T · T|A|−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Consequently, the first term equals ∂n(b) × T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Write D0 = X × {0} and D∞ = X × {∞}, and identify both divisors to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' For a subset A of A , we have ρDA∪{0}(ω ∧ dt/t) = ρDA ◦ ρD0(ω ∧ dt/t) = ρDA(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Consequently, the second term is equal to � A⊆A (−1)|A|[DA, ρDA(ω)] · T|A| = [X, ω] − ∂n(b) · T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Similarly, the third term is equal to [X, −ω] − ∂n(b) · T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Summing up these three terms, we get ∂n+1(b × T) = −∂n(b) · T + [X, ω] + [X, −ω].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' We now recall that ∂1(T) = [Spec(k), 1] + [Spec(k), −1], so that [X, ω] + [X, −ω] = [X, ω] · ∂1(T) = b · ∂1(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — Let a ∈ Burnm(k) and b ∈ Burnn(k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' we have ∂m+n(a · b) = εn · ∂m(a) · b + a · ∂n(b) − T · ∂m(a) · ∂n(b) in Burnm+n−1(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — It suffices to treat the case where a and b are classes of proper integral smooth varieties (X, ω), (Y, η), endowed with meromorphic volume forms whose polar divisors have strict normal crossings and no multiplicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Let (Dα)α∈A be the irreducible components of the polar divisor of ω, let (Eβ)β∈B be the irreducible components of the polar divisor of η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Then [X, ω]·[Y, η] is the class of [X×Y, ω ∧η];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' the polar divisor of ω ∧ η is equal to � α∈A Dα × Y + � β∈B X × Eβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' BURNSIDE RINGS AND VOLUME FORMS WITH LOGARITHMIC POLES 15 We fix a total order on the disjoint union of A and B such that the elements of A are smaller than those of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' For any subsets A, B of A and B, observe that we have ρDA∪B(ω ∧ η) = ±ρDA(ω) ∧ ρEB(η), where ρDA has to be understood as the identity when A is empty, and similarly for ρEB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' The sign is 1 when A = ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' when B = ∅, it is equal to (−1)|A|n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' we won’t need to use its explicit value in the other cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Then we can write ∂([X, ω] · [Y, η]) as � A⊆A B⊆B A∪B̸=∅ (−1)|A|+|B|−1[DA × EB, ±ρA(ω) ∧ ρEB(η)] · T|A∪B|−1 and we split it into the sum of three terms, according to which B = ∅, or A = ∅, or none of them is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' The first two terms are respectively equal to � ∅̸=A⊆A (−1)|A|−1[DA × Y, (−1)n|A|ρDA(ω) ∧ η] · T|A|−1 = ∂([X, (−1)nω]) · [Y, η] and � ∅̸=B⊆B (−1)|B|−1[X × EB, ω ∧ ρEB(η)] · T|B|−1 = [X, ω] · ∂([Y, η]), since T belongs to the center of Burn(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' As for the third one, we obtain − � ∅̸=B⊆B (−1)|B|−1 � ∅̸=A⊆A (−1)|A|−1[DA, ρDA(ω)] · [EB, ρEB(η)] · T|A|+|B|−2 which equals −∂([X, ω]) · ∂([Y, η]) · T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Finally, we get ∂m+n(a · b) = ∂m+n([X, ω] · [Y, η]) = ∂m([X, (−1)nω) · [Y, η] + [X, ω] · ∂n([Y, η]) − T · ∂m([X, ω]) · ∂n([Y, η]) = εn · ∂m(a) · b + a · ∂n(b) − T · ∂m(a) · ∂n(b) as was to be shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' In particular, using the computation of example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='8, we obtain the following generalization of proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — For any a ∈ Burnm(k) and any integer n, we have ∂m+n(a · Tn) = � ∂m(a) · Tn if n is even;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' −∂m(a) · Tn + a · ∂n(Tn) if n is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — For the variant of Burn(k) where we consider forms up to sign, the formula of theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='10 simplifies to ∂m+n(a · b) = ∂m(a) · b + a · ∂n(b) − T · ∂m(a) · ∂n(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' 16 ANTOINE CHAMBERT-LOIR, MAXIM KONTSEVICH & YURI TSCHINKEL 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' A complex of Burnside rings Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — For any integer n ⩾ 2, we have ∂n−1 ◦ ∂n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' In other words, the residue morphisms of Burnside groups give rise to a complex · · → Burnn(k) → Burnn−1(k) → · · · → Burn1(k) → Burn0(k) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — It suffices to prove the following result: Let (X, ω) be an integral proper smooth variety of dimension n equipped with a meromorphic volume form ω whose polar divisor has strict normal crossings and no multiplicities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' then ∂n−1(∂n([X, ω])) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Let (Dα)α∈A be the family of irreducible components of the polar divisor of ω in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' By definition, one has ∂n([X, ω]) = � ∅̸=A⊆A (−1)|A|−1ρDA(X, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Fix a total order on A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Let (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' , αm) be a strictly increasing sequence in A and let A = {α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' , αm}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' We have seen in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='5 that ρDA(X, ω) can be defined via iterated residue maps: ρDA([X, ω]) = [DA, ρDα1 ◦ · · · ◦ ρDαm(ω)] · T|A|−1 = [DA, ωA] · T|A|−1 where we wrote ωA for the composition ρDα1 ◦ · · · ◦ ρDαm(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' When |A| is odd, we have ∂(ρDA([X, ω])) = ∂([DA, ωA]) · T|A|−1, while when |A| is even, we have ∂(ρDA([X, ω])) = −∂([DA, ωA]) · Ta−1 + [DA, ωA] · ∂(T|A|−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Consequently, we have ∂ ◦ ∂([X, ω]) = � ∅̸=A⊆A ∂([DA, ωA]) · T|A|−1 − � ∅̸=A⊆A |A| even [DA, ωA] · ∂(T|A|−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' The polar divisor of the form ωA on DA is equal to � β̸∈A Dβ ∩ DA, so that, by definition (and computation of ∂ via iterated residues), ∂([DA, ωA]) = � ∅̸=B⊆∁A (−1)|B|−1[DAB, ωA∪B] · T|B|−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Also, when A is nonempty and of even cardinality, ∂(T|A|−1) = 2T|A|−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' When we put these two formulas into the antepenultimate one and collect the various terms, we obtain ∂ ◦ ∂([X, ω]) = � C⊆A 2⩽|C| nC[DC, ωC] · T|C|−2, BURNSIDE RINGS AND VOLUME FORMS WITH LOGARITHMIC POLES 17 where nC = − � ∅̸=A,B A∪B=C,A∩B=∅ (−1)|B| − 2δ|C| is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' In the first sum, the terms A = ∅ or B = ∅ are omitted, while if we put them in, we obtain � A∪B=C A∩B=∅ (−1)|B| = |C| � b=0 �|C| b � (−1)b = (1 − 1)|C| = 0 since |C| ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Consequently, nC = 1 + (−1)|C| − 2δ|C| is even = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Algebraic structure of Burn(k) after localization at 2 In this section, we study the algebraic structure of the Burnside ring Burn(k), endowed with its elements ε, T and the operator ∂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — By construction, Burn(k) = � n⩾0 Burnn(k) is an associative unital Z⩾0- graded ring, ε ∈ Burn0(k), T ∈ Burn1(k) and ∂ is a homogeneous additive map of degree −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' They satisfy the following relations, for homogeneous elements a, b ∈ Burn(k): b · a = ε|a||b| · a · b (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='6);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' (1) ε2 = 1 (example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' (2) T = ε · T (example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' (3) ∂(T) = 1 + ε (example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='8);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' (4) ∂(a · b) = ε|b| · ∂(a) · b + a · ∂(b) − T · ∂(a) · ∂(b) (theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='10);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' (5) ∂(∂(a)) = 0 (theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' (6) By (1), the element ε is central, and by (2), we may view Burn(k) as an algebra over Z[ε]/(ε2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' After inverting 2, the algebra Burn(k) splits into two components Burnε=1(k) and Burnε=−1(k), one over which ε = 1, and the other over which ε = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' In the rest of this section, we implicitly assume that 2 is inverted, without changing the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' 18 ANTOINE CHAMBERT-LOIR, MAXIM KONTSEVICH & YURI TSCHINKEL 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Sector ε = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — Here, we have T = −T, hence T = 0 since 2 is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' As a consequence, after replacing ∂ with ∂′ : a �→ (−1)1+|a|∂(a), one gets from (5) the usual graded Leibniz rule ∂′(a · b) = ∂′(a) · b + (−1)|a|a · ∂′(b) and therefore Burnε=−1(k) is a classical differential graded (super-)commutative algebra, similar to, eg, the de Rham complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Sector ε = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — The algebra Burnε=1 is now commutative (and not graded commutative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' This reflects the intuition in our constructions that they speak about volume forms (as opposed to top-degree differential forms) for which we have com- mutativity (as reflected by the change of order of integration in multiple integrals).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — The map F: a �→ a − T · ∂(a) is a ring endomorphism of Burnε=1(k), and F2 = id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Moreover, one has F ◦ ∂ = ∂ = −∂ ◦ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — This map is additive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' One has F(1) = 1 − T · ∂(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Let us show multiplicativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Indeed, for a, b ∈ Burnε=1(k), one has F(a) · F(b) = (a − T · ∂(a)) · (b − T · ∂(b)) = a · b − T · ∂(a) · b − T · a · ∂(b) + T2 · ∂(a) · ∂(b) = a · b − T · (∂(a) · b + a · ∂(b) − T · ∂(a) · ∂(b)) = a · b − T · ∂(a · b) (using (5)) = F(a · b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Since ∂2 = 0, one has F(∂(a)) = ∂(a) − T · ∂(∂(a)) = ∂(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' On the other hand, ∂(F(a)) = ∂(a − T · ∂(a)) = ∂(a) − ∂(T · ∂(a)) = ∂(a) − ∂(T) · ∂(a) − T · ∂(∂(a)) + T · ∂(T) · ∂(∂(a)) = −∂(a) using that ∂(T) = 2 and ∂2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Consequently, for a ∈ Burnε=1(k), we have F2(a) = F(a) − T · ∂(F(a)) = a − T · ∂(a) + T · ∂(a) = a since ∂ ◦ F = −∂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' BURNSIDE RINGS AND VOLUME FORMS WITH LOGARITHMIC POLES 19 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — To simplify the notation, write B = Burnε=1(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Since F2 = id and 2 is invertible, the algebra B splits as a direct sum B = B+ ⊕ B−, such that F acts as id on B+ and as − id on B−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Moreover, B+ is a subalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Since the operator ∂ anticommutes with F, it induces maps ∂± : B+ → B−, ∂∓ : B− → B+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Note that F(T) = T − T · ∂(T) = −T, so that T ∈ B−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Consequently, the multiplication by T map induces two maps t± : B+ → B−, t∓ : B− → B+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — The map ∂ vanishes on B+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Equivalently, ∂± = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' The maps 1 2∂∓ and t± are inverses the one of the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — For a ∈ B+, one has ∂(a) = −∂(F(a)) = −∂(a), since ∂ ◦ F = −∂, hence ∂(a) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' On the other hand, for a ∈ B+, one has ∂(T · a) = 2 · a + T · ∂(a) − 2T · ∂(a) = 2a − T · ∂(a) = a + F(a) = 2a while for a ∈ B−, we have T · ∂(a) = a − F(a) = 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' This concludes the proof of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' In particular, we see that the cohomology of the differential ∂ vanishes in the sector Burnε=1(k) = B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — It follows from the lemma that we have a ring isomorphism B = B+[t](t2 − T2), from which we see that all the algebraic structure of B+ (namely δ, T, F) can be canonically reconstructed from a unital commutative associative Z⩾0-graded ring B+ endowed with an element in degree +2 (namely, the element T2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — The situation clarifies even more if we invert the class T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Then we can write ∂(a) = (a − F(a))/T, and all relations happen to follow from the fact that F is an involution such that F(T) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Indeed, ∂2(a) = ∂(a) − F(∂(a)) T = 1 T �a − ∂(a) T − F(a − ∂(a) T � = 0 20 ANTOINE CHAMBERT-LOIR, MAXIM KONTSEVICH & YURI TSCHINKEL explains that ∂2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Moreover, for a, b ∈ B, we have ∂(a · b) = a · b − F(a · b) T = a · b − F(a) · F(b) T = a − F(a) T b + a · b − F(b) T − T · a − F(a) T b − F(b) T = ∂(a) · b + a · ∂(b) − T · ∂(a) · ∂(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Birational morphisms preserving volume forms 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — Let (X, ωX) be a smooth integral k-variety of dimension n equipped with a meromorphic form with poles of order at most one on X and let f : Y → X be a proper birational morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Let E be an exceptional divisor in Y, that is, such that dim(f(E)) < dim(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Then the Jacobian of p vanishes along E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Since ω has poles of order at most one on X, the meromorphic form f ∗ω on Y is regular at the generic point of E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' its restriction to E is a meromorphic form with poles of order at most one and we may consider the class [E, f ∗ωX|E] in Burnn−1(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' We define c(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' X, ω) to be the sum of all such classes [E, f ∗ω|E] in the free abelian group Burnn−1(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — Let g : Z → Y be a proper birational morphism of smooth integral varieties of dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Then g ◦ f is a proper birational morphism and one has c(g ◦ f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' X, ω) = c(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Y, f ∗ω) + c(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' X, ω) in Burnn−1(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — An integral divisor F in Z is exceptional for g ◦ f if and only if one of the two mutually excluding situations happen: – The divisor F is exceptional for g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' – Or g(F) is a divisor in Y which is exceptional for f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Moreover, any divisor E in Y which is exceptional for f appears once and only as a divisor of the form g(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' In the first case, F contributes to c(g ◦ f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' X, ω) by a term [F, (g ◦ f)∗ω|F], and it contributes to c(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Y, f ∗ω) by precisely the same term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' In the second case, F contributes to c(g ◦ f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' X, ω) by a term [F, g∗(f ∗ω|E)], while E contributes to c(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' X, ω) by the term [E, f ∗ω|E].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Since g is birational around the generic point of E, they coincide, and this concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — Let (X, ωX) and (Y, ωY) be proper smooth k-varieties equipped with loga- rithmic volume forms and let ϕ: (X, ωX) ��� (Y, ωY) BURNSIDE RINGS AND VOLUME FORMS WITH LOGARITHMIC POLES 21 be a birational map preserving the volume forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' By definition, this means that there exists a diagram W X Y ← → p ← → q ← → ϕ of integral k-varieties such that that p and q are proper and birational, and such that p∗ω = q∗ω′ on W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' In this situation, we may assume that W is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — With this notation, the element c(ϕ) = c(q) − c(p) ∈ Burnn−1(k) only depends on the birational map ϕ, and not on the choice of the triple (W, p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — Consider two possible diagrams X p←− V q−→ Y and X r←− W s−→ Y describ- ing ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Considering for example a resolution of singularities U of V ×X W, we can fit these two diagrams in a common commutative diagram of the following form: U V W X Y ← → u ← → v ← → p ← → q ← → r ← → s ← → ϕ The equalities p∗ωX = q∗ωY and r∗ωX = s∗ωY imply that (p ◦ u)∗ωX = u∗p∗ωX = u∗q∗ωY = (q ◦ u)∗ωY = (s ◦ v)∗ωY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' By lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='2, we then have c(p) − c(q) = c(p ◦ u) − c(q ◦ u) = c(r ◦ v) − c(s ◦ v) = c(r) − c(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — If ψ: (Y, ωY) ��� (Z, ωZ) is another birational map preserving volume forms, then one has c(ψ ◦ ϕ) = c(ψ) + c(ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — Consider two diagrams X p←− V q−→ Y and Y r←− W s−→ Y describing ϕ and ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Considering for example a resolution of singularities U of V ×Y W, we can 22 ANTOINE CHAMBERT-LOIR, MAXIM KONTSEVICH & YURI TSCHINKEL fit these two diagrams in a common commutative diagram of the following form: U V W X Y Z ← → u ← → v ← → p ← → q ← → r ← → s ← → ϕ ← → ψ and the diagram X p◦u ←−− U s◦v −−→ describes the birational map ψ◦ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Since q◦u = r◦v, we then have c(ψ ◦ ϕ) = c(p ◦ u) − c(s ◦ v) = c(p ◦ u) − c(q ◦ u) + c(r ◦ v) − c(s ◦ v) = c(p) − c(q) + +c(r) − c(s) = c(ϕ) + c(ψ), as was to be shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — Let Bir(X, ω) be the set of birational automorphisms of X pre- serving ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' The map c induces a homomorphism of abelian groups Bir(X, ω) → Burnn−1(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Its kernel contains the group of automorphisms of X that preserve ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Specialization Let K be the field of fractions of a discrete valuation ring R with residue field k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Fix a uniformizer t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' In this context, Kontsevich & Tschinkel (2019) have defined two (distinct) specialization morphisms ρt : Burnn(K) → Burnn(k), relating the Burnside groups of K and k (see 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='1), one of which is a ring homo- morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' (The latter homomorphism actually depends on the choice of t, see example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='2 of (Kresch & Tschinkel, 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=') The goal of this section is to define a similar homomorphism ρt : Burn(K) → Burn(k) for varieties with logarithmic volume forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' BURNSIDE RINGS AND VOLUME FORMS WITH LOGARITHMIC POLES 23 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — Let X be an integral proper scheme over R, of relative dimension n, whose special fiber ∆ is a divisor with strict normal crossings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Let (∆α)α∈A be the family of irreducible components of the special fiber ∆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' for α ∈ A , let eα be the multiplicity of ∆α in ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' For every nonempty subset A of A , let ∆A be the intersection of all divisors ∆α, for α ∈ A and eA be the greatest common divisor of the eα, for α ∈ A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' let also ∆◦ A be the complement ∆A � α̸∈A ∆α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' The first specialization morphism of (Kontsevich & Tschinkel, 2019) is de- fined by (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='2) ρt([XK]) = � ∅̸=A⊆A (−1)|A|−1[∆A]L|A|−1, where L ∈ Burn(k) is the class of the affine line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Although this map is not multiplicative, it proved sufficient for many applications to rationality problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' To ensure multiplicativity, a more delicate construction was necessary, valued in the Burnside ring Burn�µ(k) of varieties endowed with an action of the profinite group �µ, limit of finite groups of roots of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Fix a nonempty subset A of A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' We identify the normal bundle of ∆A in X as a direct sum of line bundles: N∆A(X ) ≃ � α∈A N∆α(X )|∆A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Let us consider its open subscheme N ◦ ∆A(X ) obtained by restricting to ∆◦ A and taking out all “coordinate” hyperplanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' This furnishes a morphism νA : N ◦ ∆A(X ) → � α∈A N∆α(X )⊗eα|∆◦ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Since the uniformizer t has divisor − � α∈A eα∆α on X , it trivializes the line bundle on the target of νA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' We set ∆′ A = ν−1 A (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' By construction, the projection ∆′ A → ∆A is a torsor with group µeA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' With this notation, the correct, multiplicative, specialization map of (Kontsevich & Tschinkel, 2019) is given by the formula �ρt(X) = � ∅̸=A⊆A (−1)|A|−1[∆′ A]L|A|−1 in Burn�µ(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — The relation between the two specialization morphisms is as fol- lows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Fix a nonempty subset A of A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' The group Gm acts diagonally on N ◦ ∆A(X ) (the factors of index α /∈ A don’t act), and this induces an action of the finite group of roots of unity of order eA on ∆′ A, hence an action of �µ, so that �ρt(X) naturally lives in the equivariant Burnside ring Burn�µ(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Moreover, taking the �µ-invariants 24 ANTOINE CHAMBERT-LOIR, MAXIM KONTSEVICH & YURI TSCHINKEL of ∆′ A, we get ∆◦ A, so that the specialization map ρt is the composition of �ρt with the map Burn�µ(k) → Burn(k) obtained by taking �µ-invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Taking invariants does not commute with taking products, in general, so that ρt is not multiplicative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — Let us explain how to define analogous specialization homomorphisms in our context of Burnside groups with volume forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' For simplicity, we only consider the case where K has transcendence degree 1 over k, in which case the idea can be explained geometrically as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' We assume that there exists an smooth integral curve C together with a k-point o ∈ C(k) such that K = k(C) and R = OC,o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' We fix a local parameter t ∈ R such that V(t) = o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Let us consider a pair (X, ω) consisting of an integral proper K-variety X of di- mension n and a logarithmic n-form ω on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — Consider a regular flat proper model X is of X over C, let ∆ = (Xo)red be its reduced special fiber, and consider a divisor D with relative normal cross- ings on X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' We assume that the divisor D + ∆ has normal crossings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' In this situation, Deligne (1970, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='2) says that a meromorphic relative differential m- form on X /C is logarithmic with respect to D + ∆ if it is (locally) the image of a logarithmic m-form ˜ω in Ωm X /k with poles D + ∆ under the natural morphism Ωm X /k → Ωm X /C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Consider a logarithmic relative n-form ω on X /C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' We consider an associated volume form ω′ on X , defined locally by ω′ = ˜ω ∧ dt/t, where ˜ω is any local lift of ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' This form ω′ is logarithmic and we can compute its “residue along ∆” as in §4, only taking into account the strata of the polar divisor of ω′ which are contained in the special fiber ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' There exists a subset Ao of A and a subset Bo of B such that the polar divisor of ω′ is given by � α∈Ao ∆α + � β∈Bo Dβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' We thus set ρt(X , ω) = � ∅̸=A⊆Ao B⊆Bo (−1)|A|+|B|−1ρ∆A∩DB(X , ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' This is an element of Burnn(Xo/k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — Let Y be an irreducible closed subscheme of X which is transverse to D + ∆ and let g : X ′ → X be the blowing-up of X along Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' The form g∗ω on X ′ is logarithmic and we have g∗ρt(X ′, g∗ω) = ρt(X , ω) BURNSIDE RINGS AND VOLUME FORMS WITH LOGARITHMIC POLES 25 in Burnn(Xo/k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — With the notation of §4, the difference ρ(X , ˜ω) − ρt(X , ω) is exactly the part of ρ(X , ˜ω) which lies over the complement of the special fiber Xo in X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' We have seen in theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='7 that g∗ρ(X ′, ˜ω′) = ρ(X , ω), and a similar formula holds over X Xo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' This implies the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — Starting from a smooth proper K-variety X and a logarithmic volume form ω on X, we can define a model X /C, with D and ∆ as above, but the form ω will not necessarily extend to a logarithmic relative form with respect to D +∆, nor does the volume form ˜ω on X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' However, this can be achieved by multiplying ω by a suitable power of the uniformizing element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Let us write the polar divisor of ˜ω on X as divX (˜ω) = D + ∆ = � α∈A dα∆α + � β∈B dβDβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' With this notation, the condition for ˜ω to be logarithmic on X is just that dα ⩾ −1, dβ ⩾ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' In particular, while the conditions at the horizontal components follow from their counterparts on the generic fiber, those for the vertical components are not auto- matic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' On the other hand, for any κ ∈ Z, the form tκ˜ω is logarithmic if and only if κeα + dα ⩾ −1 for all α ∈ A , that is, if and only if κ ⩾ κ(ω), where κ(ω) is defined by κ(ω) = inf α∈A 1 − dα eα .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Since the rational number κ(ω) is defined in terms of logarithmic forms, it only depends on the class of (X, ω) in Burnn(K), and not on the actual model which is chosen to compute it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — We assume for the moment that κ(ω) ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' This holds in particular if the special fiber Xo is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Let then Ao be the subset of A consisting of all α such that κ(ω)eα + dα = −1, and let Bo be the subset of B consisting of all β such that dβ = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' The polar divisor of tκ˜ω is equal to � α∈Ao ∆α + � β∈Bo Dβ, 26 ANTOINE CHAMBERT-LOIR, MAXIM KONTSEVICH & YURI TSCHINKEL and we set ρt(X , ω) = ρt(X , tκ(ω)ω) in Burnn(Xo/k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' In the particular case where D is empty, the strata of the Clemens complex of the special fiber that actually appear in the definition of this class are those defined by Kontsevich & Soibelman (2006), more precisely, by the adjustment provided by Mustaţă & Nicaise (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — In the general case, the rational number κ(ω) is not an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Let us consider the finite ramified extension Kd = K(t1/d) of K, whose ramification index d is a multiple of the denominator of κ(ω), but which induces an isomorphism on the residue field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Geometrically, this furnishes a morphism π: Cd → C which is ramified at the point o, together with a lift of o in Cd(k) (still denoted by o), and a distinguished uniformizing element t1/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' We consider the extension of (X, ω) to Kd and introduce a model (Xd, ωd) as above, over Cd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Now, the corresponding κ-parameter is integral, so that any choice of a uniformizing element t1/d in Rd induces a class ρt1/d(Xd, ωd) in Burn(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' In fact, we can assume that the scheme Xd carries an action of the group scheme µd of dth roots of unity induced by its action on Spec(Rd), leaving the logarithmic form ωd invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' In other words, we obtain a class in the group Burn�µ(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Combinining these classes, we obtain the desired group homomorphism �ρt : Burn(K) → Burn�µ(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' In fact, as explained in (Nicaise, 2013, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='3), especially proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='2, one can compute the normalisation of X ⊗ Rd in terms of the given model X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' This gives an explicit decomposition of �ρt(X, ω) as a sum � ∅̸=A⊆Ao (−1)|A|−1[D′ A, ν∗ AωA] · T|A|−1, where νA : D′ A → DA is the µdA-torsor introduced in §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='1 for the definition of the classical specialization map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — In the case of specialization of rationality, it has proved fruitful to consider models with singularities on the special fiber, mild enough so that the special fiber computes the specialization of the birational type of the generic fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' This is in particular the case for rational double points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' A parallel study can be developped in the context of varieties with logarithmic forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Following (Kontsevich & Tschinkel, 2019) and keeping track of the various logarithmic volume forms on the strata, we have: Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' — The morphism �ρt is a ring homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' BURNSIDE RINGS AND VOLUME FORMS WITH LOGARITHMIC POLES 27 References D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Abramovich, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Karu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Matsuki & J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Włodarczyk (2002), “Torifica- tion and factorization of birational maps”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Journal of the American Mathematical Society, 15 (3), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' 531–572 (electronic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Boucksom & M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Jonsson (2017), “Tropical and non-Archimedean limits of degenerating families of volume forms”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Journal de l’École polytechnique - Math- ématiques, 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' 87–139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Chambert-Loir & Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Tschinkel (2010), “Igusa integrals and volume asymp- totics in analytic and adelic geometry”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Confluentes Mathematici, 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' 351–429.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Deligne (1970), Equations différentielles à points singuliers réguliers, Lect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Notes Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' 163, Springer, Cham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Gorchinskiy & A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Rosly (2015), “A Polar Complex for Locally Free Sheaves”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' International Mathematics Research Notices, 2015 (10), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' 2784–2829.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Hironaka (1964), “Resolution of singularities of an algebraic variety over a field of characteristic zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' I, II”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Annals of Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Second Series, 79, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' 109–203, 205–326.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Jonsson & J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Nicaise (2020), “Convergence of p-adic pluricanonical measures to Lebesgue measures on skeleta in Berkovich spaces”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Journal de l’École poly- technique — Mathématiques, 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' 287–336.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} +page_content=' Khesin & A.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf'} diff --git a/H9E4T4oBgHgl3EQfIAys/content/tmp_files/2301.04909v1.pdf.txt b/H9E4T4oBgHgl3EQfIAys/content/tmp_files/2301.04909v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c6748124ecc071fa10f30758379d704ad5f4225c --- /dev/null +++ b/H9E4T4oBgHgl3EQfIAys/content/tmp_files/2301.04909v1.pdf.txt @@ -0,0 +1,1008 @@ +SIMPLE LYAPUNOV SPECTRUM FOR LINEAR HOMOGENEOUS +DIFFERENTIAL EQUATIONS WITH Lp PARAMETERS +DINIS AMARO, MÁRIO BESSA, AND HELDER VILARINHO +Abstract. In the present paper we prove that densely, with respect to an Lp-like +topology, the Lyapunov exponents associated to linear continuous-time cocycles +Φ : R× M → GL(2, R) induced by second order linear homogeneous differential +equations ¨x + α(ϕt(ω))˙x + β(ϕt(ω))x = 0 are almost everywhere distinct. The +coefficients α, β evolve along the ϕt-orbit for ω ∈ M and ϕt : M → M is an +ergodic flow defined on a probability space. We also obtain the corresponding +version for the frictionless equation ¨x + β(ϕt(ω))x = 0 and for a Schrödinger +equation ¨x + (E − Q(ϕt(ω)))x = 0, inducing a cocycle Φ : R × M → SL(2, R). +Keywords: Linear cocycles; Linear differential systems; Multiplicative ergodic +theorem; Lyapunov exponents; second order linear homogeneous differential equa- +tions. +2010 Mathematics Subject Classification: Primary: 34D08, 37H15, Secondary: +34A30, 37A20. +1. Introduction +1.1. Non-autonomous linear differential equations. The behaviour of the Lya- +punov exponents which are determined by the asymptotic growth of the expression +log ∥Φt +A∥1/t where Φt +A is a matricial solution of the autonomous differential equa- +tion ˙U(t) = A · U(t) and A is a square matrix of the same order as U(t), is a simple +exercise of linear algebra. Standard linear algebraic computations allows us to de- +termine the Lyapunov spectrum which is defined by the Lyapunov exponents and +its eigendirections. The dynamics of a perturbed system like ˙U(t) = B·U(t), where +B is a perturbation of A, is a problem that is well understood (see e.g. [26]). A much +more complicated and interesting situation was considered in the pioneering works +of Lyapunov and intended to consider the non-autonomous case ˙U(t) = A(t) · U(t), +where A is a matrix depending continuously on t. Not only the asymptotic de- +meanor of log ∥Φt +A∥1/t as well as its stability proves to be a substantially more +difficult issue. A standard way of looking to non-autonomous linear differential +equations is to consider the language of linear cocycles (see §2.1 for full details) +where being non-autonomous is captured by a labelling through an orbit of a given +flow ϕt on a certain phase space. +1.2. The quest for positive Lyapunov exponents. A positive (or negative) Lya- +punov exponent gives us the average exponential rate of divergence (or conver- +gence) of two neighboring trajectories whereas zero exponents give us the ab- +sence of any kind of exponential behavior. +Pesin’s theory guarantee a strong +stable/unstable manifold theory in the presence of non-zero Lyapunov exponents. +These geometric tools underlie much of the central results in today’s dynamical +Date: January 13, 2023. +1 +arXiv:2301.04909v1 [math.DS] 12 Jan 2023 + +systems. Consequently, there is no doubt that detecting non-zero Lyapunov expo- +nents is an important question in dynamics an issue dating back to the late six- +tiees and the work of Millionshchikov [28]. It the early eightees Cornelis and +Wojtkowski [11], and Ledrappier [25] obtained criteria for the positivity of the +Lyapunov exponents and in the nineties Knill [30] and Nerurkar [29] proved that +non-zero Lyapunov exponents are a C0-dense phenomena for certain cocycles. In +the late nineties Arnold and Cong [7] proved the Lp-denseness of positive Lya- +punov exponents and their strategy was widespread in [13] by two of the authors. +Using Moser-type methods based on the concept of rotation number allowed Fab- +bri and Johnson to obtain abundance of positive Lyapunov exponents for linear +differential systems evolving on SL(2, R) and based on a translation on the torus +(see [20, 21, 22] and also the work with Zampogni [23]). Clearly, finding a po- +sitive Lyapunov exponent in SL(2, R) immediately enable us to obtain a negative +Lyapunov exponent and thus the simplicity of the Lyapunov spectrum (i.e. all +Lyapunov exponents are different). Several results on the positivity of Lyapunov +exponents established in the last ten years or so bring up different new approaches +[17, 13, 34, 19, 14, 35]. As a paradigmatic example we recall [10] where Avila +obtained abundance of simple spectrum, on a quite large scope of topologies and +on the two dimensional case. +1.3. Asymptotic behaviour of second order linear homogeneous differential +equations from Lyapunov’s viewpoint. It has been known for almost two cen- +turies that there are serious constraints when we try to apply analytic methods to +integrate most functions. Indeed, Liouville theory (see e.g [32]) explicitly describes +what kind of problems can arise when solving differential equations. The qualita- +tive theory of differential equations created by Poincaré and Lyapunov turn out to +be a clever approach to deal with this setback. Here we intend to analyze the as- +ymptotic behavior of the solutions of second order homogeneous linear differential +equations of the form +¨x(t) + α(ϕt(ω))˙x(t) + β(ϕt(ω))x(t) = 0, +(1) +with coefficients α and β displaying Lp regularity, varying in time along the orbits +of a flow ϕt and allowing an Lp-small perturbation on the parameters. Namely, we +will describe its Lyapunov spectrum taking into account the possibility of making +a Lp-type perturbation on its coefficients. Instead of deal with a single equation +we will consider infinite equations simultaneously as explained now: we consider +a time-continuous cocycle based on an ergodic flow ϕt : M → M with respect to a +probability measure in M and with a dynamics on the fiber defined by a linear flow +Φt +A which is solution of the linear variational equation ˙U(ω, t) = A(ϕt(ω)) · U(ω, t) +with generator +A: +M +−→ +R2×2 +ω +�−→ +� +0 +1 +−β(ω) +−α(ω) +� +(2) +Differential equations like (1) appear in large scale in physics, engineering, com- +plex biological systems and numerous applications of mathematics. The quintes- +sential example is the simple damped pendulum free from external forces where α +and β are functions depending on ω ∈ M evolving along a flow ϕt : M → M for +t ∈ R. When α and β are first integrals (i.e. functions that are constant along the +orbits of the flow ϕt) related with ϕt, then (1) can be solved by simple algorithms of +2 + +an elementary course on differential equations. When the parameters vary in time, +explicit solutions could be hard to get. This is the case when the frictional force α +and the frequency of the oscillator β change over time which, we must admit, is the +most plausible to happen in nature. Notice that generators like A in (2) generate a +particular class of solutions. Clearly, when α � 0 the solutions evolve on a sub- +class of the general linear group GL(2, R) and when α = 0 the solutions evolve on +a subclass of the special linear group SL(2, R). Therefore, a specific study should +be made taking into consideration that perturbations must belong to our class and +not to the wider class of generators of cocycles evolving in GL(2, R) or even in +SL(2, R). Questions related to this particular class were treated in several works +like e.g. [8, 9, 12, 24, 27, 3]. +Fixing position and momentum (x(0), ˙x(0)) we intend to study the asymptotic +behavior when t → ∞ of the pair (x(t), ˙x(t)) namely asymptotic exponential growth +rate given by the Lyapunov exponent. In the present work and broadly speaking we +intend to answer the following question: +Is it possible to perturb the coefficients α and β, in an Lp-topology, +in order to obtain two distinct Lyapunov exponents? +Of course that, when considering the autonomous case in (2), say α and β not +depending on ω previous question is easily answered. Indeed, consider Aβ in (2) +with α = 0, then A0 has a solution with trivial Lyapunov spectrum (a single Lya- +punov exponent equal to 0) but any Aβ with small β � 0 will produce a solution +with simple Lyapunov spectrum (two Lyapunov exponents equal to ± √β). The +difficulty increases significantly when we consider the non-autonomous case. +The precise concepts that allow an adequate formalisation to express the above +question will be presented in Theorem 1 and Corollaries 1 and 2. +2. Definitions and statement of the results +2.1. Linear cocycles. In this section we present some definitions that will be use- +ful in the sequel. Let (M, M, µ) be a probability space and let ϕ: R × M → M be a +metric dynamical system (or flow) in the sense that is a measurable map and +(1) ϕt : M → M given by ϕt(ω) = ϕ(t, ω) preserves the measure µ for all t ∈ R; +(2) ϕ0 = IdM and ϕt+s = ϕt ◦ ϕs for all t, s ∈ R. +Unless stated otherwise we will consider along the text that the flow is ergodic +in the usual sense that there exist no invariant sets except zero measure sets and +their complements. Let B(X) be the Borel σ-algebra of a topological space X. +A (continuous-time) linear random dynamical system (RDS) on (R2, B(R2)), or a +(continuous-time) linear cocycle, over ϕ is a (B(R) ⊗ M/B(GL(2, R))-measurable +map +Φ : R × M → GL(2, R) +such that the mappings Φ(t, ω) forms a cocycle over ϕ, i.e., +(1) Φ(0, ω) = Id for all ω ∈ M; +(2) Φ(t + s, ω) = Φ(t, ϕs(ω)) ◦ Φ(s, ω), for all s, t ∈ R and ω ∈ M, +and t �→ Φ(t, ω) is continuous for all ω ∈ M. We recall that having ω �→ Φ(t, ω) +measurable for each t ∈ R and t �→ Φ(t, ω) continuous for all ω ∈ M implies that +Φ is measurable in the product measure space. These objects are also called linear +differential systems (LDS) in the literature. +3 + +2.2. Kinetic linear cocycles. We begin by considering as motivation the non- +autonomous linear differential equation which describes a motion of the damped +harmonic oscillator as the simple pendulum along the path (ϕt(ω))t∈R, with ω ∈ M +described by the flow ϕ. Let K ⊂ R2×2 be the set of matrices 2 × 2 of type +�0 +1 +b +a +� +(3) +with a, b ∈ R. Denote by G the set of measurable applications A : M → R2×2 +and by K ⊂ G the set of kinetic measurable applications A : M → K. As usual +we identify two applications on G that coincide on a µ full measure subset of M. +Consider measurable maps α: M → R and β: M → R. Take the differential +equation given in (1). Considering y(t) = ˙x(t) we may rewrite (1) as the following +vectorial first order linear system +˙X = A(ϕt(ω)) · X, +(4) +where X = X(t) = (x(t), y(t))T = (x(t), ˙x(t))T and A ∈ K is given by (2). For all +1 ≤ p < ∞ we define +Gp = +� +A ∈ G: +� +M +∥A∥pdµ < ∞ +� +, +where ∥ · ∥ denotes de standard Euclidean matrix norm. It is clear that for all +1 ≤ p < q < ∞ we have Gq ⊂ Gp. It follows from [5, Thm. 2.2.2] (see also +Lemma 2.2.5 and Example 2.2.8 in this reference) that if A ∈ G1 then it generates +a unique (up to indistinguishability) linear RDS ΦA satisfying +ΦA(t, ω) = Id + +� t +0 +A(ϕs(ω)) · ΦA(s, ω) ds. +(5) +The solution ΦA(t, ω) defined in (5) is called the Carathéodory solution or weak +solution. Given an initial condition X(0) = v ∈ R2, we say that t �→ ΦA(t, ω)v +solves or is a solution of (4), or that (4) generates ΦA(t, ω). Note that ΦA(0, ω)v = v +for all ω ∈ M and v ∈ R2. If the solution (5) is differentiable in time (i.e. with +respect to t) and satisfies for all t +d +dtΦA(t, ω)v = A(ϕt(ω)) · ΦA(t, ω)v +and +ΦA(0, ω)v = v, +(6) +then it is called a classical solution of (4). Of course that t �→ ΦA(t, ω)v is con- +tinuous for all ω and v. Due to (6) we call A : M → K a (kinetic) ‘infinitesimal +generator’ of ΦA. Sometimes, due to the relation between A and ΦA, we refer +to both A and ΦA as a kinetic linear cocyle/RDS/LDS. If (4) has initial condition +X(0) = v then ΦA(0, ω)v = v and X(t) = ΦA(t, ω)v. +Let K0 ⊂ K stand for the traceless kinetic cocycles derived from matrices as +in (3) but with a = 0. For 1 ≤ p < ∞ set K p = K ∩ Gp and K p +0 = K0 ∩ Gp ⊂ K p. +2.3. The Lp topology. We begin by defining an Lp-like topology generated by a +metric that compares the infinitesimal generators on G. Given 1 ≤ p < ∞ and +A, B ∈ G we set +ˆσp(A, B) := +����������� +�� +M +∥A(ω) − B(ω)∥p dµ(ω) +� 1 +p +, +∞ if the above integral does not exists, +4 + +and define +σp(A, B) := +������� +ˆσp(A,B) +1+ ˆσp(A,B), +if ˆσp(A, B) < ∞ +1, +if ˆσp(A, B) = ∞ . +Clearly, σp is a distance in G. It can be understood has a version of the Lp-distance. +Next topological content results were mainly proved in [3]. The remaining state- +ments follow straightforwardly. +Proposition 2.1. Consider 1 ≤ p < ∞. Then: +(i) σp(A, B) ≤ σq(A, B) for all 1 ≤ p ≤ q < ∞ and all A, B ∈ G. +(ii) If A ∈ G1 then sup0≤t≤1 log+ ∥ΦA(t, ω)±1∥ ∈ L1(µ). +(iii) If A ∈ Gp then for any B ∈ G satisfying σp(A, B) < p we have B ∈ Gp. +(iv) The sets (K p, σp) and (K p +0 , σp) are closed, for all 1 ≤ p < ∞. +(v) For all 1 ≤ p < ∞, (K p, σp) and (K p +0 , σp) are complete metric spaces +and, therefore Baire spaces. +Next results are elementary in measure theory nevertheless we will use it often. +They capture the whole idea of making huge perturbations on the uniform norm +but small perturbations in the σp-distance as long the support is small in measure. +Lemma 2.2. Let 1 ≤ p < ∞. Given A ∈ Gp and ϵ > 0 there exists δ > 0 such that +if F ∈ M and µ(F ) < δ, then +� +F ∥A(ω)∥p dµ(ω) < ϵ. +Proof. The proof is made by contradiction. Suppose that exists ϵ > 0 and Fn ∈ M, +for each n ∈ N, such that µ(Fn) < 1 +2n and +� +Fn +∥A(ω)∥p dµ(ω) ≥ ϵ. +(7) +Letting F = lim supn Fn, by the Borel-Cantelli lemma µ(F ) = 0, and so +� +F +∥A(ω)∥p dµ(ω) = 0. +(8) +The following leads to a contradiction: +ϵ +(7) +≤ +lim sup +� +Fn +∥A(ω)∥p dµ(ω) = lim sup +� +∥A(ω)∥pχFn(ω) dµ(ω) +⋆≤ +� +lim sup ∥A(ω)∥pχFn(ω) dµ(ω) = +� +∥A(ω)∥pχF (ω) dµ(ω) += +� +F +∥A(ω)∥p dµ(ω) +(8)= 0, +where in ⋆ we used the reverse Fatou lemma. +□ +Corollary 2.3. Let 1 ≤ p < ∞, A ∈ Gp and ϵ > 0 be given. Consider B ∈ Gp such +that A(ω) � B(ω) if and only if ω ∈ F for some F ∈ M (that is, B only differs from +A in F ). Then there exists δ > 0 such that if µ(F ) < δ we have σp(A, B) < ϵ. +Proof. Is is enough to prove that ˆσp(A, B) < ϵ. For that, apply Lemma 2.2 for +(A − B) ∈ Gp and ϵ p. +□ +5 + +2.4. Statement of Theorem 1 and a tour on its proof. Let 1 ≤ p < ∞ and +A ∈ K p. Since K p ⊂ K1 ⊂ G1, from Proposition 2.1 the cocycle ΦA satisfies the +following integrability condition +sup +0≤t≤1 +log+ ∥ΦA(t, ω)±1∥ ∈ L1(µ). +(9) +Hence, under condition (9) Oseledets theorem (see e.g. [31, 5]) guarantees that +for µ almost every ω ∈ M, there exists a ΦA-invariant splitting, called Oseledets +splitting, of the fiber R2 +ω = E1 +ω ⊕ E2 +ω and real numbers λ1(A, ω) ≥ λ2(A, ω), called +Lyapunov exponents, such that: +λ(A, ω, vi) := lim +t→±∞ +1 +t log ∥ΦA(t, ω)vi∥ = λi(A, ω), +for any vi ∈ Ei +ω \ {⃗0} and i = 1, 2. If the flow ϕt is ergodic, then the Lyapunov +exponents (and the dimensions of the associated subbundles) are constant µ almost +everywhere, and we refer to them as λ1(A) and λ2(A), with λ1(A) ≥ λ2(A). We say +that A (or ΦA) has one-point Lyapunov spectrum or trivial Lyapunov spectrum if +for µ a.e. ω ∈ M, λ1(A, ω) = λ2(A, ω). Otherwise we say A (or ΦA) has simple +Lyapunov spectrum. For details on these results see [5] (in particular, Example +3.4.15). +We are now in conditions to state our main result that establishes the existence +of a σp-dense subset of K p displaying simple spectrum: +Theorem 1. Let ϕt : M → M be ergodic. For any 1 ≤ p < ∞, A ∈ K p and ϵ > 0, +there exists B ∈ K p exhibiting simple Lyapunov spectrum satisfying σp(A, B) < ϵ. +This result shows in particular that the σp-generic subset of K p in which the +trivial spectrum prevails, obtained in [3], can not contain σp-open sets. The stra- +tegy to prove that for each kinetic cocycle satisfying the integrability condition +there is another kinetic cocycle, arbitrarily close with a simple spectrum, borrow +some ideas of [7, 13] where the authors obtained a similar result for the discrete +time case and for more general cocycles. However, the context of continuous-time +cocycles and the restriction to a very particular family of cocycles, such as the one +we are considering in this paper, bring several difficulties that have no similarities +in previous works. We have to face the situation that kinetic cocycles are rigid1 +and to obtain the desired perturbation we will make a step-by-step perturbation +algorithm that we now describe: +(1) We begin by coding ϕ by a special flow to avoid overlaps and then consider +a thin time-1 flowbox VR concatenated to an also thin time-1 flowbox VS , +so that o VR ∪ VS will be a time-2 flowbox; +(2) We cut the original dynamics in VR (respectively VS ) and paste a simple +constant traceless infinitesimal generator R2π, whose solution basically ro- +tates an angle 2πη in time-η. Outside VR ∪ VS we keep the same dynamic +of A. By simple we mean that we can easily obtain the identity by just +doing a time-1 iteration. Call A0 this new cocycle; +1 The pertubative arguments in [7, 13] were easier to make because since dim SL(2, R) = 3 three +degrees of freedom were available. In our kinetic scenario we have to perform the same perturbations +but with only a single degree of freedom. +6 + +(3) Since VR ∪ VS is a thin flowbox, A0 will be arbitrarily σp-near A. If A0 +has simple spectrum we are over, otherwise we prove Theorem 1 for A0 +instead of A; +(4) Inside VR we cut the dynamics of A0 and paste a tailor-made rotation R +such that for each ω entering in VR we rotate in time-1 a vector vω into a +fixed special direction given by v = (1, 1). The vector vω will be used to +forcefully create an Oseledets direction so we can calculate the Lyapunov +exponents. Here we rotate any angle by a small σp-perturbation since by +(1) VR is thin. A key observation is that the trace keeps unchanged, and +that is the main motivation to the previous placement of R2π on VR. Call +B0 this new cocycle. If B0 has simple spectrum we are over, otherwise we +prove Theorem 1 for B0 instead of A0; +(5) Inside VS we cut the dynamics of B0 and paste a constant infinitesimal +generator S which stretch the vector v in time-1 by a known magnitude e. +No problem arises with the (eventually large) size of the uniform norm of +the perturbation because the σp-distance is small due to the thickness of +VS . Again the trace keeps unchanged. Call B this new cocycle; +(6) Now we use ergodicity and compute the Lyapunov exponents of points +who will inevitably have to return to VR ∪ VS infinitely many times; +(7) The stretch S is a perturbation that is concerned with providing an expan- +sion along an invariant direction. As it is difficult to find different kinetic +cocycles which keep the same invariant directions here it becomes clear +why we have chosen back there the identity after time 1 (more precisely a +rotation by 2π) given by R2π; +(8) Finally, the concern to keep the trace constant in (4) and (5) will bear fruit +since if a perturbation increases a Lyapunov exponent and simultaneously +the sum of the two Lyapunov exponents of the original cocycle and the +perturbed one remains the same, then only one thing could have happened: +the perturbed cocycle cannot have trivial spectrum but instead must display +a Lyapunov exponent smaller than the Lyapunov exponent of the original +cocycle. +The following table summarises the step-by-step construction from the linear +differential systems A to B: +Table 1. Step-by-step description of the several perturbations +Cocycle +M \ (VR ∪ VS ) +VR +VS +A +A +A +A +A0 +A +R2π +R2π +B0 +A +R +R2π +B +A +R +S +We use an approach slightly different from the previous works [7, 13, 6, 18]. +Moreover, to avoid overlapping in the perturbations, we will encode the base flow +through a special flow in a Kakutani Castle (as in [2, 33]). On the other hand, to +estimate the proximity of the perturbed cocycle to the original one, we also use a +control over the measure of VR ∪VS that support the two perturbations taking into +account Corollary 2.3. +7 + +It should be noted that, in addition to the difficulties inherent in the context +of continuous-time cocycles, performing these perturbations (rotation and stretch) +are not trivial, as we do not have the usual mechanisms like those that exist in +the context in cocycles that evolve in GL(2, R) or SL(2, R), or, more generally, +cocycles that satisfy the accessibility condition (also recognized as twisting) and +saddle-conservative (also known as pinching), which allow the realization of these +processes in a less demanding way, as, for example, in [4, 7, 15, 16, 13]. +As our perturbations are all traceless we get from Theorem 1 that conservative +kinetic cocycles have non-zero Lyapunov exponents σp-densely. +Corollary 1. Let ϕt : M → M be ergodic. For any 1 ≤ p < ∞, A ∈ K p +0 and +ϵ > 0, there exists B ∈ K p +0 exhibiting non-zero Lyapunov exponents satisfying +σp(A, B) < ϵ. +Finally, we present Corollary 1 with a somewhat different look, namely by con- +sidering the one-dimensional Schrödinger operator on L2(R) and with an Lp poten- +tial Q: M → R given by: +Hω : +L2(R) +−→ +L2(R) +φ +�−→ +� +− d2 +dt2 + Q(ϕt(ω)) +� +φ +(10) +In particular we like to describe the Lyapunov spectrum of the time-independent +Schrödinger equation +Hωφ = Eφ, +(11) +where E ∈ R is a given energy. Putting together (10) and (11) we deduce a kinetic +cocycle as in (2) but with α(ω) = 0 and β(ω) = E − Q(ω) for all ω ∈ M. We fix the +energy E and focus on the LDS +AE : +M +−→ +R2×2 +ω +�−→ +� +0 +1 +−E + Q(ω) +0 +� +(12) +called one-dimensional Schrödinger LDS with potential Q. As a direct conse- +quence of Corollary 1 we have: +Corollary 2. Let ϕt : M → M be ergodic. Given 1 ≤ p < ∞, ϵ > 0 and a one- +dimensional Schrödinger LDS with a fixed energy E as in (12) and with potential +Q, there exists ˜Q such that the one-dimensional Schrödinger LDS with the same +energy E and potential ˜Q exhibits non-zero Lyapunov exponents and ∥ ˜Q−Q∥Lp < ϵ. +3. On the perturbations +3.1. Special flows. Consider a measure space Σ, a map T : Σ → Σ, a T -invariant +probability measure ˜µ defined in Σ and a roof function h: Σ → R+ satisfying +h(ω) ≥ H > 0, for some H > 0 and all ω ∈ Σ, and +� +Σ h(ω)d˜µ(ω) < ∞. Define the +space Mh ⊆ Σ × R+ by +Mh = �(ω, t) ∈ Σ × R+ : 0 ≤ t ≤ h(ω)� +with the identification between the pairs (ω, h(ω)) and (T (ω), 0). The semiflow +defined on Mh by S s(ω, r) = (T n(ω), r + s − �n−1 +i=0 h(T i(ω))), where n ∈ N is +uniquely defined by +n−1 +� +i=0 +h(T i(ω)) ≤ r + s < +n +� +i=0 +h(T i(ω)) +8 + +is called a suspension semiflow. If T is invertible then (S t)t is a flow. Furthermore, +if ℓ denotes the one dimensional Lebesgue measure the measure µ = (˜µ×ℓ)/ +� +h d˜µ +defined on Mh by +� +g dµ = +1 +� +h d˜µ +� �� h(ω) +0 +g(ω, t)dt +� +d˜µ(ω), +∀g ∈ C0(Mh) +is a probability measure and it is invariant by the suspension semiflow (S t)t. Flows +with such representation are called special flows (or flows built under a function) +and are denoted by (ϕt, Σ, T , h). It is well-known (see [1, Theorem 2]) that any +ergodic flow is isomorphic to a special flow. Along this work we assume that the +base flow is a special flow (ϕt, Σ, T , h) and, without any loss of generality, that +H > 2. To avoid overloading the notation we write M instead of Mh. +3.2. Perturbations supported in time-τ flowboxes. Take A ∈ G and a non- +periodic orbit ω ∈ M. +We will consider a perturbation B = Bω,τ of A only +along a segment of the orbit of ω with extremes ω and ϕτ(ω) for τ > 0. Let +P ∈ G be given and define B: M → R2×2 such that B( ˆω) = A( ˆω) for all ˆω outside +ϕ[0,τ](ω) = {ϕs(ω) : s ∈ [0, τ]} and B( ˆω) = P( ˆω) otherwise. The map B is called a +(local) perturbation of A by P supported on ϕ[0,τ](ω). Given Σ0 ⊂ Σ and 0 ≤ a < b +we define the set +ϕ[a,b](Σ0) = +� +ϕt(ω): ω ∈ Σ0, t ∈ [a, b] +� +. +Given A ∈ G1, P ∈ G, Σ0 ⊂ Σ and a > 0, we may extend the local perturbations +of A by P to be supported on the flowbox ϕ[a,b](Σ0), with 0 ≤ a < b < H, in the +following way: for ω ∈ ϕ[a,b](Σ0) we project ω in ˜ω ∈ ϕa(Σ0) i.e. ω = ϕr( ˜ω), for +some 0 ≤ r ≤ b−a, and let B ˜ω,b−a be (local) perturbation of A by P = P ˜ω supported +on ϕ[0,b−a]( ˜ω) and define +B(ω) := +� A(ω), +if ω � ϕ[a,b](Σ0) +B ˜ω,b−a(ω), +if ω ∈ ϕ[a,b](Σ0) . +To distinguish the situations we refer for B(ω) as a global perturbation of A by +P supported in ϕ[a,b](Σ0), where we always suppose that P(ω) = P ˜ω(ω) for all +ω ∈ ϕ[a,b](Σ0). +3.3. Rotating and Stretching. Next two results provide local and global argu- +ments to rotate over prescribed directions under a small σp-perturbation. This will +be used to generate a suitable invariant direction. The first one allows us to perform +a uniform bounded kinetic perturbation in a local segment of orbit which rotates +a given vector. The second one thickens Lemma 3.1 by broaden the rotation in a +single orbit to rotations in a flowbox. +Lemma 3.1. Given ω ∈ M, u, v ∈ R2 \ {0}, A ∈ K p, there is γ � 0, and a +perturbation Bω,1 ∈ K p of A supported on ϕ[0,1](ω) such that: +(i) ∥Bω,1( ˆω)∥ ≤ 4π2 for all ˆω on ϕ[0,1](ω), and +(ii) ΦBω,1(1, ω)u = γ v. +Proof. Let θ = ∡(Ru, Rv) ∈ ]0, 2π] measured clockwise. Set a constant infinitesi- +mal generator R: M → R2×2 given by +R(ω) = Rθ(ω) = +� 0 +1 +−θ2 +0 +� +. +(13) +9 + +We consider the perturbation B = Bω,1 ∈ K p of A by R supported on ϕ[0,1](ω). +The infinitesimal generator in (13) generates a linear differential system with fun- +damental classical solution (6) given, for all ω ∈ M and t ∈ R by the ‘clockwise +elliptical rotation’ defined by: +ΦR(t, ω) = +� +cos(θt) +θ−1 sin(θt) +−θ sin(θt) +cos(θt) +� +, +(14) +and such that ΦB(1, ω)u = ΦR(1, ω)u = γv, for some γ � 0 fulfilling (ii). +□ +From Corollary 2.3 it follows that we may extend the local perturbation Bω,1 +given by the rotation Rθ(ω) as in Lemma 3.1, to a global perturbation, tuned for +each orbit segment, to obtain a new generator that is σp-close to the original, once +we have a smaller measure of the flowbox were the perturbation takes place. This +is pointed in the next basic measure theoretic result which is an immediate conse- +quence of Corollary 2.3. +Lemma 3.2 (Global). For all 1 ≤ p < ∞, A ∈ Gp, a > 0 and ϵ > 0, there exists +a measurable set Σ0 ⊂ Σ with ˜µ(Σ0) > 0 such that for any global perturbation +B ∈ Gp of A supported in the flowbox ϕ[a,a+1](Σ0), with ∥B(ϕt(ω))∥ ≤ 4π2 for all +ω ∈ Σ0 and t ∈ [a, a + 1], we have that σp(A, B) < ϵ. +Let us fix a suitable constant and traceless infinitesimal generator +S = +�0 +1 +1 +0 +� +. +(15) +As S has simple expression we integrate it obtaining: +ΦS (t, ω) = eS t = +�cosh t +sinh t +sinh t +cosh t +� +(16) +We notice that (16) has eigenvalues σS +1 = et and σS +2 = e−t with associated eigen- +vectors vS +1 = (1, 1) and vS +2 = (−1, 1), respectively. Observe that ES +1 = R · vS +1 is a +unstable direction and ES +2 = R · vS +2 is a stable direction. +Next trivial remark will be of utmost importance in the sequel because it com- +bines three main ingredients: invariance of certain 1-dimensional directions, some +expansiveness along this direction and all this done in traceless kinetic infinitesi- +mal generators. +Remark 3.1 (Invariance and stretch). Considering θ = 2π in (14), say R2π, we get +e · vS +1 = e · ΦR2π(1, ω) vS +1 = ΦS (1, ω) vS +1 . +(17) +4. Proof of Theorem 1 +Let A ∈ K p, 1 ≤ p < ∞ and ϵ > 0 be given. We assume that ΦA has a single +Lyapunov exponent λ(A). The sequence of perturbations are summarized in Table +1. +10 + +4.1. Defining A0 (picking out good coordinates): Let Σ0 ⊂ Σ be as in Lemma 3.2. +For r > 0 we assume that we have flowboxes defined by VR := ϕ[0,1](Br) and +VS := ϕ[1,2](Br), where Br ∈ Σ0 is such that 0 < ˜µ(Br) ≤ r. Consider A0 ∈ K1 +defined as: +A0(ω) := +� A(ω), +if ω � VR ∪ VS +R2π, +if ω ∈ VR ∪ VS +. +By Corollary 2.3 if r is sufficiently small when compared with ϵ we get +σp(A, A0) < ϵ +3. +(18) +If ΦA0 has simple spectrum we are over. Otherwise, we prove the theorem for A0 +instead of A. +4.2. Defining B0 (rotating on VR): Set +k(ω) = inf +t≥0 +� +t: ϕ−t(ω) ∈ ϕ1(Br) +� +. +We will define the a random vector field g(ω). We start with the normalized image +under the cocycle associated with ΦA0 of the vector v = +vS +1 +∥vS +1 ∥ = +� √ +2 +2 , +√ +2 +2 +� +: +g(ω) := +��������� +v, +if +ω ∈ ϕ1(Br) +ΦA0(k(ω),ϕ−k(ω)(ω))v +∥ΦA0(k(ω),ϕ−k(ω)(ω))v∥, +if +ω � (VR \ Br) +and set from now on E(ω) = span {g(ω)}. +Let B0 be a perturbation of A0 supported in the flowbox VR as in Lemma 3.2 +such that for all ω ∈ Br we have ΦB0(1, ω)g(ω) = κv for some κ ∈ R, that is: +B0(ω) := +� R(ω), +if ω ∈ VR +A0(ω), +otherwise +. +Observe that the rotation must be tuned for each ω0 ∈ Br, in the sense that for +ω = ϕt(ω0) ∈ VR, with 0 ≤ t ≤ 1, we set R(ω) = Rθ(ω0) with θ = ∡(g(ω0), v). In +particular, for all ω0 ∈ Br we have Φ(1, ω0)g(ω) = κv, for some κ ∈ R. Moreover, +A0 and B0 have the same trace. Indeed, A0 = B0 outside VR and in VR we have +B0 = R and A0 = R2π, which are both traceless (see (13)). Therefore, by Liouville’s +formula for all ω and t ≥ 0 +det ΦB0(t, ω) = det ΦA0(t, ω). +(19) +For ω ∈ VR \ Br define +g(ω) = ΦB0(k(ω), ϕ−k(ω)(ω))v +∥ΦB0(k(ω), ϕ−k(ω)(ω))v∥. +(20) +Notice that for ω ∈ Br, since ΦB0(1, ω)Rg(ω) = Rv we get +ΦB0(1, ω)Rg(ω)(ω) = Rg(ϕ1(ω)). +(21) +Let ˜ω ∈ ϕ1(Br) and τ > 0 be such that ϕt( ˜ω) � VR for all t ∈]0, τ[. Then, for all +t ∈ [0, τ] we have the ΦB0-invariance of g: +ΦB0(t, ˜ω)Rg( ˜ω) = ΦB0(t, ˜ω)Rv = ΦA0(t, ˜ω)Rv = Rg(ϕt( ˜ω)). +(22) +11 + +If ϕt( ˜ω) ∈ VR for some t ∈]0, τ[ then considering s > 0 such that ϕs(ω) ∈ Br we +get: +ΦB0(t, ˜ω)Rg( ˜ω) += +ΦB0(t − s, ϕs( ˜ω))ΦA0(s, ˜ω)Rv += +ΦB0(t − s, ϕs( ˜ω))Rg(ϕs( ˜ω)) +(20) += +Rg(ϕt( ˜ω)). +Finally, (21), (26) and last equality gives that the vector field g is ΦB0-invariant. +Again by Corollary 2.3 if r is sufficiently small we get +σp(A0, B0) < ϵ +3. +(23) +If ΦB0 has simple spectrum we are over. Otherwise, we prove the theorem for +B0 instead of A0. +4.3. Defining B (stretching on VS ): We define +B(ω) := +� B0(ω), +if ω � VS +S, +if ω ∈ VS +. +Observe that B and B0 have the same trace. Indeed, B = B0 outside VS and in +VS we have B0 = R2π which are both traceless (see (13) and (15)). Therefore, by +Liouville’s formula and (19) for all ω and t ≥ 0 +det ΦB(t, ω) = det ΦB0(t, ω) = det ΦA0(t, ω). +(24) +From Corollary 2.3, once more, if r is sufficiently small we get +σp(B0, B) < ϵ +3. +(25) +Notice that the invariance of the direction E(ω) under ΦB fails when ϕt(ω) enters +VS . However, for ˜ω ∈ ϕ1(Br) we have by (17) and (26) +ΦB(1, ˜ω)Rg( ˜ω) = ΦS (1, ˜ω)Rv = Rv = RΦR2π(1, ˜ω)v = ΦA0(1, ˜ω)Rv = Rg(ϕ1( ˜ω)) +and so +ΦS (1, ˜ω)E( ˜ω) = E(ϕ1( ˜ω)), +(26) +which will be enough for our purposes; see Figure 1. +Figure 1. The traceless perturbation scheme with the invariant di- +rections and the stretch effect. +Let λ1(B) ≥ λ2(B) be the Lyapunov exponents of ΦB. We assume that ΦB0 has +one-point spectrum, say λ1(B0) = λ2(B0) = λ(B0), because otherwise the theorem +12 + +dB(T, W)g() +ΦB(1, +B(1, T()) +P Bo +3is proved. Let λ(B0) be the single Lyapunov exponent of ΦB0. Hence we have +λ(B0) = λ(B0, ω, vS +1 ) for a.e. ω. By the Oseledets theorem we have +2λ(B0) = +� +log +���det(ΦB0(1, ω)) +���dµ +(27) +and +λ1(B) + λ2(B) = +� +log +���det(ΦB(1, ω)) +���dµ. +(28) +The two previous equalities together with (24) allows us to conclude that +2λ(B0) = λ1(B) + λ2(B) +(29) +and so, if we show that λ1(B) > λ(B0) then we get λ1(B) > λ2(B) and Theorem 1 +is proved. Recall that the random vector field g is invariant by ΦB0 but in what ΦB +concerns, the invariance fails as the base dynamics enters VS . However, by (26) +the invariance is recovered in the moment the base dynamics is leaving VS . +For ω ∈ M let us consider the real map b0(·, ω) for all t ∈ R in such a way that +b0(t, ω)g(ϕt(ω)) = ΦB0(t, ω)g(ω). +(30) +Claim 4.1. The map b0(t, ω) forms a cocycle over ϕt. +Indeed, since ΦB0(0, ω) = Id for all ω ∈ M we have b0(0, ω) = 1 and for all s, t, +evaluating b0(t + s, ω) at g(ϕt+s(ω)), we have +b0(t + s, ω)g(ϕt+s(ω)) +(30) += +ΦB0(t + s, ω)g(ω) += +ΦB0(t, ϕs(ω)) · ΦB0(s, ω)g(ω) +(30) += +ΦB0(t, ϕs(ω)) · b0(s, ω)g(ϕs(ω)) += +b0(s, ω) ΦB0(t, ϕs(ω))g(ϕs(ω)) += +b0(t, ϕs(ω))b0(s, ω)g(ϕt+s(ω)), +and so b0(t + s, ω) = b0(t, ϕs(ω))b0(s, ω). +Since the random vector field g is not completely invariant by ΦB we consider +two distinct situations. Set ϕ{1,2}(Br) = ϕ1(Br) ∪ ϕ2(Br). For ω ∈ M and τ ≥ 0 such +that ϕt(ω) � VS \ ϕ{1,2}(Br), for all 0 ≤ t ≤ τ, we consider the real map b(·, ω) for +all t ∈ [0, τ] in such a way that +b(t, ω)g(ϕt(ω)) = ΦB(t, ω)g(ω) +(31) +and, for all ω ∈ ϕ1(Br), we set b(1, ω) ∈ R in such a way that +ΦB(1, ω)g(ω) = b(1, ω)g(ϕ1(ω)). +(32) +If ϕt(ω) � VS \ ϕ{1,2}(Br), for all 0 ≤ t ≤ τ, we have B(ϕt(ω)) = B0(ϕt(ω)) and +b(t, ω)g(ϕt(ω)) = ΦB(t, ω)g(ω) = ΦB0(t, ω)g(ω) = b0(t, ω)g(ϕt(ω)). +(33) +In particular this holds between the output of VS to the next input in VS . +Claim 4.2. If ϕt(ω), ϕs(ω) � VS \ ϕ{1,2}(Br), b(t, ω) forms a cocycle over ϕt in the +sense that b(t + s, ω) = b(t, ϕs(ω))b(s, ω). +13 + +The proof follows similarly to Claim 4.1 taking also into account (32). +Pick ω in a full measure subset of points that visits infinitely often Br and for +which the conclusion of Birkhoff’s Ergodic theorem holds. Without loss of gene- +rality we may assume that ω � Vr ∪ VS . For t ≥ 0 set +Jt(ω) = # +� +j ∈ N: j ≤ t, ϕ j(ω) ∈ ϕ2(Br) +� +. +Recall that +λ(B, ω, g(ω)) += +lim +t→∞ +1 +t log ∥ΦB(t, ω)g(ω)∥, +and we may split the previous orbit in the limit by considering the time for ϕt(ω) +to enter VS , the time-1 moment crossing the flowbox VS , where we use (17), and, +again, the time it takes to return to VS and so on. For simplicity, let us define +recursively +s0 = s0(ω) = min{t: ϕt(ω) ∈ ϕ1(Br)}, +ℓ0 = ℓ0(ω) = s0 + 1, +sn = s0(ϕℓn−1(ω)) and ℓn = sn + 1, for n ≥ 1, +∆n = sn − ℓn−1, for n ≥ 1, +˜ωn = ϕsn(ω) ∈ ϕ1(Br) and ˆωn = ϕℓn(ω) ∈ ϕ2(Br), for n ≥ 1. +Now, in one hand, since B0 has one-point spectrum, for µ-a.e. ω, +λ(B0, ω) += +λ(B0, ω, g(ω)) += +lim +t→∞ +1 +t log ∥ΦB0(t, ω)g(ω)∥ +(30) += +lim +t→∞ +1 +t log |b0(t, ω)|. +(34) +On the other hand, by Remark 3.1 and (32) we have for ˜ω ∈ ϕ1(Br) that +b(1, ˜ω)g(ϕ1( ˜ω)) = ΦB(1, ˜ω)g( ˜ω) +(17) += e·ΦB0(1, ˜ω)g( ˜ω) = e·b0(1, ˜ω)g(ϕ1( ˜ω)). (35) +Without loss of generality, we can consider the following limits over the un- +bounded set {t ≥ 0: ϕt(ω) ∈ ϕ1(Br)}. From Birkhoff’s Ergodic theorem we have +λ(B, ω, g(ω)) += +lim +t→∞ +1 +t log ∥ΦB(t, ω)g(ω)∥ +(31)+(32) += +lim +t→∞ +1 +t +���������log |b(s0, ω)| + +Jt(ω)−1 +� +j=0 +log |b(∆j+1, ˆωsj)b(1, ˜ωsj)| +��������� +(33)+(35) += +lim +t→∞ +1 +t +���������log |b0(s0, ω)| + +Jt(ω)−1 +� +j=0 +log |b0(∆j+1, ˆωsj) · e · b0(1, ˜ωsj)| +��������� +Claim 4.1 += +lim +t→∞ +1 +t log |b0(t, ω)| + lim +t→∞ +Jt(ω) +t +(30) += +lim +t→∞ +1 +t log ∥ΦB0(t, ω)g(ω)∥ + lim +t→∞ +1 +t +� t +0 +1VS (ϕt(ω)) dt += +λ(B0, ω, g(ω)) + µ(VS ), +which implies λ1(B, ω) > λ(B0, ω), hence λ1(B) > λ(B0). From (29), we get +λ1(B) > λ(B0) > λ2(B) so that B has simple spectrum. Moreover, by (18), (23) and +(25) we have σp(A, B) < ϵ and Theorem 1 is now proved. □ +14 + +Clearly when considering the set K1 +0 on Corollary 1 the equalities (27) and (28) +become 2λ(B0) = λ1(B) + λ2(B) = 0. Hence the conclusion this time will be that +λ1(B) > 0 for B ∈ K1 +0 arbitrarily σp-close to A and also λ2(B) = −λ1(B) < 0. +Acknowledgements: The authors were partially supported by FCT - ‘Fundação +para a Ciência e a Tecnologia’, through Centro de Matemática e Aplicações (CMA- +UBI), Universidade da Beira Interior, project UIDB/MAT/00212/2020. MB was +partially supported by the Project ‘Means and Extremes in Dynamical Systems’ +(PTDC/MAT-PUR/4048/2021). MB also like to thank CMUP for providing the +necessary conditions in which this work was developed. +References +[1] W. Ambrose, Representation of ergodic flows, Annals of Mathematics 42 (1941), 3, 723–739. +[2] W. Ambrose, S. Kakutani, Structure and continuity of measure preserving transformations, +Duke Math. J., 9: (1942), 25–42. +[3] D. Amaro, M. Bessa, H. Vilarinho Genericity of trivial Lyapunov spectrum for Lp-cocycles +derived from second order linear homogeneous differential equations (Submitted). +[4] A. Arbieto, J. 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Viana, Almost all cocycles over any hyperbolic system have nonvanishing Lyapunov expo- +nents, Ann. of Math. 167 (2) (2008) 643–680. +[35] D. Xu, Density of positive Lyapunov exponents for symplectic cocycles, J. Eur. Math. Soc., 21, +10, (2019), 3143–3190. +Centro de Matem´atica e Aplica¸c˜oes (CMA-UBI), Universidade da Beira Interior, Rua Marquˆes +d’Ávila e Bolama, 6201-001, Covilh˜a, Portugal. +Email address: dinis.amaro@ubi.pt +Email address: bessa@ubi.pt +Email address: helder@ubi.pt +16 + diff --git a/H9E4T4oBgHgl3EQfIAys/content/tmp_files/load_file.txt b/H9E4T4oBgHgl3EQfIAys/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9c0e3279c3a5bf83b41338be819fff7b925f0fe4 --- /dev/null +++ b/H9E4T4oBgHgl3EQfIAys/content/tmp_files/load_file.txt @@ -0,0 +1,587 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf,len=586 +page_content='SIMPLE LYAPUNOV SPECTRUM FOR LINEAR HOMOGENEOUS DIFFERENTIAL EQUATIONS WITH Lp PARAMETERS DINIS AMARO, MÁRIO BESSA, AND HELDER VILARINHO Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' In the present paper we prove that densely, with respect to an Lp-like topology, the Lyapunov exponents associated to linear continuous-time cocycles Φ : R× M → GL(2, R) induced by second order linear homogeneous differential equations ¨x + α(ϕt(ω))˙x + β(ϕt(ω))x = 0 are almost everywhere distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' The coefficients α, β evolve along the ϕt-orbit for ω ∈ M and ϕt : M → M is an ergodic flow defined on a probability space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' We also obtain the corresponding version for the frictionless equation ¨x + β(ϕt(ω))x = 0 and for a Schrödinger equation ¨x + (E − Q(ϕt(ω)))x = 0, inducing a cocycle Φ : R × M → SL(2, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Keywords: Linear cocycles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Linear differential systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Multiplicative ergodic theorem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Lyapunov exponents;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' second order linear homogeneous differential equa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' 2010 Mathematics Subject Classification: Primary: 34D08, 37H15, Secondary: 34A30, 37A20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Non-autonomous linear differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' The behaviour of the Lya- punov exponents which are determined by the asymptotic growth of the expression log ∥Φt A∥1/t where Φt A is a matricial solution of the autonomous differential equa- tion ˙U(t) = A · U(t) and A is a square matrix of the same order as U(t), is a simple exercise of linear algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Standard linear algebraic computations allows us to de- termine the Lyapunov spectrum which is defined by the Lyapunov exponents and its eigendirections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' The dynamics of a perturbed system like ˙U(t) = B·U(t), where B is a perturbation of A, is a problem that is well understood (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' [26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' A much more complicated and interesting situation was considered in the pioneering works of Lyapunov and intended to consider the non-autonomous case ˙U(t) = A(t) · U(t), where A is a matrix depending continuously on t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Not only the asymptotic de- meanor of log ∥Φt A∥1/t as well as its stability proves to be a substantially more difficult issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' A standard way of looking to non-autonomous linear differential equations is to consider the language of linear cocycles (see §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='1 for full details) where being non-autonomous is captured by a labelling through an orbit of a given flow ϕt on a certain phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' The quest for positive Lyapunov exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' A positive (or negative) Lya- punov exponent gives us the average exponential rate of divergence (or conver- gence) of two neighboring trajectories whereas zero exponents give us the ab- sence of any kind of exponential behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Pesin’s theory guarantee a strong stable/unstable manifold theory in the presence of non-zero Lyapunov exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' These geometric tools underlie much of the central results in today’s dynamical Date: January 13, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='04909v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='DS] 12 Jan 2023 systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Consequently, there is no doubt that detecting non-zero Lyapunov expo- nents is an important question in dynamics an issue dating back to the late six- tiees and the work of Millionshchikov [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' It the early eightees Cornelis and Wojtkowski [11], and Ledrappier [25] obtained criteria for the positivity of the Lyapunov exponents and in the nineties Knill [30] and Nerurkar [29] proved that non-zero Lyapunov exponents are a C0-dense phenomena for certain cocycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' In the late nineties Arnold and Cong [7] proved the Lp-denseness of positive Lya- punov exponents and their strategy was widespread in [13] by two of the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Using Moser-type methods based on the concept of rotation number allowed Fab- bri and Johnson to obtain abundance of positive Lyapunov exponents for linear differential systems evolving on SL(2, R) and based on a translation on the torus (see [20, 21, 22] and also the work with Zampogni [23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Clearly, finding a po- sitive Lyapunov exponent in SL(2, R) immediately enable us to obtain a negative Lyapunov exponent and thus the simplicity of the Lyapunov spectrum (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' all Lyapunov exponents are different).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Several results on the positivity of Lyapunov exponents established in the last ten years or so bring up different new approaches [17, 13, 34, 19, 14, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' As a paradigmatic example we recall [10] where Avila obtained abundance of simple spectrum, on a quite large scope of topologies and on the two dimensional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Asymptotic behaviour of second order linear homogeneous differential equations from Lyapunov’s viewpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' It has been known for almost two cen- turies that there are serious constraints when we try to apply analytic methods to integrate most functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Indeed, Liouville theory (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='g [32]) explicitly describes what kind of problems can arise when solving differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' The qualita- tive theory of differential equations created by Poincaré and Lyapunov turn out to be a clever approach to deal with this setback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Here we intend to analyze the as- ymptotic behavior of the solutions of second order homogeneous linear differential equations of the form ¨x(t) + α(ϕt(ω))˙x(t) + β(ϕt(ω))x(t) = 0, (1) with coefficients α and β displaying Lp regularity, varying in time along the orbits of a flow ϕt and allowing an Lp-small perturbation on the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Namely, we will describe its Lyapunov spectrum taking into account the possibility of making a Lp-type perturbation on its coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Instead of deal with a single equation we will consider infinite equations simultaneously as explained now: we consider a time-continuous cocycle based on an ergodic flow ϕt : M → M with respect to a probability measure in M and with a dynamics on the fiber defined by a linear flow Φt A which is solution of the linear variational equation ˙U(ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' t) = A(ϕt(ω)) · U(ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' t) with generator A: M −→ R2×2 ω �−→ � 0 1 −β(ω) −α(ω) � (2) Differential equations like (1) appear in large scale in physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' com- plex biological systems and numerous applications of mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' The quintes- sential example is the simple damped pendulum free from external forces where α and β are functions depending on ω ∈ M evolving along a flow ϕt : M → M for t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' When α and β are first integrals (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' functions that are constant along the orbits of the flow ϕt) related with ϕt, then (1) can be solved by simple algorithms of 2 an elementary course on differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' When the parameters vary in time, explicit solutions could be hard to get.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' This is the case when the frictional force α and the frequency of the oscillator β change over time which, we must admit, is the most plausible to happen in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Notice that generators like A in (2) generate a particular class of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Clearly, when α � 0 the solutions evolve on a sub- class of the general linear group GL(2, R) and when α = 0 the solutions evolve on a subclass of the special linear group SL(2, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Therefore, a specific study should be made taking into consideration that perturbations must belong to our class and not to the wider class of generators of cocycles evolving in GL(2, R) or even in SL(2, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Questions related to this particular class were treated in several works like e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' [8, 9, 12, 24, 27, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Fixing position and momentum (x(0), ˙x(0)) we intend to study the asymptotic behavior when t → ∞ of the pair (x(t), ˙x(t)) namely asymptotic exponential growth rate given by the Lyapunov exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' In the present work and broadly speaking we intend to answer the following question: Is it possible to perturb the coefficients α and β, in an Lp-topology, in order to obtain two distinct Lyapunov exponents?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Of course that, when considering the autonomous case in (2), say α and β not depending on ω previous question is easily answered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Indeed, consider Aβ in (2) with α = 0, then A0 has a solution with trivial Lyapunov spectrum (a single Lya- punov exponent equal to 0) but any Aβ with small β � 0 will produce a solution with simple Lyapunov spectrum (two Lyapunov exponents equal to ± √β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' The difficulty increases significantly when we consider the non-autonomous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' The precise concepts that allow an adequate formalisation to express the above question will be presented in Theorem 1 and Corollaries 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Definitions and statement of the results 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Linear cocycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' In this section we present some definitions that will be use- ful in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Let (M, M, µ) be a probability space and let ϕ: R × M → M be a metric dynamical system (or flow) in the sense that is a measurable map and (1) ϕt : M → M given by ϕt(ω) = ϕ(t, ω) preserves the measure µ for all t ∈ R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' (2) ϕ0 = IdM and ϕt+s = ϕt ◦ ϕs for all t, s ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Unless stated otherwise we will consider along the text that the flow is ergodic in the usual sense that there exist no invariant sets except zero measure sets and their complements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Let B(X) be the Borel σ-algebra of a topological space X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' A (continuous-time) linear random dynamical system (RDS) on (R2, B(R2)), or a (continuous-time) linear cocycle, over ϕ is a (B(R) ⊗ M/B(GL(2, R))-measurable map Φ : R × M → GL(2, R) such that the mappings Φ(t, ω) forms a cocycle over ϕ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=', (1) Φ(0, ω) = Id for all ω ∈ M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' (2) Φ(t + s, ω) = Φ(t, ϕs(ω)) ◦ Φ(s, ω), for all s, t ∈ R and ω ∈ M, and t �→ Φ(t, ω) is continuous for all ω ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' We recall that having ω �→ Φ(t, ω) measurable for each t ∈ R and t �→ Φ(t, ω) continuous for all ω ∈ M implies that Φ is measurable in the product measure space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' These objects are also called linear differential systems (LDS) in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Kinetic linear cocycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' We begin by considering as motivation the non- autonomous linear differential equation which describes a motion of the damped harmonic oscillator as the simple pendulum along the path (ϕt(ω))t∈R, with ω ∈ M described by the flow ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Let K ⊂ R2×2 be the set of matrices 2 × 2 of type �0 1 b a � (3) with a, b ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Denote by G the set of measurable applications A : M → R2×2 and by K ⊂ G the set of kinetic measurable applications A : M → K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' As usual we identify two applications on G that coincide on a µ full measure subset of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Consider measurable maps α: M → R and β: M → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Take the differential equation given in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Considering y(t) = ˙x(t) we may rewrite (1) as the following vectorial first order linear system ˙X = A(ϕt(ω)) · X, (4) where X = X(t) = (x(t), y(t))T = (x(t), ˙x(t))T and A ∈ K is given by (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' For all 1 ≤ p < ∞ we define Gp = � A ∈ G: � M ∥A∥pdµ < ∞ � , where ∥ · ∥ denotes de standard Euclidean matrix norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' It is clear that for all 1 ≤ p < q < ∞ we have Gq ⊂ Gp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' It follows from [5, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='2] (see also Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='5 and Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='8 in this reference) that if A ∈ G1 then it generates a unique (up to indistinguishability) linear RDS ΦA satisfying ΦA(t, ω) = Id + � t 0 A(ϕs(ω)) · ΦA(s, ω) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' (5) The solution ΦA(t, ω) defined in (5) is called the Carathéodory solution or weak solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Given an initial condition X(0) = v ∈ R2, we say that t �→ ΦA(t, ω)v solves or is a solution of (4), or that (4) generates ΦA(t, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Note that ΦA(0, ω)v = v for all ω ∈ M and v ∈ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' If the solution (5) is differentiable in time (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' with respect to t) and satisfies for all t d dtΦA(t, ω)v = A(ϕt(ω)) · ΦA(t, ω)v and ΦA(0, ω)v = v, (6) then it is called a classical solution of (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Of course that t �→ ΦA(t, ω)v is con- tinuous for all ω and v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Due to (6) we call A : M → K a (kinetic) ‘infinitesimal generator’ of ΦA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Sometimes, due to the relation between A and ΦA, we refer to both A and ΦA as a kinetic linear cocyle/RDS/LDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' If (4) has initial condition X(0) = v then ΦA(0, ω)v = v and X(t) = ΦA(t, ω)v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Let K0 ⊂ K stand for the traceless kinetic cocycles derived from matrices as in (3) but with a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' For 1 ≤ p < ∞ set K p = K ∩ Gp and K p 0 = K0 ∩ Gp ⊂ K p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' The Lp topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' We begin by defining an Lp-like topology generated by a metric that compares the infinitesimal generators on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Given 1 ≤ p < ∞ and A, B ∈ G we set ˆσp(A, B) := ����������� �� M ∥A(ω) − B(ω)∥p dµ(ω) � 1 p , ∞ if the above integral does not exists, 4 and define σp(A, B) := ������� ˆσp(A,B) 1+ ˆσp(A,B), if ˆσp(A, B) < ∞ 1, if ˆσp(A, B) = ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Clearly, σp is a distance in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' It can be understood has a version of the Lp-distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Next topological content results were mainly proved in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' The remaining state- ments follow straightforwardly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Consider 1 ≤ p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Then: (i) σp(A, B) ≤ σq(A, B) for all 1 ≤ p ≤ q < ∞ and all A, B ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' (ii) If A ∈ G1 then sup0≤t≤1 log+ ∥ΦA(t, ω)±1∥ ∈ L1(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' (iii) If A ∈ Gp then for any B ∈ G satisfying σp(A, B) < p we have B ∈ Gp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' (iv) The sets (K p, σp) and (K p 0 , σp) are closed, for all 1 ≤ p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' (v) For all 1 ≤ p < ∞, (K p, σp) and (K p 0 , σp) are complete metric spaces and, therefore Baire spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Next results are elementary in measure theory nevertheless we will use it often.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' They capture the whole idea of making huge perturbations on the uniform norm but small perturbations in the σp-distance as long the support is small in measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Let 1 ≤ p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Given A ∈ Gp and ϵ > 0 there exists δ > 0 such that if F ∈ M and µ(F ) < δ, then � F ∥A(ω)∥p dµ(ω) < ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' The proof is made by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Suppose that exists ϵ > 0 and Fn ∈ M, for each n ∈ N, such that µ(Fn) < 1 2n and � Fn ∥A(ω)∥p dµ(ω) ≥ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' (7) Letting F = lim supn Fn, by the Borel-Cantelli lemma µ(F ) = 0, and so � F ∥A(ω)∥p dµ(ω) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' (8) The following leads to a contradiction: ϵ (7) ≤ lim sup � Fn ∥A(ω)∥p dµ(ω) = lim sup � ∥A(ω)∥pχFn(ω) dµ(ω) ⋆≤ � lim sup ∥A(ω)∥pχFn(ω) dµ(ω) = � ∥A(ω)∥pχF (ω) dµ(ω) = � F ∥A(ω)∥p dµ(ω) (8)= 0, where in ⋆ we used the reverse Fatou lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' □ Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Let 1 ≤ p < ∞, A ∈ Gp and ϵ > 0 be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Consider B ∈ Gp such that A(ω) � B(ω) if and only if ω ∈ F for some F ∈ M (that is, B only differs from A in F ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Then there exists δ > 0 such that if µ(F ) < δ we have σp(A, B) < ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Is is enough to prove that ˆσp(A, B) < ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' For that, apply Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='2 for (A − B) ∈ Gp and ϵ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' □ 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Statement of Theorem 1 and a tour on its proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Let 1 ≤ p < ∞ and A ∈ K p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Since K p ⊂ K1 ⊂ G1, from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='1 the cocycle ΦA satisfies the following integrability condition sup 0≤t≤1 log+ ∥ΦA(t, ω)±1∥ ∈ L1(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' (9) Hence, under condition (9) Oseledets theorem (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' [31, 5]) guarantees that for µ almost every ω ∈ M, there exists a ΦA-invariant splitting, called Oseledets splitting, of the fiber R2 ω = E1 ω ⊕ E2 ω and real numbers λ1(A, ω) ≥ λ2(A, ω), called Lyapunov exponents, such that: λ(A, ω, vi) := lim t→±∞ 1 t log ∥ΦA(t, ω)vi∥ = λi(A, ω), for any vi ∈ Ei ω \\ {⃗0} and i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' If the flow ϕt is ergodic, then the Lyapunov exponents (and the dimensions of the associated subbundles) are constant µ almost everywhere, and we refer to them as λ1(A) and λ2(A), with λ1(A) ≥ λ2(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' We say that A (or ΦA) has one-point Lyapunov spectrum or trivial Lyapunov spectrum if for µ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' ω ∈ M, λ1(A, ω) = λ2(A, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Otherwise we say A (or ΦA) has simple Lyapunov spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' For details on these results see [5] (in particular, Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' We are now in conditions to state our main result that establishes the existence of a σp-dense subset of K p displaying simple spectrum: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Let ϕt : M → M be ergodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' For any 1 ≤ p < ∞, A ∈ K p and ϵ > 0, there exists B ∈ K p exhibiting simple Lyapunov spectrum satisfying σp(A, B) < ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' This result shows in particular that the σp-generic subset of K p in which the trivial spectrum prevails, obtained in [3], can not contain σp-open sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' The stra- tegy to prove that for each kinetic cocycle satisfying the integrability condition there is another kinetic cocycle, arbitrarily close with a simple spectrum, borrow some ideas of [7, 13] where the authors obtained a similar result for the discrete time case and for more general cocycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' However, the context of continuous-time cocycles and the restriction to a very particular family of cocycles, such as the one we are considering in this paper, bring several difficulties that have no similarities in previous works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' We have to face the situation that kinetic cocycles are rigid1 and to obtain the desired perturbation we will make a step-by-step perturbation algorithm that we now describe: (1) We begin by coding ϕ by a special flow to avoid overlaps and then consider a thin time-1 flowbox VR concatenated to an also thin time-1 flowbox VS , so that o VR ∪ VS will be a time-2 flowbox;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' (2) We cut the original dynamics in VR (respectively VS ) and paste a simple constant traceless infinitesimal generator R2π, whose solution basically ro- tates an angle 2πη in time-η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Outside VR ∪ VS we keep the same dynamic of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' By simple we mean that we can easily obtain the identity by just doing a time-1 iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Call A0 this new cocycle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' 1 The pertubative arguments in [7, 13] were easier to make because since dim SL(2, R) = 3 three degrees of freedom were available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' In our kinetic scenario we have to perform the same perturbations but with only a single degree of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' 6 (3) Since VR ∪ VS is a thin flowbox, A0 will be arbitrarily σp-near A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' If A0 has simple spectrum we are over, otherwise we prove Theorem 1 for A0 instead of A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' (4) Inside VR we cut the dynamics of A0 and paste a tailor-made rotation R such that for each ω entering in VR we rotate in time-1 a vector vω into a fixed special direction given by v = (1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' The vector vω will be used to forcefully create an Oseledets direction so we can calculate the Lyapunov exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Here we rotate any angle by a small σp-perturbation since by (1) VR is thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' A key observation is that the trace keeps unchanged, and that is the main motivation to the previous placement of R2π on VR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Call B0 this new cocycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' If B0 has simple spectrum we are over, otherwise we prove Theorem 1 for B0 instead of A0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' (5) Inside VS we cut the dynamics of B0 and paste a constant infinitesimal generator S which stretch the vector v in time-1 by a known magnitude e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' No problem arises with the (eventually large) size of the uniform norm of the perturbation because the σp-distance is small due to the thickness of VS .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Again the trace keeps unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Call B this new cocycle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' (6) Now we use ergodicity and compute the Lyapunov exponents of points who will inevitably have to return to VR ∪ VS infinitely many times;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' (7) The stretch S is a perturbation that is concerned with providing an expan- sion along an invariant direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' As it is difficult to find different kinetic cocycles which keep the same invariant directions here it becomes clear why we have chosen back there the identity after time 1 (more precisely a rotation by 2π) given by R2π;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' (8) Finally, the concern to keep the trace constant in (4) and (5) will bear fruit since if a perturbation increases a Lyapunov exponent and simultaneously the sum of the two Lyapunov exponents of the original cocycle and the perturbed one remains the same, then only one thing could have happened: the perturbed cocycle cannot have trivial spectrum but instead must display a Lyapunov exponent smaller than the Lyapunov exponent of the original cocycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' The following table summarises the step-by-step construction from the linear differential systems A to B: Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Step-by-step description of the several perturbations Cocycle M \\ (VR ∪ VS ) VR VS A A A A A0 A R2π R2π B0 A R R2π B A R S We use an approach slightly different from the previous works [7, 13, 6, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Moreover, to avoid overlapping in the perturbations, we will encode the base flow through a special flow in a Kakutani Castle (as in [2, 33]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' On the other hand, to estimate the proximity of the perturbed cocycle to the original one, we also use a control over the measure of VR ∪VS that support the two perturbations taking into account Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' 7 It should be noted that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' in addition to the difficulties inherent in the context of continuous-time cocycles,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' performing these perturbations (rotation and stretch) are not trivial,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' as we do not have the usual mechanisms like those that exist in the context in cocycles that evolve in GL(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' R) or SL(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' R),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' or,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' more generally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' cocycles that satisfy the accessibility condition (also recognized as twisting) and saddle-conservative (also known as pinching),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' which allow the realization of these processes in a less demanding way,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' as,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' for example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' in [4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' 15,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' As our perturbations are all traceless we get from Theorem 1 that conservative kinetic cocycles have non-zero Lyapunov exponents σp-densely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Let ϕt : M → M be ergodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' For any 1 ≤ p < ∞, A ∈ K p 0 and ϵ > 0, there exists B ∈ K p 0 exhibiting non-zero Lyapunov exponents satisfying σp(A, B) < ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Finally, we present Corollary 1 with a somewhat different look, namely by con- sidering the one-dimensional Schrödinger operator on L2(R) and with an Lp poten- tial Q: M → R given by: Hω : L2(R) −→ L2(R) φ �−→ � − d2 dt2 + Q(ϕt(ω)) � φ (10) In particular we like to describe the Lyapunov spectrum of the time-independent Schrödinger equation Hωφ = Eφ, (11) where E ∈ R is a given energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Putting together (10) and (11) we deduce a kinetic cocycle as in (2) but with α(ω) = 0 and β(ω) = E − Q(ω) for all ω ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' We fix the energy E and focus on the LDS AE : M −→ R2×2 ω �−→ � 0 1 −E + Q(ω) 0 � (12) called one-dimensional Schrödinger LDS with potential Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' As a direct conse- quence of Corollary 1 we have: Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Let ϕt : M → M be ergodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Given 1 ≤ p < ∞, ϵ > 0 and a one- dimensional Schrödinger LDS with a fixed energy E as in (12) and with potential Q, there exists ˜Q such that the one-dimensional Schrödinger LDS with the same energy E and potential ˜Q exhibits non-zero Lyapunov exponents and ∥ ˜Q−Q∥Lp < ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' On the perturbations 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Special flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Consider a measure space Σ, a map T : Σ → Σ, a T -invariant probability measure ˜µ defined in Σ and a roof function h: Σ → R+ satisfying h(ω) ≥ H > 0, for some H > 0 and all ω ∈ Σ, and � Σ h(ω)d˜µ(ω) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Define the space Mh ⊆ Σ × R+ by Mh = �(ω, t) ∈ Σ × R+ : 0 ≤ t ≤ h(ω)� with the identification between the pairs (ω, h(ω)) and (T (ω), 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' The semiflow defined on Mh by S s(ω, r) = (T n(ω), r + s − �n−1 i=0 h(T i(ω))), where n ∈ N is uniquely defined by n−1 � i=0 h(T i(ω)) ≤ r + s < n � i=0 h(T i(ω)) 8 is called a suspension semiflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' If T is invertible then (S t)t is a flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Furthermore, if ℓ denotes the one dimensional Lebesgue measure the measure µ = (˜µ×ℓ)/ � h d˜µ defined on Mh by � g dµ = 1 � h d˜µ � �� h(ω) 0 g(ω, t)dt � d˜µ(ω), ∀g ∈ C0(Mh) is a probability measure and it is invariant by the suspension semiflow (S t)t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Flows with such representation are called special flows (or flows built under a function) and are denoted by (ϕt, Σ, T , h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' It is well-known (see [1, Theorem 2]) that any ergodic flow is isomorphic to a special flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Along this work we assume that the base flow is a special flow (ϕt, Σ, T , h) and, without any loss of generality, that H > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' To avoid overloading the notation we write M instead of Mh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Perturbations supported in time-τ flowboxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Take A ∈ G and a non- periodic orbit ω ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' We will consider a perturbation B = Bω,τ of A only along a segment of the orbit of ω with extremes ω and ϕτ(ω) for τ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Let P ∈ G be given and define B: M → R2×2 such that B( ˆω) = A( ˆω) for all ˆω outside ϕ[0,τ](ω) = {ϕs(ω) : s ∈ [0, τ]} and B( ˆω) = P( ˆω) otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' The map B is called a (local) perturbation of A by P supported on ϕ[0,τ](ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Given Σ0 ⊂ Σ and 0 ≤ a < b we define the set ϕ[a,b](Σ0) = � ϕt(ω): ω ∈ Σ0, t ∈ [a, b] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Given A ∈ G1, P ∈ G, Σ0 ⊂ Σ and a > 0, we may extend the local perturbations of A by P to be supported on the flowbox ϕ[a,b](Σ0), with 0 ≤ a < b < H, in the following way: for ω ∈ ϕ[a,b](Σ0) we project ω in ˜ω ∈ ϕa(Σ0) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' ω = ϕr( ˜ω), for some 0 ≤ r ≤ b−a, and let B ˜ω,b−a be (local) perturbation of A by P = P ˜ω supported on ϕ[0,b−a]( ˜ω) and define B(ω) := � A(ω), if ω � ϕ[a,b](Σ0) B ˜ω,b−a(ω), if ω ∈ ϕ[a,b](Σ0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' To distinguish the situations we refer for B(ω) as a global perturbation of A by P supported in ϕ[a,b](Σ0), where we always suppose that P(ω) = P ˜ω(ω) for all ω ∈ ϕ[a,b](Σ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Rotating and Stretching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Next two results provide local and global argu- ments to rotate over prescribed directions under a small σp-perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' This will be used to generate a suitable invariant direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' The first one allows us to perform a uniform bounded kinetic perturbation in a local segment of orbit which rotates a given vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' The second one thickens Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='1 by broaden the rotation in a single orbit to rotations in a flowbox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Given ω ∈ M, u, v ∈ R2 \\ {0}, A ∈ K p, there is γ � 0, and a perturbation Bω,1 ∈ K p of A supported on ϕ[0,1](ω) such that: (i) ∥Bω,1( ˆω)∥ ≤ 4π2 for all ˆω on ϕ[0,1](ω), and (ii) ΦBω,1(1, ω)u = γ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Let θ = ∡(Ru, Rv) ∈ ]0, 2π] measured clockwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Set a constant infinitesi- mal generator R: M → R2×2 given by R(ω) = Rθ(ω) = � 0 1 −θ2 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' (13) 9 We consider the perturbation B = Bω,1 ∈ K p of A by R supported on ϕ[0,1](ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' The infinitesimal generator in (13) generates a linear differential system with fun- damental classical solution (6) given, for all ω ∈ M and t ∈ R by the ‘clockwise elliptical rotation’ defined by: ΦR(t, ω) = � cos(θt) θ−1 sin(θt) −θ sin(θt) cos(θt) � , (14) and such that ΦB(1, ω)u = ΦR(1, ω)u = γv, for some γ � 0 fulfilling (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' □ From Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='3 it follows that we may extend the local perturbation Bω,1 given by the rotation Rθ(ω) as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='1, to a global perturbation, tuned for each orbit segment, to obtain a new generator that is σp-close to the original, once we have a smaller measure of the flowbox were the perturbation takes place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' This is pointed in the next basic measure theoretic result which is an immediate conse- quence of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='2 (Global).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' For all 1 ≤ p < ∞, A ∈ Gp, a > 0 and ϵ > 0, there exists a measurable set Σ0 ⊂ Σ with ˜µ(Σ0) > 0 such that for any global perturbation B ∈ Gp of A supported in the flowbox ϕ[a,a+1](Σ0), with ∥B(ϕt(ω))∥ ≤ 4π2 for all ω ∈ Σ0 and t ∈ [a, a + 1], we have that σp(A, B) < ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Let us fix a suitable constant and traceless infinitesimal generator S = �0 1 1 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' (15) As S has simple expression we integrate it obtaining: ΦS (t, ω) = eS t = �cosh t sinh t sinh t cosh t � (16) We notice that (16) has eigenvalues σS 1 = et and σS 2 = e−t with associated eigen- vectors vS 1 = (1, 1) and vS 2 = (−1, 1), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Observe that ES 1 = R · vS 1 is a unstable direction and ES 2 = R · vS 2 is a stable direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Next trivial remark will be of utmost importance in the sequel because it com- bines three main ingredients: invariance of certain 1-dimensional directions, some expansiveness along this direction and all this done in traceless kinetic infinitesi- mal generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='1 (Invariance and stretch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Considering θ = 2π in (14), say R2π, we get e · vS 1 = e · ΦR2π(1, ω) vS 1 = ΦS (1, ω) vS 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' (17) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Proof of Theorem 1 Let A ∈ K p, 1 ≤ p < ∞ and ϵ > 0 be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' We assume that ΦA has a single Lyapunov exponent λ(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' The sequence of perturbations are summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Defining A0 (picking out good coordinates): Let Σ0 ⊂ Σ be as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' For r > 0 we assume that we have flowboxes defined by VR := ϕ[0,1](Br) and VS := ϕ[1,2](Br), where Br ∈ Σ0 is such that 0 < ˜µ(Br) ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Consider A0 ∈ K1 defined as: A0(ω) := � A(ω), if ω � VR ∪ VS R2π, if ω ∈ VR ∪ VS .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' By Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='3 if r is sufficiently small when compared with ϵ we get σp(A, A0) < ϵ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' (18) If ΦA0 has simple spectrum we are over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Otherwise, we prove the theorem for A0 instead of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Defining B0 (rotating on VR): Set k(ω) = inf t≥0 � t: ϕ−t(ω) ∈ ϕ1(Br) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' We will define the a random vector field g(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' We start with the normalized image under the cocycle associated with ΦA0 of the vector v = vS 1 ∥vS 1 ∥ = � √ 2 2 , √ 2 2 � : g(ω) := ��������� v, if ω ∈ ϕ1(Br) ΦA0(k(ω),ϕ−k(ω)(ω))v ∥ΦA0(k(ω),ϕ−k(ω)(ω))v∥, if ω � (VR \\ Br) and set from now on E(ω) = span {g(ω)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Let B0 be a perturbation of A0 supported in the flowbox VR as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='2 such that for all ω ∈ Br we have ΦB0(1, ω)g(ω) = κv for some κ ∈ R, that is: B0(ω) := � R(ω), if ω ∈ VR A0(ω), otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Observe that the rotation must be tuned for each ω0 ∈ Br, in the sense that for ω = ϕt(ω0) ∈ VR, with 0 ≤ t ≤ 1, we set R(ω) = Rθ(ω0) with θ = ∡(g(ω0), v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' In particular, for all ω0 ∈ Br we have Φ(1, ω0)g(ω) = κv, for some κ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Moreover, A0 and B0 have the same trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Indeed, A0 = B0 outside VR and in VR we have B0 = R and A0 = R2π, which are both traceless (see (13)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Therefore, by Liouville’s formula for all ω and t ≥ 0 det ΦB0(t, ω) = det ΦA0(t, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' (19) For ω ∈ VR \\ Br define g(ω) = ΦB0(k(ω), ϕ−k(ω)(ω))v ∥ΦB0(k(ω), ϕ−k(ω)(ω))v∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' (20) Notice that for ω ∈ Br, since ΦB0(1, ω)Rg(ω) = Rv we get ΦB0(1, ω)Rg(ω)(ω) = Rg(ϕ1(ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' (21) Let ˜ω ∈ ϕ1(Br) and τ > 0 be such that ϕt( ˜ω) � VR for all t ∈]0, τ[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Then, for all t ∈ [0, τ] we have the ΦB0-invariance of g: ΦB0(t, ˜ω)Rg( ˜ω) = ΦB0(t, ˜ω)Rv = ΦA0(t, ˜ω)Rv = Rg(ϕt( ˜ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' (22) 11 If ϕt( ˜ω) ∈ VR for some t ∈]0, τ[ then considering s > 0 such that ϕs(ω) ∈ Br we get: ΦB0(t, ˜ω)Rg( ˜ω) = ΦB0(t − s, ϕs( ˜ω))ΦA0(s, ˜ω)Rv = ΦB0(t − s, ϕs( ˜ω))Rg(ϕs( ˜ω)) (20) = Rg(ϕt( ˜ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Finally, (21), (26) and last equality gives that the vector field g is ΦB0-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Again by Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='3 if r is sufficiently small we get σp(A0, B0) < ϵ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' (23) If ΦB0 has simple spectrum we are over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Otherwise, we prove the theorem for B0 instead of A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Defining B (stretching on VS ): We define B(ω) := � B0(ω), if ω � VS S, if ω ∈ VS .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Observe that B and B0 have the same trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Indeed, B = B0 outside VS and in VS we have B0 = R2π which are both traceless (see (13) and (15)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Therefore, by Liouville’s formula and (19) for all ω and t ≥ 0 det ΦB(t, ω) = det ΦB0(t, ω) = det ΦA0(t, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' (24) From Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='3, once more, if r is sufficiently small we get σp(B0, B) < ϵ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' (25) Notice that the invariance of the direction E(ω) under ΦB fails when ϕt(ω) enters VS .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' However, for ˜ω ∈ ϕ1(Br) we have by (17) and (26) ΦB(1, ˜ω)Rg( ˜ω) = ΦS (1, ˜ω)Rv = Rv = RΦR2π(1, ˜ω)v = ΦA0(1, ˜ω)Rv = Rg(ϕ1( ˜ω)) and so ΦS (1, ˜ω)E( ˜ω) = E(ϕ1( ˜ω)), (26) which will be enough for our purposes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' see Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' The traceless perturbation scheme with the invariant di- rections and the stretch effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Let λ1(B) ≥ λ2(B) be the Lyapunov exponents of ΦB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' We assume that ΦB0 has one-point spectrum, say λ1(B0) = λ2(B0) = λ(B0), because otherwise the theorem 12 dB(T, W)g() ΦB(1, B(1, T()) P Bo 3is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Let λ(B0) be the single Lyapunov exponent of ΦB0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Hence we have λ(B0) = λ(B0, ω, vS 1 ) for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' By the Oseledets theorem we have 2λ(B0) = � log ���det(ΦB0(1, ω)) ���dµ (27) and λ1(B) + λ2(B) = � log ���det(ΦB(1, ω)) ���dµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' (28) The two previous equalities together with (24) allows us to conclude that 2λ(B0) = λ1(B) + λ2(B) (29) and so, if we show that λ1(B) > λ(B0) then we get λ1(B) > λ2(B) and Theorem 1 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Recall that the random vector field g is invariant by ΦB0 but in what ΦB concerns, the invariance fails as the base dynamics enters VS .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' However, by (26) the invariance is recovered in the moment the base dynamics is leaving VS .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' For ω ∈ M let us consider the real map b0(·, ω) for all t ∈ R in such a way that b0(t, ω)g(ϕt(ω)) = ΦB0(t, ω)g(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' (30) Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' The map b0(t, ω) forms a cocycle over ϕt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Indeed, since ΦB0(0, ω) = Id for all ω ∈ M we have b0(0, ω) = 1 and for all s, t, evaluating b0(t + s, ω) at g(ϕt+s(ω)), we have b0(t + s, ω)g(ϕt+s(ω)) (30) = ΦB0(t + s, ω)g(ω) = ΦB0(t, ϕs(ω)) · ΦB0(s, ω)g(ω) (30) = ΦB0(t, ϕs(ω)) · b0(s, ω)g(ϕs(ω)) = b0(s, ω) ΦB0(t, ϕs(ω))g(ϕs(ω)) = b0(t, ϕs(ω))b0(s, ω)g(ϕt+s(ω)), and so b0(t + s, ω) = b0(t, ϕs(ω))b0(s, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Since the random vector field g is not completely invariant by ΦB we consider two distinct situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Set ϕ{1,2}(Br) = ϕ1(Br) ∪ ϕ2(Br).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' For ω ∈ M and τ ≥ 0 such that ϕt(ω) � VS \\ ϕ{1,2}(Br), for all 0 ≤ t ≤ τ, we consider the real map b(·, ω) for all t ∈ [0, τ] in such a way that b(t, ω)g(ϕt(ω)) = ΦB(t, ω)g(ω) (31) and, for all ω ∈ ϕ1(Br), we set b(1, ω) ∈ R in such a way that ΦB(1, ω)g(ω) = b(1, ω)g(ϕ1(ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' (32) If ϕt(ω) � VS \\ ϕ{1,2}(Br), for all 0 ≤ t ≤ τ, we have B(ϕt(ω)) = B0(ϕt(ω)) and b(t, ω)g(ϕt(ω)) = ΦB(t, ω)g(ω) = ΦB0(t, ω)g(ω) = b0(t, ω)g(ϕt(ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' (33) In particular this holds between the output of VS to the next input in VS .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' If ϕt(ω), ϕs(ω) � VS \\ ϕ{1,2}(Br), b(t, ω) forms a cocycle over ϕt in the sense that b(t + s, ω) = b(t, ϕs(ω))b(s, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' 13 The proof follows similarly to Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='1 taking also into account (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Pick ω in a full measure subset of points that visits infinitely often Br and for which the conclusion of Birkhoff’s Ergodic theorem holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Without loss of gene- rality we may assume that ω � Vr ∪ VS .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' For t ≥ 0 set Jt(ω) = # � j ∈ N: j ≤ t, ϕ j(ω) ∈ ϕ2(Br) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Recall that λ(B, ω, g(ω)) = lim t→∞ 1 t log ∥ΦB(t, ω)g(ω)∥, and we may split the previous orbit in the limit by considering the time for ϕt(ω) to enter VS , the time-1 moment crossing the flowbox VS , where we use (17), and, again, the time it takes to return to VS and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' For simplicity, let us define recursively s0 = s0(ω) = min{t: ϕt(ω) ∈ ϕ1(Br)}, ℓ0 = ℓ0(ω) = s0 + 1, sn = s0(ϕℓn−1(ω)) and ℓn = sn + 1, for n ≥ 1, ∆n = sn − ℓn−1, for n ≥ 1, ˜ωn = ϕsn(ω) ∈ ϕ1(Br) and ˆωn = ϕℓn(ω) ∈ ϕ2(Br), for n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Now, in one hand, since B0 has one-point spectrum, for µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' ω, λ(B0, ω) = λ(B0, ω, g(ω)) = lim t→∞ 1 t log ∥ΦB0(t, ω)g(ω)∥ (30) = lim t→∞ 1 t log |b0(t, ω)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' (34) On the other hand, by Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='1 and (32) we have for ˜ω ∈ ϕ1(Br) that b(1, ˜ω)g(ϕ1( ˜ω)) = ΦB(1, ˜ω)g( ˜ω) (17) = e·ΦB0(1, ˜ω)g( ˜ω) = e·b0(1, ˜ω)g(ϕ1( ˜ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' (35) Without loss of generality, we can consider the following limits over the un- bounded set {t ≥ 0: ϕt(ω) ∈ ϕ1(Br)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' From Birkhoff’s Ergodic theorem we have λ(B, ω, g(ω)) = lim t→∞ 1 t log ∥ΦB(t, ω)g(ω)∥ (31)+(32) = lim t→∞ 1 t ���������log |b(s0, ω)| + Jt(ω)−1 � j=0 log |b(∆j+1, ˆωsj)b(1, ˜ωsj)| ��������� (33)+(35) = lim t→∞ 1 t ���������log |b0(s0, ω)| + Jt(ω)−1 � j=0 log |b0(∆j+1, ˆωsj) · e · b0(1, ˜ωsj)| ��������� Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='1 = lim t→∞ 1 t log |b0(t, ω)| + lim t→∞ Jt(ω) t (30) = lim t→∞ 1 t log ∥ΦB0(t, ω)g(ω)∥ + lim t→∞ 1 t � t 0 1VS (ϕt(ω)) dt = λ(B0, ω, g(ω)) + µ(VS ), which implies λ1(B, ω) > λ(B0, ω), hence λ1(B) > λ(B0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' From (29), we get λ1(B) > λ(B0) > λ2(B) so that B has simple spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Moreover, by (18), (23) and (25) we have σp(A, B) < ϵ and Theorem 1 is now proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' □ 14 Clearly when considering the set K1 0 on Corollary 1 the equalities (27) and (28) become 2λ(B0) = λ1(B) + λ2(B) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Hence the conclusion this time will be that λ1(B) > 0 for B ∈ K1 0 arbitrarily σp-close to A and also λ2(B) = −λ1(B) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Acknowledgements: The authors were partially supported by FCT - ‘Fundação para a Ciência e a Tecnologia’, through Centro de Matemática e Aplicações (CMA- UBI), Universidade da Beira Interior, project UIDB/MAT/00212/2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' MB was partially supported by the Project ‘Means and Extremes in Dynamical Systems’ (PTDC/MAT-PUR/4048/2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' MB also like to thank CMUP for providing the necessary conditions in which this work was developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' References [1] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Ambrose, Representation of ergodic flows, Annals of Mathematics 42 (1941), 3, 723–739.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' [2] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Ambrose, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Kakutani, Structure and continuity of measure preserving transformations, Duke Math.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=', 21, 10, (2019), 3143–3190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Centro de Matem´atica e Aplica¸c˜oes (CMA-UBI), Universidade da Beira Interior, Rua Marquˆes d’Ávila e Bolama, 6201-001, Covilh˜a, Portugal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content=' Email address: dinis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='amaro@ubi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='pt Email address: bessa@ubi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='pt Email address: helder@ubi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} +page_content='pt 16' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E4T4oBgHgl3EQfIAys/content/2301.04909v1.pdf'} diff --git a/L9FAT4oBgHgl3EQfxB4e/content/tmp_files/2301.08684v1.pdf.txt b/L9FAT4oBgHgl3EQfxB4e/content/tmp_files/2301.08684v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8d0ab9cd20fad679c9485245577f7dd737fcd471 --- /dev/null +++ b/L9FAT4oBgHgl3EQfxB4e/content/tmp_files/2301.08684v1.pdf.txt @@ -0,0 +1,2278 @@ +STATE-CONSTRAINED OPTIMIZATION PROBLEMS UNDER +UNCERTAINTY: A TENSOR TRAIN APPROACH +HARBIR ANTIL, SERGEY DOLGOV, AND AKWUM ONWUNTA +Abstract. We propose an algorithm to solve optimization problems constrained by partial +(ordinary) differential equations under uncertainty, with almost sure constraints on the state +variable. +To alleviate the computational burden of high-dimensional random variables, +we approximate all random fields by the tensor-train decomposition. To enable efficient +tensor-train approximation of the state constraints, the latter are handled using the Moreau- +Yosida penalty, with an additional smoothing of the positive part (plus/ReLU) function by +a softplus function. We derive theoretical bounds on the constraint violation in terms of +the Moreau-Yosida regularization parameter and smoothing width of the softplus function. +This result also proposes a practical recipe for selecting these two parameters. When the +optimization problem is strongly convex, we establish strong convergence of the regularized +solution to the optimal control. We develop a second order Newton type method with a +fast matrix-free action of the approximate Hessian to solve the smoothed Moreau-Yosida +problem. This algorithm is tested on benchmark elliptic problems with random coefficients, +optimization problems constrained by random elliptic variational inequalities, and a real- +world epidemiological model with 20 random variables. These examples demonstrate mild +(at most polynomial) scaling with respect to the dimension and regularization parameters. +1. Introduction +Over last two decades optimization problems constrained by physical laws, such as partial +(ordinary) differential equations (PDEs/ODEs), have emerged as a prominent research area. +This is fueled by many applications in science and engineering, such as controlling pathogen +propagation in built environment [26, 25], shape and topology optimization [36, 28], optimal +strategies to predict shutdowns due to pandemics [11]. The optimization variables consist of +state (y) and control/design (u). However, often due to noisy measurements and ambiguous +models due to incomplete physics, the underlying physical laws contain uncertainty. This +has led to significant theoretical and algorithmic developments in the area of optimization +problems constrained by physical laws under uncertainty. See for instance [23, 4, 14, 3] and +the references therein. These papers focus on problems with control constraints. +The literature on state-constrained optimization problems under uncertainty is scarce. +For instance, [12, 17] use probability constraints, and [15, 13, 16] consider almost surely +Date: 20 January 2023. +2020 Mathematics Subject Classification. +49J55, 93E20, 49K20, 49K45, 90C15, 65D15, 15A69, 15A23 . +Key words and phrases. almost surely constraints, state constraints, risk neutral, tensor train, reduced +space, preconditioner, variational inequality. +HA is partially supported by NSF grant DMS-2110263 and the AirForce Office of Scientific Research under +Award NO: FA9550-22-1-0248. SD is thankful for the support from Engineering and Physical Sciences Re- +search Council (EPSRC) New Investigator Award EP/T031255/1 and New Horizons grant EP/V04771X/1. +1 +arXiv:2301.08684v1 [math.OC] 20 Jan 2023 + +2 +HARBIR ANTIL, SERGEY DOLGOV, AND AKWUM ONWUNTA +type constraints. +It is well-known that even in the deterministic setting, the state con- +strained problems are highly challenging. One of the fundamental difficulties is that the +state constraints are imposed in the sense of continuous functions. As a result, the Lagrange +multipliers corresponding to those constraints are Radon measures that exhibit low regu- +larity [6]. The situation is much more delicate in the stochastic setting. We refer to the +aforementioned references for a detailed discussion on this topic. Motivated by the deter- +ministic setting, [13] introduces a Moreau-Yosida based approximation scheme to solve the +state-constrained optimization problems when the PDE constraints are given by an elliptic +equation with random coefficients. Further extensions of this work are considered in [1, 16]. +However, all of these papers approximate expectations of random fields by Monte-Carlo-type +methods, which may converge slowly. +In [3], we introduced an algorithm (TTRISK) based on the tensor train (TT) decom- +position [30] to solve risk-averse optimization problems with control constraints, and the +conditional value-at-risk (CVaR) [32] risk measure. We demonstrated that the extra com- +putational cost due to the uncertainty can scale proportionally to error−0.5 when the TT +approximation is used, in contrast to a error−2 scaling of Monte Carlo quadratures. +In +the current paper, we continue this program and develop a TT based algorithm for state- +constrained optimization problems. +For simplicity of presentation, we only consider the +risk-neutral setting, i.e., the objective function is given by the expected value of a quantity +of interest. Similarly to [13, 16], we tackle the state constrains using Moreau-Yosida based +relaxation with a softplus smoothing. The main contributions of this paper are listed next: +(i) We consider an ε-softplus regularization of the positive part function (·)+ = max{·, 0} +and derive a probabilistic estimate of state constraint violation in terms of Moreau- +Yosida regularization parameter γ and ε. In particular, we show that selecting ε ∝ γ−1/2 +ensures the convergence of the constraint violation with a rate γ−1/2. This result is +motivated by [13, Prop. 2]. Notice that the ε-smoothing is carried out because the +irregular function (·)+ may lack an efficient TT decomposition. +(ii) When the optimization problem is strongly convex, we establish strong convergence +of the regularized solution to the optimal control. Our final results can be seen as +generalizations of the results in deterministic setting. +(iii) We derive a second order Newton type method to solve the regularized problem with +a fast matrix-free action of the approximate Hessian. +(iv) We test the proposed method on elliptic equations in one and two physical dimensions +and random coefficients, as well as an ODE example (motivated by a realistic applica- +tion) with 20 random variables, and show that the algorithm is free from the curse of +dimensionality. +(v) The proposed approach has been also successfully applied to an example where the +PDE constraint is given by an elliptic variational inequality. +Outline: The remainder of the paper is organized as follows. In Section 2, we provide a +rigorous mathematical formulation of the problem under consideration. Section 3 is devoted +to the Moreau-Yosida approximation, derivation of the second order Newton method and +approximation error estimates due to the Moreau-Yosida approximation. In Section 4, we +provide a brief description of the TT format. This is followed by practical aspects of Moreau- +Yosida approximations in Section 5. Finally, in Section 6, we provide a series of numerical + +STATE-CONSTRAINED OPTIMIZATION PROBLEMS UNDER UNCERTAINTY +3 +experiments. At first, we consider an optimization problem with an elliptic PDE in one +spatial dimension as constraints. This is followed by a two-dimensional case. After these +benchmarks, an optimization problem with an elliptic variational inequality as constraint is +considered in Section 6.3. The numerical experiments conclude with a realistic ODE example +for designing optimal lockdown strategies in Section 6.4. +2. Problem Formulation +Let (Ω, F, P) denote a complete probability space, where Ω represents the sample space, +F is the Borel σ-algebra of events on the power set of Ω, and P : Ω → [0, 1] is an appropriate +probability measure. We denote by E[·] the expectation with respect to P. Let U be a real +deterministic reflexive Banach space of optimization variables (control or design) defined on +an open, bounded and connected set D ⊂ Rn with Lipschitz boundary. We denote by ∥ · ∥U +the norm on U, and the duality pairing between U and U ∗ as ⟨·, ·⟩U∗,U. Let Y = L2(Ω, F, P; ˆY) +and Z = L2(Ω, F, P; ˆZ) be Bochner spaces of random fields, based on deterministic Banach +spaces ˆY �→ L2(D) �→ ˆY∗ and ˆZ, with corresponding norms and duality pairings +∥y∥2 +Y = E[∥y(ω)∥2 +ˆY], +⟨y, v⟩Y∗,Y = E +� +⟨y(ω), v(ω)⟩ ˆY∗, ˆY +� +, +and similarly for Z. Let Uad ⊆ U be a closed convex nonempty subset and let c : Y×Uad×Ω → +Z denote, e.g., a partial differential operator, then consider the equality constraint +c(y, u; ω) = 0, +in Z, +a.s. ω ∈ Ω, +where a.s. indicates “almost surely” with respect to the probability measure P. +In this paper, we consider the optimization problems of the form +min +y,u R[J(y, u; ω)] +(2.1) +s.t +c(y, u; ω) = 0, +in Z, +a.s. ω ∈ Ω, +(2.2) +where R represents the risk measure and R[J(y, u; ω)] is a deterministic cost function. More +precisely, we will focus on the so-called risk-neutral formulation; that is, R is simply the +expectation, denoted by E. We are particularly interested in the case in which the state +variable y is constrained by a random variable: +y ≤ ymax(ω) +a.s., +(2.3) +where we assume that ymax ∈ Y. +In what follows, we discuss the Moreau-Yosida approximation for (2.1)-(2.3) and derive a +Newton type method. Throughout the paper, without explicitly stating, we will make use +of the following assumption. +Assumption 2.1 (unique forward solution). There exists an injective operator S(ω) : Uad → +Y (maybe nonlinear) such that c(S(ω)u, u; ω) = 0 ∀u ∈ Uad a.s. +This allows us to define the reduced-space cost function +j(u) := R[J(S(ω)u, u; ω)]. +(2.4) + +4 +HARBIR ANTIL, SERGEY DOLGOV, AND AKWUM ONWUNTA +The resulting reduced optimization problem is given by +min +u∈Uad j(u) +s.t +y ≤ ymax(ω) +a.s. +(2.5) +3. Smoothed Moreau-Yosida approximation +Solving (2.5) with state constraints involve computation of the indicator function of an +active set and/or Lagrange multiplier as a random field that is nonnegative on a compli- +cated high-dimensional domain. +This may be difficult for many function approximation +methods, especially for tensor decompositions that are considered in this paper. We tackle +this difficulty by first turning the constrained optimization problem (2.5) into an uncon- +strained optimization problem with the Moreau-Yosida penalty, and further by smoothing +the indicator function in the penalty term. +The classical Moreau-Yosida problem reads, with γ ≥ 0 denoting the regularization pa- +rameter, +min +u∈Uad jγ(u), +where +jγ(u) := j(u) + γ +2E +���(Su − ymax(ξ))+ +��2 +L2(D) +� +, +(3.1) +where the so-called positive part or ReLU function (·)+ reads (s)+ = s if s ≥ 0 and 0, oth- +erwise. Here, we have removed the need to optimize the Lagrange multiplier (corresponding +to the inequality constraints) over the nonnegative cone, but the function approximation of +a nonsmooth high-dimensional random field (Su − ymax(ξ))+ (and derivatives thereof) may +be still inefficient. +For this reason, we replace the ReLU function in the penalty term by a smoothed version. +In this paper, we use the softplus function +gε(s) = ε · log(1 + exp(s/ε)) ∈ C∞(R), +g0(s) = lim +ε→0 gε(x) = (s)+, +(3.2) +although other (e.g. piecewise polynomial) functions are also possible [24, 1]. Now, the cost +function becomes +jγ,ε(u) := j(u) + γ +2E +���gε(Su − ymax) +��2 +L2(D) +� +. +(3.3) +3.1. Discretization and Derivatives of the Cost. In practice, the operator S involves +the solution of a differential equation, which needs to be discretized (using e.g. +Finite +Element methods and/or time integration schemes). For a given mesh parameter h > 0, we +introduce the discretized (maybe nonlinear) operator Sh(ω) : Uad → Rny, where ny is the +total number of degrees of freedom in the discrete solution. We denote the induced Bochner +space Yh ∼= L2 +h(Ω, D) := L2(Ω, F, P; Rny). The L2-norm can be written as an expectation of +a vector quadratic form, +∥y∥2 +L2 +h(Ω,D) = E +� +y(ω)⊤My(ω) +� +, +∀y ∈ L2 +h(Ω, D), +where M = M⊤ > 0 ∈ Rny×ny is a mass matrix. The discretized problem cost is denoted +by jh(u) ≈ j(u), and the discretized constraint is yh +max ∈ Yh. Now, the semi-discretized +Moreau-Yosida cost function (3.3) becomes +jγ,ε,h(u) := jh(u) + γ +2E +� +∥gε(Shu − yh +max)∥2 +M +� +. +(3.4) + +STATE-CONSTRAINED OPTIMIZATION PROBLEMS UNDER UNCERTAINTY +5 +To derive a Newton type method, we compute the expressions of gradient and Hessian: +∇ujγ,ε,h = ∇ujh + γE +� +S∗ +h · diag(g′ +ε(Shu − yh +max)) · Mgε(Shu − yh +max) +� +, +(3.5) +∇uujγ,ε,h = ∇2 +uujh + γE +� +S∗ +h · diag(g′ +ε)Mdiag(g′ +ε) · S′ +h +� +(3.6) ++ γE +� +S∗ +h · (tendiag(g′′ +ε) ×3 (Mgε)) · S′ +h +� +(3.7) ++ γE +� +∇uS∗ +h ×3 (diag(g′ +ε(Shu − yh +max)) · Mgε(Shu − yh +max)) +� +, +(3.8) +where tendiag(·) is producing a 3-dimensional tensor out of vector by putting the vector +elements along the diagonal, and zero elements otherwise, and ×3 is the tensor-vector con- +traction product over the 3d mode of the tensor. If Sh is a nonlinear operator, S′ +h = ∇uSh(u) +denotes the gradient of an image of u, and S∗ +h is the adjoint of S′ +h. +3.2. Matrix-free Fixed Point Gauss-Newton Hessian. The exact assembly of all terms +of the Hessian (3.6)–(3.8) can be too computationally expensive, since this involves dense +tensor-valued random fields (such as ∇uS∗ +h). To simplify the computations, we can firstly +omit the terms (3.7) and (3.8) which contain order-3 tensors. Secondly, we can replace the +exact expectation by a fixed-point evaluation. Rewriting (2.1) using Assumption 2.1 we can +define J(u; ω) = J(S(ω)u, u; ω) and its discretized version Jh(u; ω) = J(Sh(ω)u, u; ω). The +Hessian of jh can then be written as +∇2 +uujh = E +� +∇2 +uuJh(u; ω) +� +. +For practical computations, it is convenient to parametrize all random fields with inde- +pendent identically distributed (i.i.d.) random variables with a known probability density +function. Those variables can then be sampled independently, and an expectation can be +computed simply by quadrature. Therefore, we will use the following assumption. +Assumption 3.1 (finite noise). There exists a d-dimensional random vector ξ(ω) ∈ Rd +with a product probability density function π(ξ) = π(ξ1) · · · π(ξd), such that any random field +y ∈ Y can be expressed as a function of ξ, y(ω) = y(ξ(ω)) a.s., and +E[y] = +ˆ +Rd y(ξ)π(ξ)dξ. +In particular, the vector ξ can often be derived from a parametrization of the forward +solution operator Sh(ω) = Sh(ξ(ω)), and/or the constraint yh +max(ω) = yh +max(ξ(ω)). +Example 3.2. Let y = S(ν(ω))u be the resolution of an elliptic PDE +−∇(κ(x; ν(ω))∇y) = u, +where the diffusivity +κ(x; ν(ω)) = κ0(x) + +p +� +k=1 +ψk(x)νk(ω) +and the constraint +ymax(x; η(ω)) = y0(x) + +q +� +k=1 +φk(x)ηk(ω) + +6 +HARBIR ANTIL, SERGEY DOLGOV, AND AKWUM ONWUNTA +are given by Karhunen-Loeve expansions (see e.g., [27]), where ν and η are independent +random variables. Then, we can define ξ = (ν1, . . . , νp, η1, . . . , ηq). +Now we can replace ∇2 +uujh = E[∇2 +uuJh(u; ξ)] by +˜∇2 +uujh = ∇2 +uuJh(u; E[ξ]). +This is exact if ∇2 +uuJh is linear in ξ, but we can take it as an approximation in the general case +too. Now to apply ˜∇2 +uujh to a vector we just need to apply one deterministic ∇2 +uuJh(u; E[ξ]), +which involves solving one forward, one adjoint, and two linear sensitivity (of state and +adjoint) deterministic problems in the most general setting [4, Ch. 1, Algo. 2]. +Similarly we approximate the second term in (3.6) by +γS∗ +h(ξ∗)MS′ +h(ξ∗), +where +ξ∗ = E +� +ξ · 1⊤g′ +ε(Shu − yh +max(ξ)) +� +E +� +1⊤g′ +ε(Shu − yh +max(ξ)) +� +is the mean of the random variable with respect to the probability density πg′ ∝ π · +(1⊤g′ +ε(Shu − yh +max)), and 1 ∈ Rny is the constant vector, averaging the spatial components. +Note that 1⊤g′ +ε(Shu − yh +max) is a nonnegative function bounded by ny, so π1⊤g′ +ε(Shu − yh +max) +is nonnegative and normalizable, and πg′ is indeed a probability density. +Finally, we obtain a deterministic approximate Hessian +˜H = ∇2 +uuJh(u; E[ξ]) + γS∗ +h(ξ∗)MS′ +h(ξ∗), +(3.9) +which can be applied to a vector by solving 2 forward, 2 adjoint, and 2 sensitivity problems. +3.3. Probability of the Constraint Violation. In the rest of this section, we prove certain +properties about the quality of the solution of the smoothed problem (3.3) with respect to +the constraint, and the exact solution of (2.1)–(2.3). This needs a few properties of the +softplus smoothing function. +Lemma 3.3. For any ε ≥ 0, the softplus function (3.2) satifies: gε(s) ≥ (s)+ for any s ∈ R, +g′ +ε(s) ≥ 0.5 for s ≥ 0, and g′ +ε(s) ≤ 0.5 for s ≤ 0. +Proof. Using the monotonicity of the logarithm, +gε(s) = ε log +� +1 + exp(s/ε) +� +≥ +� +ε log +� +exp(s/ε) +� += s = (s)+, +s ≥ 0, +0 = (s)+, +s < 0. +The remaining inequalities follow simply from the monotonicity of the sigmoid function +g′ +ε(s) = 1/(1 + exp(−s/ε)) and that g′ +ε(0) = 0.5. +□ +Theorem 3.4. Let uγ,ε be a minimizer of (3.3), and assume that j(u) ≥ 0 for any u ∈ Uad. +Then for any δ > 0, we have +P +� +∥(S(ω)uγ,ε − ymax(ω))+∥2 +L2(D) > δ +� +≤ C1 + C2γε2 +γδ +, +where C1 = 2j(u∗), C2 = log2 2 · ∥1∥2 +L2(D), and u∗ is a minimizer of (2.1)–(2.3). +Remark 3.5. This motivates the condition ε ≲ 1/√γ to overcome the effect of smoothing. + +STATE-CONSTRAINED OPTIMIZATION PROBLEMS UNDER UNCERTAINTY +7 +Proof. Using Markov’s inequality, we obtain +P +� +∥(Suγ,ε − ymax(ω))+∥2 +L2(D) > δ +� +≤ +E +���(Suγ,ε − ymax(ω))+ +��2 +L2(D) +� +δ +≤ +E +���gε(Suγ,ε − ymax(ω)) +��2 +L2(D) +� +δ +, +where in the second inequality we used Lemma 3.3. Since uγ,ε minimizes (3.3), it holds +j(uγ,ε) + γ +2E[∥gε(Suγ,ε − ymax(ω))∥2 +L2(D)] ≤ j(u∗) + γ +2E[∥gε(Su∗ − ymax(ω))∥2 +L2(D)] +for any u∗ ∈ Uad such as the minimizer of (2.1) constrained to (2.3). Dividing by γ/2 and +neglecting j(uγ,ε) ≥ 0, we get +E[∥gε(Suγ,ε − ymax(ω))∥2 +L2(D)] ≤ C1 +γ + E[∥gε(Su∗ − ymax(ω))∥2 +L2(D)]. +For the latter term, (2.3) implies Su∗ − ymax(ω) ≤ 0 a.s., and due to monotonicity of gε, +gε(Su∗ − ymax(ω)) ≤ gε(0) = ε · log 2 +a.s. +Taking this upper bound out of the expectation and norm, we obtain +E[∥gε(Suγ,ε−ymax(ω))∥2 +L2(D)] ≤ C1 +γ +ε2·log2 2·E[∥1∥2 +L2(D)] = C1 +γ +ε2·log2 2·∥1∥2 +L2(D), (3.10) +and the estimate on probability follows by the Markov’s inequality. +□ +3.4. Strong Convergence with Strongly Convex Cost. To prove the strong conver- +gence of the minimizer of (3.3) to the minimizer of (2.1)–(2.3) we need further assumptions +on the cost and smoothing functions. +Assumption 3.6 (Bounded derivative of the cost). There exists L < ∞ such that +∥j′(u)∥U∗ ≤ L +∀u ∈ Uad. +Assumption 3.7 (α-strong convexity of the cost). There exists α > 0 such that +⟨j′(u) − j′(v), u − v⟩U∗,U ≥ α∥u − v∥2 +U, +∀u, v ∈ Uad. +Assumption 3.8 (Smoothing function). The smoothing function gε possesses the following +properties +g′ +ε(s) ≥ 0.5, +gε(s) ≥ s, +for +s ≥ 0, +g′ +ε(s) ≤ 0.5, +for +s ≤ 0, +(3.11) +and either: +gε(s)s ≥ −ηmax(ε), +for +s ≤ 0, +(3.12) +or, for any random field y(ω) ∈ Y such that y(ω) ≤ 0 a.s., +⟨y, gε(y)⟩Y∗,Y ≥ −ηint(ε), +(3.13) +where ηmax(ε), ηint(ε) ≥ 0, ∀ε > 0, ηmax(ε), ηint(ε) → 0 as ε → 0. +Notice that all the conditions in (3.11) are satisfied by the softplus function (3.2) (see +Lemma 3.3). We only need to check (3.12) or alternatively (3.13). + +8 +HARBIR ANTIL, SERGEY DOLGOV, AND AKWUM ONWUNTA +Conjecture 3.9. Our numerical experiments demonstrate that for the softplus function +(3.2) it holds ηmax(ε) = O(ε2) and ηint(ε) = O(ε3), although we are only able to prove the +latter estimate under specific conditions (Lemma 3.11 and Theorem 3.12). +Now we are able to prove the strong convergence of the smoothed optimal control. +Theorem 3.10. Under Assumptions 2.1 and 3.6–3.8, linear operator S, and ε = εγ depen- +dent on γ in such a way that +γ min{ηmax(εγ), ηint(εγ)} → 0, +as +γ → ∞, +and ⟨f, f⟩Y∗,Y = ∥f∥2 +L2(Ω,D) for any f ∈ Y, the minimizer uγ of (3.3) converges to the +solution u∗ of the exact problem (2.1)–(2.3), +α∥uγ − u∗∥2 +U + γ +2∥(Suγ − ymax)+∥2 +L2(Ω,D) → 0, +γ → ∞. +Proof. The optimality condition for the smoothed problem, ⟨∇ujγ,ε(uγ), v − uγ⟩U∗,U ≥ 0, +∀v ∈ Uad, can be expanded by introducing an auxiliary variable λγ to match the gradient of +the Moreau-Yosida term: +⟨j′(uγ) + S∗λγ, v − uγ⟩U∗,U ≥ 0, +(3.14) +γg′ +ε(Suγ − ymax)gε(Suγ − ymax) = λγ. +(3.15) +In turn, the KKT conditions for the original problem read +⟨j′(u∗) + S∗λ∗, v − u∗⟩U∗,U ≥ 0 +∀v ∈ Uad +(3.16) +λ∗ ≥ 0 +Su∗ − ymax ≤ 0 +⟨λ∗, Su∗ − ymax⟩Y∗,Y = 0. +(3.17) +Adding (3.16) with v = uγ to (3.14) with v = u∗, and casting S∗ onto another side of the +duality pairing, we get +0 ≥ ⟨j′(uγ) + S∗λγ − j′(u∗) − S∗λ∗, uγ − u∗⟩U∗,U += ⟨j′(uγ) − j′(u∗), uγ − u∗⟩U∗,U + ⟨λγ, Suγ − Su∗⟩Y∗,Y + ⟨j′(u∗), uγ − u∗⟩U∗,U. +(3.18) +Due to the strong convexity, (3.18), and Assumption 3.6 we arrive at +α∥uγ − u∗∥2 +U + ⟨λγ, Suγ − Su∗⟩Y∗,Y ≤ ⟨j′(u∗), u∗ − uγ⟩U∗,U ≤ ∥j′(u∗)∥U∗∥u∗ − uγ∥U. (3.19) +The second term on the left hand side can be bounded as follows. +Using the fact that +ymax − Su∗ ≥ 0 a.s. and the definition of λγ, we obtain that +⟨λγ, Suγ − Su∗⟩Y∗,Y = ⟨λγ, (Suγ − ymax) + (ymax − Su∗)⟩Y∗,Y +≥ ⟨λγ, Suγ − ymax⟩Y∗,Y += γ⟨g′ +ε(Suγ − ymax)gε(Suγ − ymax), Suγ − ymax⟩Y∗,Y += γ⟨g′ +ε(Suγ − ymax)(Suγ − ymax), gε(Suγ − ymax)⟩Y∗,Y += γ⟨g′ +ε(Suγ − ymax)(Suγ − ymax)+, gε(Suγ − ymax)⟩Y∗,Y ++ γ⟨g′ +ε(Suγ − ymax)(Suγ − ymax)−, gε(Suγ − ymax)⟩Y∗,Y, +(3.20) + +STATE-CONSTRAINED OPTIMIZATION PROBLEMS UNDER UNCERTAINTY +9 +where we have split Suγ − ymax into positive and negative parts, with (s)− = min(s, 0) +denoting the negative part. Next using Assumption 3.8 in (3.20), we readily obtain that +⟨λγ, Suγ − Su∗⟩Y∗,Y ≥ γ⟨0.5(Suγ − ymax)+, (Suγ − ymax)+⟩Y∗,Y ++ γ⟨0.5(Suγ − ymax)−, gε(Suγ − ymax)⟩Y∗,Y +(3.21) +≥ γ +� +0.5∥(Suγ − ymax)+∥2 +L2(Ω,D) − 0.5ηint(ε) +� +. +(3.22) +Alternatively, we can bound (3.21) using (3.12) to arrive at +⟨λγ, Suγ − Su∗⟩Y∗,Y ≥ γ +� +0.5∥(Suγ − ymax)+∥2 +L2(Ω,D) − 0.5ηmax(ε)∥1∥2 +L2(Ω,D) +� +. +In either case, (3.19) implies that uγ is bounded in Uad. Therefore, there exists a weakly +converging subsequence uγ ⇀ ˆu in U as γ → ∞. Since, Uad is closed convex, therefore +ˆu ∈ Uad. If ε = εγ → 0 as γ → ∞, Assumption 3.8 (for both ηmax and ηint) implies that +0.5γ∥(Suγ −ymax)+∥2 +L2(Ω,D) is bounded, which means ∥(Suγ −ymax)+∥2 +L2(Ω,D) → 0 as γ → ∞. +Since S is injective and linear, ∥(Suγ − ymax)+∥2 +L2(Ω,D) is continuous and convex, hence [38, +Theorem 2.12]: +0 = lim inf +γ→∞ ∥(Suγ − ymax)+∥2 +L2(Ω,D) ≥ ∥(Sˆu − ymax)+∥2 +L2(Ω,D). +Since D is a connected domain of positive measure, this yields |(Sˆu − ymax)+| = 0, that is, +Sˆu ≤ ymax a.s. Adding again (3.16) and (3.14) and using strong convexity of j, but keeping +both λγ and λ∗, we get +α∥uγ − u∗∥2 +U ≤ ⟨λ∗ − λγ, Suγ − Su∗⟩Y∗,Y +(3.23) +≤ ⟨λ∗, (Suγ − ymax) + (ymax − Su∗)⟩Y∗,Y +(3.24) +− γ +2∥(Suγ − ymax)+∥2 +L2(Ω,D) + γ +2 min{∥1∥2 +L2(Ω,D)ηmax(εγ), ηint(εγ)}, +(3.25) +where we used (3.22) with the negative sign. If γηmax(εγ) → 0 or γηint(εγ) → 0, then +0 ≤ lim +γ→∞[α∥uγ − u∗∥2 +U] ≤ lim +γ→∞⟨λ∗, Suγ − ymax⟩Y∗,Y = ⟨ λ∗ +���� +≥0 +, Sˆu − ymax +� +�� +� +≤0 +⟩Y∗,Y ≤ 0 +(3.26) +due to (3.17), so uγ → u∗, thereby completing the proof of the theorem. +□ +Lemma 3.11. For the softplus function (3.2) it holds for any ε ≥ 0: +ˆ 0 +−∞ +sgε(s)ds ≥ −ε3. +Proof. The proof uses elementary calculus and is given in Appendix A. +□ +In order to search for a rate of convergence, we establish the following result: +Theorem 3.12. Suppose Assumptions 2.1, 3.1 and 3.6–3.8 hold, ˆY is a space of scalar +functions, the operator S is linear, and |∂(Su − ymax)/∂ξ1| ≥ c > 0 a.s. ∀u ∈ Uad. Suppose +that ⟨f, g⟩ ˆY∗, ˆY = +´ +D f(x)g(x)dx ∀f, g ∈ ˆY, and maxξ1∈R π(ξ1) = P < ∞. Let ε = ε0/√γ + +10 +HARBIR ANTIL, SERGEY DOLGOV, AND AKWUM ONWUNTA +with any ε0 > 0. Then the minimizer uγ of (3.3) converges to the solution u∗ of the exact +problem (2.1)–(2.3), and +∥uγ − u∗∥2 +U ≤ Cε3 +0γ−1/2 + 1 +α⟨λ∗, Suγ − ymax⟩Y∗,Y → 0, +γ → ∞, +where C > 0 is independent of γ and ε0. +Remark 3.13. For the classical Moreau-Yosida penalty with ε0 = 0, we recover existing +convergence estimates [21, 2] that depend only on ⟨λ∗, Suγ − ymax⟩Y∗,Y. This term converges +to 0 as shown in (3.26), but the rate of this convergence can be estimated only if bounds on +∥λ∗∥L2(Ω,D) or ∥Suγ −ymax∥Y can be established from other sources, such as the discretization +of Y [21, Theorem 3.7]. +Proof. We aim at refining the estimate (3.22). Specifically, we need to lower-bound ⟨(Suγ − +ymax)−, gε(Suγ − ymax)⟩Y∗,Y, where (y)− = min(y, 0). +For brevity, let f(x, ξ) = Suγ − +ymax(x, ξ). Using the particular form of duality pairing and Assumption 3.1, we can write +⟨(Suγ − ymax)−, gε(Suγ − ymax)⟩Y∗,Y = +ˆ +Rd +ˆ +D +(f)−gε(f)dxπ(ξ1) · · · π(ξd)dξ += +ˆ +D +ˆ +f(x,ξ)≤0 +fgε(f)π(ξ1) · · · π(ξd)dξdx. +(3.27) +Introduce a change of variables +� +���� +ξ1 +ξ2... +ξd +� +���� → +� +���� +f(x, ξ) +ξ2... +ξd +� +���� +with the Jacobian +J := +��������� +det +� +���� +∂f +∂ξ1 +∂f +∂ξ2 +· · · +∂f +∂ξd +0 +1 +· · · +0 +... +0 +· · · +0 +1 +� +���� +��������� += +���� +∂f +∂ξ1 +���� ≥ c > 0. +Now we can express (3.27) using univariate integration, +⟨(Suγ − ymax)−, gε(Suγ − ymax)⟩Y∗,Y = +ˆ +D +ˆ 0 +min f +ˆ +Rd−1 fgε(f)J−1π(ξ1(f)) · · · π(ξd)dξ2 · · · dξddfdx +≥ +ˆ +D +ˆ 0 +−∞ +fgε(f)1 +cPdfdx +≥ −|D|P 1 +cε3, +where in the second line we used that the expression under the integral is nonpositive, and +´ +π(x2)dx2 = · · · = +´ +π(xd)dxd = 1, and in the third line we used Lemma 3.11. + +STATE-CONSTRAINED OPTIMIZATION PROBLEMS UNDER UNCERTAINTY +11 +Now we can replace (3.22) as follows: +⟨λγ, Suγ − Su∗⟩Y∗,Y ≥ γ +� +0.5∥(Suγ − ymax)+∥2 +L2(Ω,D) − 0.5|D|P 1 +cε3 +� +. +Proceeding as in Theorem 3.10, we replace (3.25) by +α∥uγ − u∗∥2 +U ≤ ⟨λ∗, Suγ − ymax⟩Y∗,Y + γ +2|D|P 1 +cε3. +Setting ε = ε0/√γ, we obtain that +∥uγ − u∗∥2 +U ≤ 1 +α⟨λ∗, Suγ − ymax⟩Y∗,Y + |D|P +2cα +� �� � +C +ε3 +0 +γ1/2. +Thus the proof is complete. +□ +Remark 3.14. This theorem can be generalized to vector-valued functions straightforwardly. +Indeed, if fi(x, ξ) denotes the ith component of a vector function, the duality pairing (3.27) +reads +⟨(f)−, gε(f)⟩Y∗,Y = +ˆ +Rd +ˆ +D +� +i +(fi)−gε(fi)dxπ(ξ)dξ = +� +i +ˆ +D +ˆ +fi(x,ξ)≤0 +figε(fi)π(ξ)dξdx, +and ξ1 can be changed to fi for each term of the sum over i. +The assumption of a lower bound of the Jacobian is practical. +The Karhunen-Loeve +expansion as in Example 3.2 is normally derived as the eigenvalue expansion of the covariance +function of e.g. +κ. +By the Perron-Frobenius theorem, ψ1(x) = ∂κ/∂ξ1 > 0. +Further, +∂y/∂κ ̸= 0 due to ellipticity. Hence ∂(Su)/∂ξ1 ̸= 0 whenever either u or boundary conditions +or source term are nonzero. The remaining assumptions of Thm. 3.12 are also reasonable for +practical solutions of regularized optimization problems. A convenient observation is that +ε = ε0/√γ is the sufficient condition on the law of decay of the smoothing parameter for +both Theorems 3.4 and 3.12. +4. Tensor-Train decomposition +Throughout this section, we use Assumption 3.1. Recall that the bottleneck is the com- +putation of the expectation in e.g. gradient (3.5). While it may be possible to use a Monte +Carlo quadrature, its convergence is usually slow, which may make estimates of small values +of the gradient near the optimum particularly inaccurate. In this section, we describe the +Tensor-Train (TT) decomposition as a function approximation technique that allows fast +computation of the expectation. The original TT decomposition [30] was proposed for ten- +sors (such as tensors of expansion coefficients), and the functional TT (FTT) decomposition +[5, 19] has extended this idea to multivariate functions. +Let us introduce a basis {ℓi(ξk)} +nξ +i=1 in each random variable ξk, k = 1, . . . , d, and a +quadrature with nodes Z = {zj} and weights {wj} which is exact on this basis, +E[ℓi] = +nξ +� +j=1 +wjℓi(zj). + +12 +HARBIR ANTIL, SERGEY DOLGOV, AND AKWUM ONWUNTA +For example, we can take Lagrange interpolation polynomials built upon a Gaussian quadra- +ture, or orthogonal polynomials up to degree nξ − 1 together with the roots of the degree-nξ +polynomial, or Fourier modes and the rectangular quadrature with the number of nodes +corresponding to the highest frequency. Then we can approximate any random field y ∈ Y +in the tensor product basis, +y(ξ) ≈ +nξ +� +i1=1 +· · · +nξ +� +id=1 +Yi1,...,idℓi1(ξ1) · · · ℓid(ξd). +Note that the expansion coefficients Y form a tensor of nd +ξ entries, which is impossible to +store directly if d is large. The TT decomposition aims to factorize this tensor further to a +product of tensors of manageable size. +Definition 4.1. A tensor Y ∈ Rnξ×···×nξ is said to be approximated by the TT decomposition +with a relative approximation error ϵ if there exist 3-dimensional tensors Y(k) ∈ Rrk−1×nξ×rk, +k = 1, . . . , d, such that +˜Yi1,...,id := +r0,...,rd +� +s0,...,sd=1 +Y(1) +s0,i1,s1Y(2) +s1,i2,s2 · · · Y(d) +sd−1,id,sd, +(4.1) +and ∥Y− ˜Y∥F = ϵ∥Y∥F. The factors Y(k) are called TT cores, and the ranges of summation +indices r0, . . . , rd ∈ N are called TT ranks. Note that without loss of generality we can let +r0 = rd = 1. +Plugging in the basis and redistributing the summations we obtain the FTT approximation +˜y(ξ) := +r0,...,rd +� +s0,...,sd=1 +y(1) +s0,s1(ξ1)y(2) +s1,s2(ξ2) · · · y(d) +sd−1,sd(ξd), +where +y(k) +sk−1,sk(ξk) = +nξ +� +i=1 +Y(k) +sk−1,i,skℓi(ξk), +k = 1, . . . , d. +Smooth [35], weakly correlated [33] or certainly structured [20] functions have been shown +to induce rapidly converging TT approximations. +Given the TT decomposition, its expectation can be computed by first integrating each +TT core, and then multiplying the TT cores one by one. Let +V(k) +sk−1,sk = +nξ +� +j=1 +wjy(k) +sk−1,sk(zj) = +nξ +� +i,j=1 +wjLi,jY(k) +sk−1,i,sk, +where +Li,j = ℓi(zj). +(4.2) +Now we multiply the matrices V(k) ∈ Rrk−1×rk in order: +E[˜y] = +��� +V(1)V(2)� +V(3) +� +· · · V(d) +� +. +(4.3) +Note that each step in (4.3) is a product of 1 × rk−1 vector by rk−1 × rk matrix. In turn, the +univariate quadrature (4.2) requires n2 +ξrk−1rk floating point operations if the Vandermonde +matrix L is dense, and nξrk−1rk if it’s sparse, for example, if Lagrange polynomials are used. + +STATE-CONSTRAINED OPTIMIZATION PROBLEMS UNDER UNCERTAINTY +13 +Introducing r := maxk rk, we conclude that the expectation of a TT decomposition can be +computed with a complexity O(dr2) which is linear in the dimension. +To compute a TT approximation, we employ the TT-Cross algorithm [31]. We start with +an empirical risk minimization problem +min +Y(1),...,Y(d) +N +� +j=1 +� +˜y(ξj) − y(ξj) +�2 +, +where Ξ = {ξj} is a certain set of samples. To avoid minimization over all Y(1), . . . , Y(d) +simultaneously (which is non-convex), we switch to an alternating direction approach: iterate +over k = 1, . . . , d, solving in each step +min +Y(k) +N +� +j=1 +� +˜y(ξj) − y(ξj) +�2 +. +(4.4) +This problem can be solved by linear normal equations. Indeed, introduce a matrix Y̸=k ∈ +RN×(rk−1nξrk) with elements +(Y̸=k)j,t = +� +s0,...,sk−2 +y(1) +s0,s1(ξj +1) · · · y(k−1) +sk−2,sk−1(ξj +k−1)ℓi(ξj +k) +� +sk+1,...,sd +y(k+1) +sk,sk+1(ξj +k+1) · · · y(d) +sd−1,sd(ξj +d), +where t = (sk−1 − 1)nξrk + (i − 1)rk + sk, and a vector y(k) ∈ Rrk−1nξrk with elements +y(k) +t += Y(k) +sk−1,i,sk. Now ˜y(Ξ) = Y̸=ky(k), and (4.4) is minimized by +y(k) = (Y⊤ +̸=kY̸=k)−1(Y⊤ +̸=ky(Ξ)). +(4.5) +To both select “good” sample set Ξ and simplify the assembly of Y̸=k, we restrict the set +to have the Cartesian form +Ξ = Ξk, +where Ξk = {(ξk+1, . . . , ξd)} with nestedness conditions +(ξ1, . . . , ξk−1, ξk) ∈ Ξk−1 ⇒ (ξk+1, . . . , ξd) ∈ Ξ>k. +This makes +Y̸=k = Yk, +where +(Yk)j,s = +� +sk+1,...,sd +y(k+1) +s,sk+1(ξj +k+1) · · · y(d) +sd−1,sd(ξj +d), +(ξj +k+1, . . . , ξj +d) ∈ Ξ>k. +Moreover, Yk−1 are submatrices of +Y≤k := +� +�� +Yk +· · · +y(k)(znξ)Y>k +� +, +(4.6) + +14 +HARBIR ANTIL, SERGEY DOLGOV, AND AKWUM ONWUNTA +respectively. This allows us to build the sampling sets by selecting rk rows of Y≤k (resp. +columns of Y≥k) by the maximum volume principle [18], which needs only O(nξr3) floating +point operations per single matrix Y≤k or Y≥k. The rk indices of e.g. rows of Y≤k con- +stituting the maximum volume submatrix Yk−1 is constructed analogously. +This closes the recursion and allows us to carry out the alternating iteration in either di- +rection, k = 1, . . . , d or k = d, . . . , 1. By this construction, the cardinality of Ξk−1 is rk. Hence, the cardinality of Ξ is rk−1nξrk, and one full iteration of the TT-Cross +algorithm needs O(dnξr2) samples of y. +One drawback of the “naive” TT-Cross algorithm outlines above is that the TT ranks are +fixed. To adapt them to a desired error tolerance, several modifications have been proposed: +merge ξk, ξk+1 into one variable, optimize the corresponding larger TT core, and separate it +into two actual TT cores using truncated singular value decomposition (SVD) [34] or matrix +adaptive cross approximation [8]; oversample Ξk with random or error-targeting +points [10]; oversample the selection of submatrices from (4.6) by using the rectangular +maximum volume principle [29]. +However, in this paper we can pursue a somewhat more natural regression approach [7]. We +will always need to approximate a vector function, where different components correspond +to different degrees of freedom of an ODE or a PDE solution, or different components of +a gradient. Since the procedure to evaluate y is now taking two arguments (ξ and, say, +m = 1, . . . , M indexing extra degrees of freedom), we can replace the normal equations (4.5) +by +y(k)(m) = (Y⊤ +̸=kY̸=k)−1(Y⊤ +̸=ky(Ξ, m)), +which can be reshaped into a 4-dimensional tensor ˆY(k) ∈ Rrk−1×nξ×rk×M with elements +ˆY(k) +sk−1,i,sk,m = y(k) +t (m). To compute the usual 3-dimensional TT core, we can use a simple +Principal Component Analysis (PCA), which selects ˆr slices Y(k) +sk−1,i,1, . . . , Y(k) +sk−1,i,ˆr with the +minimal ˆr such that +min +W +� +sk−1,i,sk,m +� +� +ˆr +� +s=1 +Y(k) +sk−1,i,sWs,sk,m − ˆY(k) +sk−1,i,sk,m +� +� +2 +≤ tol2 · ∥ ˆY(k)∥2 +F. +Note that this problem is solved easily by the truncated SVD, where the new TT rank ˆr +can be chosen anywhere between 1 and min{rk−1nξ, rkM} to satisfy the error tolerance tol. +After replacing rk with ˆr, the TT-Cross iteration k = 1, . . . , d can proceed as previously. +In the last step (k = d), the PCA step is omitted, and we obtain the so-called block TT +decomposition [9], which in the functional form reads +˜y(ξ, m) = +� +s0,...,sd +y(1) +s0,s1(ξ1) · · · y(d−1) +sd−2,sd−1(ξd−1)ˆy(d) +sd−1,sd(ξd, m). +The “backward” iteration k = d, . . . , 1 can be generalized similarly. + +STATE-CONSTRAINED OPTIMIZATION PROBLEMS UNDER UNCERTAINTY +15 +5. Practical computation of the smoothed Moreau-Yosida optimization +To compute the gradient of the cost function (3.5), we need to approximate the function +under the expectation, +Gε,h +u (ξ) := Sh(ξ)∗ · diag(g′ +ε(Sh(ξ)u − yh +max(ξ))) · Mgε(Sh(ξ)u − yh +max(ξ)), +(5.1) +using the TT-Cross, followed by taking the expectation of the TT decomposition1 This can be +performed in two ways. To begin with, we can apply the TT-Cross algorithm to approximate +directly Gε,h +u (ξ). For each sample ξj ∈ Ξ, one needs to solve one forward problem to compute +Sh(ξj)u, and one adjoint problem to apply Sh(ξj)∗ to the rest of the function. Recall that +the TT-Cross needs O(dnξr2) samples, hence O(dnξr2) solutions of the forward, adjoint +and sensitivity problems. However, the maximal TT rank r of the softplus and sigmoid +functions typically grows proportional to 1/ε. When the solution of the forward and adjoint +problem is expensive (for example, in the PDE-constrained optimization), this may result in +an excessive computational complexity. +Alternatively, we can first compute TT approximations ˜y(ξ) ≈ Sh(ξ)u and ˜Sh(ξ)∗ ≈ +Sh(ξ)∗, followed by TT approximations ˜gε(ξ) :≈ gε(˜y(ξ) − yh +max(ξ)), ˜g′ +ε(ξ) :≈ g′ +ε(˜y(ξ) − +yh +max(ξ)), and finally ˜Gε,h +u (ξ) ≈ ˜Sh(ξ)∗diag(˜g′ +ε(ξ))˜gε(ξ) using the approximate solution ˜y(ξ), +which does not require the solution of the PDE anymore. The bottleneck now is the ap- +proximation of the matrix-valued function Sh(ξ)∗ ∈ Rnu×ny. If both ny and nu are large (for +example, in a case of a distributed control), the computation of Sh(ξ)∗ for each sample of ξ +requires assembling this large dense matrix, equivalent to the solution of the adjoint prob- +lem with nu right hand sides. Nevertheless, the tensor approximation of Sh(ξ)∗ converges +usually much faster (e.g. exponentially) compared to the approximation of Gε,h +u (ξ) directly, +hence the TT approximation of Sh(ξ)∗ may need much smaller TT ranks compared to the +TT approximation of Gε,h +u (ξ). In turn, the TT-Cross applied to Sh(ξ)∗ requires much fewer +solutions of the forward problem. For a moderate nu this makes it faster to precompute ˜y(ξ) +and ˜Sh(ξ)∗. The entire pseudocode of the smoothed Moreau-Yosida optimization is listed in +Algorithm 1. +6. Numerical examples +We start with γ0 = 1 and double γℓ+1 = 2γℓ in the course of the Newton iterations until +a desired value of γ∗ is reached. According to Theorem 3.4, we choose εℓ = 0.5/√γℓ. The +iteration is stopped when γL has reached the maximal desired value γ∗, and the step size has +become smaller than δmin = 10−3. We always take a zero control as the initial guess u0, and +θ = 10−4. All computations are carried out in MATLAB 2020b on a Intel Xeon E5-2640 v4 +CPU, using TT-Toolbox (https://github.com/oseledets/TT-Toolbox). +6.1. One-dimensional Elliptic PDE. We consider an elliptic PDE example from [22, 13]. +Here, a misfit functional +j(u) = 1 +2E +� +∥y(u, ω, x) − yd(x)∥2 +L2(D) +� ++ α +2 ∥u(x)∥2 +L2(D) +1Note that Gε,h +u (ξ) is a vector function with M being the number of degrees of freedom in the discretized +u. + +16 +HARBIR ANTIL, SERGEY DOLGOV, AND AKWUM ONWUNTA +Algorithm 1 Inexact projected Newton optimization with smoothed a.s. constraints +Require: Procedures to compute Shu, jh(u), ∇ujh(u), constraint yh +max, initial and maximal +Moreau-Yosida parameters γ0, γ∗, initial smoothing parameter ε0, initial control u0, ap- +proximation and stopping tolerance tol, maximal number of iterations L, Armijo tuning +parameter θ ∈ (0, 1), minimal step size δmin ∈ (0, 1). +Ensure: Optimized control uγ∗,h. +1: Set iteration number ℓ = 0, step size δ = 1, u−1 = u0. +2: while ℓ < L and δ > δmin and ∥uℓ − uℓ−1∥U > tol · ∥uℓ∥U or ℓ = 0 or γℓ < γ∗ do +3: +Set ε = ε0/√γℓ. +4: +Approximate ˜Gε,h +uℓ (ξ) ≈ Gε,h +uℓ (ξ) as shown in (5.1) using TT-Cross up to tolerance tol. +5: +Approximate ˜g′ +ε(ξ) ≈ g′ +ε(Sh(ξ)uℓ − yh +max(ξ)) using TT-Cross up to tolerance tol. +6: +Compute the gradient ∇ujγℓ,ε,h = ∇ujh(uℓ) + γℓE[ ˜Gε,h +uℓ (ξ)] +7: +Compute the anchor point ξ∗ = E[ξ · 1⊤˜g′ +ε(ξ)]/E[1⊤˜g′ +ε(ξ)]. +8: +Compute the Newton direction v = − ˜H−1∇ujγℓ,ε,h using (3.9). +9: +Set step size δ = 1. +10: +while jh(PUad(uℓ + δv)) > jh(uℓ) + δθ⟨v, ∇ujγℓ,ε,h⟩U∗,U and δ > δmin do +11: +Set δ = δ/2. +12: +end while +13: +Set uℓ+1 = PUad(uℓ + δv). +14: +Set γℓ+1 = min{2γℓ, γ∗}. +15: +Set ℓ = ℓ + 1. +16: end while +17: return uγ∗,h = uℓ. +is optimized subject to the stochastic PDE constraint2 +ν(ω)∆y(u, ω, x) = g(ω, x) + u(x), +(ω, x) ∈ Ω × D, +ν(ω) = 10ξ1(ω)−2, +g(ω, x) = ξ2(ω) +100 , +y|x=0 = −1 − ξ3(ω) +1000 , +y|x=1 = −2 + ξ4(ω) +1000 +(6.1) +where D = (0, 1), and ξ(ω) = (ξ1(ω), . . . , ξ4(ω)) ∼ U(−1, 1)4 is uniformly distributed. We +take the desired state yd(x) = − sin(50x/π) and the regularization parameter α = 10−2. +Moreover, we add the constraints +y(u, ω, x) ≤ ymax ≡ 0 +a.s., +and +− 0.75 ≤ u(x) ≤ 0.75 +a.e. +We discretize (6.1) in the spatial coordinate x using linear finite elements on a uniform +grid with ny interior points, and in each random variable ξi(ω) using nξ Gauss-Legendre +quadrature nodes on (−1, 1). Note that we exclude the boundary points x = 0 and x = 1 +due to the Dirichlet boundary conditions. This spatial discretization is used for both y and +u. +2Note that [22, 13] considered the constraint y ≥ 0, so here we reverse the sign of y to make the constraint +in the form (2.3). + +STATE-CONSTRAINED OPTIMIZATION PROBLEMS UNDER UNCERTAINTY +17 +Firstly, we study precomputation of the surrogate solution ˜y(ξ) and adjoint operator +˜S∗ +h(ξ). +We fix ny = 63, nξ = 65, the TT approximation tolerance 10−7 and the final +Moreau-Yosida regularization parameter γ∗ = 1000. +The direct computation of the TT +approximation of (5.1) requires 995 seconds of the CPU time due to the maximal TT rank +of 87. In contrast, ˜S∗ +h has the maximal TT rank of 8, and the computation of ˜S∗ +h requires +only 64 seconds despite a larger ny × ny TT core carrying the spatial variables. Using the +surrogates ˜y and ˜S∗ +h, the remaining computation of ∇ujγ,ε,h can be completed in less than 15 +seconds. The relative difference between the two approximations of ∇ujγ,ε,h is below the TT +approximation tolerance. This shows that the surrogate forward solution can significantly +speed up Algorithm 1 without degrading its convergence, so we use it in all remaining +experiments in this subsection. +0 +0.2 +0.4 +0.6 +0.8 +1 +−0.6 +−0.4 +−0.2 +0 +0.2 +0.4 +0.6 +x +u +γ∗ = 3000 +γ∗ = 1000 +γ∗ = 300 +γ∗ = 100 +0 +0.2 +0.4 +0.6 +0.8 +1 +−1.6 +−1.4 +−1.2 +−1 +−0.8 +−0.6 +−0.4 +−0.2 +0 +ymax = 0 +x +y +Figure 1. Left: control signals uγ∗(x) for different γ∗. Right: mean (solid +lines) and 95% confidence interval (shaded area, for γ∗ = 3000 only) of the +state y(uγ∗, ω, x). +In Figure 1 we show the solutions (control and state) for varying final Moreau-Yosida +penalty parameter γ∗, fixing ny = 63, nξ = 129 and the TT approximation tolerance of +10−6. We see that the solution converges with increasing γ∗, and larger γ∗ yields a smaller +probability of the constraint violation, albeit at a larger misfit cost j(u), as shown in Figure 2. +In particular, γ∗ > 300 gives a solution with less than 1% of the constraint violation, such +that the empirical 95% confidence interval computed using 1000 samples of the converged +field y(uγ∗) (see Fig. 1, right) is entirely within the constraint. +Finally, we study the convergence in the approximation parameters more systematically +in Figure 3. +In each plot we fix two out of three parameters: the final Moreau-Yosida +penalty γ∗, the number of discretization points in the random variables nξ, and the number + +18 +HARBIR ANTIL, SERGEY DOLGOV, AND AKWUM ONWUNTA +102 +103 +104 +10−3 +10−2 +γ∗ +102 +103 +104 +0.35 +0.36 +0.37 +0.38 +0.39 +γ∗ +Figure 2. Left: probability of the constraint violation, P(y(uγ∗, ω, x) > 0). +Right: total final cost j(uγ∗). +102 +102.5 +103 +10−1.5 +10−1 +10−0.5 +γ∗ +relative error +u +y +γ−0.75 +∗ +γ−0.5 +∗ +20 +40 +10−5 +10−4 +10−3 +10−2 +nξ +relative error +u +y +31 +65 +127 +255 +10−3 +10−2 +10−1 +ny +relative error +u +y +n−1.5 +y +Figure 3. Relative L2-norm difference from y and u to the reference solutions +with γ∗ = 104 with fixed nξ = 257, ny = 63 (left), nξ = 129 with fixed γ∗ = 100, +ny = 63 (middle) and ny = 511 with fixed γ∗ = 100, nξ = 25 (right). +of discretization points in space ny. In addition, we fix the TT approximation threshold +to 10−8 to reduce its influence. We observe a convergence in line with the γ−1/2 +∗ +rate of +Theorem 3.4, exponential in nξ (which is often the case for a polynomial approximation of +smooth functions [37]) until the tensor approximation error is hit, and between first and +second order in ny, which seems to be an interplay of the discretization consistency of the +linear finite elements (second order) and box constraints (first order). + +STATE-CONSTRAINED OPTIMIZATION PROBLEMS UNDER UNCERTAINTY +19 +6.2. Two-dimensional elliptic PDE. Now consider a two-dimensional extension of the +previous problem, +ν(ω)∆y(u, ω, x) = g(ω, x) + u(x), +(ω, x) ∈ Ω × D, +(6.2) +y|x1=0 = b1(ω)(1 − x2) + b2(ω)x2, +y|x2=1 = b2(ω)(1 − x1) + b3(ω)x1 +(6.3) +y|x1=1 = b4(ω)(1 − x2) + b3(ω)x2, +y|x2=0 = b1(ω)(1 − x1) + b4(ω)x1, +(6.4) +ν(ω) = 10ξ1(ω)−2, +g(ω, x) = ξ2(ω) +100 , +(6.5) +b1(ω) = −1 − ξ3(ω) +1000 , +b2(ω) = −2 + ξ4(ω) +1000 +, +(6.6) +b3(ω) = −1 − ξ5(ω) +1000 , +b4(ω) = −2 + ξ6(ω) +1000 +, +(6.7) +where D = (0, 1)2, and ξ(ω) = (ξ1(ω), . . . , ξ6(ω)) ∼ U(−1, 1)6 is uniformly distributed. We +optimize the regularized misfit functional +j(u) = 1 +2E +� +∥y(u, ω, x) − yd(x)∥2 +L2(D) +� ++ α +2 ∥u(x)∥2 +L2(D) +with the desired state yd(x) = − sin(50x1/π) cos(50x2/π) and the regularization parameter +α = 10−2, subject to constraints +y(u, ω, x) ≤ ymax ≡ 0 +a.s., +and +− 0.75 ≤ u(x) ≤ 0.75 +a.e. +We smooth the almost sure constraint by the Moreau-Yosida method with the ultimate +penalty parameter γ∗ = 102. +We discretize both y and u in (6.2) using bilinear finite elements on a ny × ny rectangular +grid. For the two-dimensional problem, the operator ˜S∗ +h is a dense matrix of size n2 +y × n2 +y, +which we are unable to precompute. Therefore, we use the TT-Cross to approximate Gε,h +u (ξ) +directly. +In Figure 4 we show the optimal control, mean and standard deviation of the solution for +ny = 63 and nξ = 17. We see that the mean solution reflects the desired state subject to the +constraints. The final cost j(uγ∗) is about 0.222634, and the probability of the constraint +violation is 0.0139223. The Newton method took L = 37 iterations to converge, the maximal +TT rank of ˜y(ξ) was 10 which was the same in all iterations, the maximal rank of g′ +ε(˜y−yh +max) +was 300, attained at the iteration after reaching γ∗ (iteration 9), and the maximal rank of +˜Gε,h +u (ξ) was 56 (in the final iterations). The computation took about a day of CPU time. +However, these TT ranks are comparable to those in the one-dimensional example. This +shows that the proposed technique can be also applied to a high-dimensional physical space, +including complex domains and non-uniform grids, since the TT structure is independent of +the spatial discretization. +6.3. Variational inequality constraints. In this section we minimize the regularized mis- +fit +j(u) = 1 +2E[∥y(u, ω, x) − yd(x)∥2 +L2(D)] + 1 +2∥u(x)∥2 +L2(D) +(6.8) + +20 +HARBIR ANTIL, SERGEY DOLGOV, AND AKWUM ONWUNTA +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +-0.9 +-0.8 +-0.7 +-0.6 +-0.5 +-0.4 +-0.3 +-0.2 +-0.1 +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +0.05 +0.1 +0.15 +0.2 +0.25 +Figure 4. Left: control signal uγ∗(x). Middle: mean E[y(uγ∗, ω, x)]. Right: +standard deviation +� +E[(y(uγ∗, ω, x) − E[y(uγ∗, ω, x)])2]. +subject to a random elliptic variational inequality (VI) constraint, +y(u, ω, x) ≤ 0 : +⟨A(ω)y(u, ω, x)−f(ω, x)−B(ω, x)u, y(u, ω, x)−v⟩ ≤ 0, +∀v : v ≤ 0. (6.9) +We use Example 5.1 from [1] (with the reversed sign of y), where D = (0, 1)2, A = −∆, +B = Id, and deterministic functions constructing the desired state: +ˆy(x) = +� +160(x3 +1 − x2 +1 + 0.25x1)(x3 +2 − x2 +2 + 0.25x2) +in (0, 0.5)2, +0, +otherwise, +ˆζ(x) = max(0, −2|x1 − 0.8| − 2|x1x2 − 0.3| + 0.5), +yd(x) = −ˆy − ˆζ + ∆ˆy. +In contrast, the right hand side depends on the random variables, +f(ξ(ω), x) = ∆ˆy + ˆy + ˆζ + b(ξ(ω), x), +b(ξ(ω), x) = +��d +i=1 +√λiφi(x)ξi(ω), +in (0, 0.5) × (0, 1), +0, +otherwise. +The Karhunen-Loeve expansion in b(ξ, x) is an affine-uniform random field, with ξi(ω) ∼ +U(−1, 1), φi(x) = 2 cos(πjx2) cos(πkx1) and λi = +1 +100 exp(− π +4(j2 + k2)), where the pairs +(j, k), j, k = 1, 2, . . . , are permuted such that λ1 ≥ λ2 ≥ · · · . +The VI (6.9) is replaced by the penalized problem +Ay + 1 +εgε(y) = f(ξ, x) + Bu, +(6.10) +so we minimize (6.8) with y(u, ξ, x) plugged in from (6.10). The latter equation is solved +via the Newton method, initialized with y = 0 as the initial guess, and stopped when the +relative difference between two consecutive iterations of y falls below 10−12. The problem is +discretized in x via the piecewise bilinear finite elements on a uniform ny × ny grid with cell +size h = 1/(ny + 1). The homogeneous Dirichlet boundary conditions y = 0 on ∂D allow us +to store only interior grid points. This gives us a discrete problem of minimizing +jh(u) = 1 +2E[∥y(u, ξ) − yd∥2 +Mh] + 1 +2∥u∥2 +Mh +(6.11) + +STATE-CONSTRAINED OPTIMIZATION PROBLEMS UNDER UNCERTAINTY +21 +subject to +Ahy + 1 +εgε(y) = f(ξ) + u, +(6.12) +where Ah, Mh ∈ Rn2 +y×n2 +y are the stiffness and mass matrices, respectively. +The state part of the cost +jy(u, ξ) = 1 +2∥y(u, ξ) − yd∥2 +Mh +and its gradient +∇ujy(u, ξ) = S∗ +h(ξ)Mh(y(u, ξ) − yd) +are approximated by the TT-Cross (as functions of ξ), which allows one to compute the ex- +pectation of ˜jy(u, ξ) ≈ jy(u, ξ) and ∇u˜jy(u, ξ) ≈ ∇ujy(u, ξ) easily. The forward model (6.12) +is solved at each evaluation of ξ in the TT-Cross. However, to avoid excessive computations, +the Hessian of (6.11) is approximated by that anchored at the mean point ξ = 0: +∇uujh(u) ≈ ˜H := S∗ +h(0)MhS′ +h(0) + Mh. +The Newton system ˜H−1∇ujh is solved iteratively by using the CG method, since the matrix- +vector product with ˜H requires the solution of only one forward and one adjoint problem, +S∗ +h · v = S′ +h · v = +� +Ah + diag +�1 +εg′ +ε(y) +��−1 +v, +∀v ∈ Rn2 +y. +(6.13) +In Table 1 we vary the dimension of the random variable d, the number of quadrature +points in each random variable nξ, and the approximation tolerance in the TT-Cross (tol). +The spatial grid size is fixed to ny = 31, which is comparable with the resolution in [1], +and the smoothing parameter ε = 10−6. As a reference solution u∗, we take the control +computed with d = 20, nξ = 5 and tol = 10−4. We see that the control and the cost can be +approximated quite accurately even with a very low order of the polynomial approximation +in ξ. It also seems unnecessary to keep 20 terms in the Karhunen-Loeve expansion. +The computation complexity is dominated by the solutions of the forward and adjoint +problems. The article [1] reports a “# PDE solves” in a path-following stochastic variance +reduced gradient method solving (6.8)–(6.9). We believe this indicates the number of the +complete solutions of the PDE (6.12). However, each solution of (6.12) to the increment +tolerance 10−12 requires 23–25 Newton iterations, each of which requires the linear system +solution of the form (6.13), Moreover, the anchored outer Hessian ˜H requires two extra linear +solves. Therefore, in Table 1, we show both the number of PDE solutions till convergence, +Npde, and the number of all linear system solutions Nlin, occurred during the optimization +of (6.11) till the relative increment of u falls below the TT-Cross tolerance. In addition, +we report the maximal TT ranks of the state cost gradient and the state itself. Note that +assembly of the full state is not needed during the optimization of (6.11) – only certain +samples of y(u, ξ) are needed in the TT-Cross approximation of ∇ujh. To save the computing +time, the TT tensor of the entire state is computed only after the optimization of u has +converged. +In Figure 5 we show the mean optimized forward state and the control. +The results +coincide qualitatively with those in [1]. If we consider the computational cost necessary to + +22 +HARBIR ANTIL, SERGEY DOLGOV, AND AKWUM ONWUNTA +Table 1. Cost, error in the control, number of solutions of n2 +y × n2 +y linear +system as in (6.13), number of complete forward PDE solutions (6.12), and +the TT ranks of the cost gradient and forward solution. +d +nξ +tol +jh(u) +∥u−u∗∥Mh +∥u∗∥Mh +Nlin +Npde +r(∇u˜jy) +r(˜y) +10 +5 +10−4 +1.261333069 +1.1473e-06 +1070007 +44584 +85 +316 +20 +3 +10−3 +1.261333069 +2.9012e-05 +46312 +1976 +7 +29 +20 +3 +10−4 +1.261333069 +4.2713e-06 +433134 +18153 +56 +183 +20 +5 +10−4 +1.261333069 +— +1840467 +76243 +102 +402 +Figure 5. Left: mean optimised state E[−y] with d = 20, nξ = 3 and tol = +10−3. Middle: variance of the optimized state E[(y −E[y])2]. Right: optimised +control u. +compute the optimal control only, we can notice that Npde is significantly lower than the +291808 PDE solves in the stochastic variance reduced gradient method of [1]. +6.4. SEIR ODE model. Now consider a slightly simplified version of the epidemiological +ODE model used for the propagation of COVID-19 in the UK using the data from March- +May 2020 [11]. +This is a compartmental differential equation model with the following +compartments. +• Susceptible (S). +• Exposed (E), but not yet infectious. +• Infected SubClinical type 1 (ISC1): may require hospitalization in the future. +• Infected SubClinical type 2 (ISC2): will recover without hospitalization. +• Infected Clinical type 1 (IC1): individuals in the hospital who may decease. +• Infected Clinical type 2 (IC2): individuals in the hospital who will recover. +• Recovered (R) and immune to reinfections. +• Deceased (D). +In turn, each of these compartments are split into 5 further sub-compartments corresponding +to age bands: 0-19, 20-39, 40-59, 60-79 and 80+. The number of individuals in each com- +partment is denoted by the name of the compartment and age band index, For example, Si +denotes the number of susceptible individuals in the ith age band (i = 1, . . . , 5), Ei denotes +the number of exposed individuals in the ith age band, and so on. Variables corresponding + +0.06 +0.04 +0.02 +0 +0.2 +0.4 +0.5 +0.6 +0.810-7 +.5 +0.5 +O +0 +0.5 +0.5 +1.50.06 +0.04 +0.02 +0 +0.2 +0.4 +0.5 +0.6 +0.8STATE-CONSTRAINED OPTIMIZATION PROBLEMS UNDER UNCERTAINTY +23 +to different age bands but same compartment are collected into vectors, S = (S1, . . . , S5), +E = (E1, . . . , E5) and so on. +Some of the variables introduced above are coupled to others only one way, and can be +removed from the actual simulations. +First, when the number of infected individuals is +small compared to the population size (which is typically the case in the early stages of the +epidemic), the relative variation of S is small. Hence, S can be taken constant instead of +solving an ODE on it. Similarly, none of the variables depend on R and D, so they can be +excluded from a coupled system of ODEs too, and computed separately after the solution of +the ODEs. With these considerations in mind, the forward model reads as follows: +d +dt +� +����� +E +ISC1 +ISC2 +IC1 +IC2 +� +����� +− +� +����� +−κI +Au +Au +0 +0 +κ · diag(ρ) +−ηCI +0 +0 +0 +κ · diag(1 − ρ) +0 +−ηRI +0 +0 +0 +ηC · diag(ρ′) +0 +−νI +0 +0 +ηC · diag(1 − ρ′) +0 +0 +−ηR,CI +� +����� +� +����� +E +ISC1 +ISC2 +IC1 +IC2 +� +����� += 0. +(6.14) +Here I ∈ R5×5 is the identity matrix and diag(·) produces a diagonal matrix from a vec- +tor. The control is defined in terms of the intensity of lockdown measures, and affects the +susceptible-infected interaction matrix Au = χ · diag(S) · Cu · diag( 1 +N ), where +Cu = diag(chome)Chome + diag(cwork +u +)Cwork + diag(cschool +u +)Cschool + diag(cother +u +)Cother (6.15) +is the matrix of contact intensities between the age compartments. The total contact inten- +sity is a sum of pre-pandemic contact intensity matrices in the four setting Chome, Cwork, Cschool +and Cother, multiplied by the reduction factors chome, cwork +u +, cschool +u +and cother +u +due to the lock- +down measures. +Since home contacts cannot be controlled, chome = (1, . . . , 1), but the +remaining factors vary proportionally to the lockdown control applied from day 17 onwards, +cµ +u(t) = +� +� +� +(1, 1, 1, 1, 1)⊤, +t < 17, +(c123(1 − uµ(t)), c123(1 − uµ(t)), c123(1 − uµ(t)), c4, c5)⊤, +17 ≤ t ≤ 90, +(c123(1 − uµ(90)), c123(1 − uµ(90)), c123(1 − uµ(90)), c4, c5)⊤, +t > 90, +(6.16) +where µ ∈ {work, school, other}, uµ are the intensities of lockdown measures applied to each +setting µ, and c123, c4, c5 are the initial contact intensities in the corresponding age groups. +Note that the control will be optimized only on the time interval [17, 90]. Before day 17 +the contact intensities are not reduced (no lockdown). From day 90 onwards we continue +applying the last value of the control. +In addition, the model depends on the following parameters: +• χ: probability of S–ISC interactions. +• κ = 1/dL: average rate of an Exposed individual becoming SubClinical. It is inversely +proportional to the average number of days dL an individual stays in the Exposed +state. +• ηC = 1/dC: average rate of a SubClinical individual becoming Clinical. Similarly, dC +is the average time spent in the SubClinical state. +• ηR = 1/dR: rate of recovery from ISC2. +• ηR,C = 1/dR,C: rate of recovery from IC2. + +24 +HARBIR ANTIL, SERGEY DOLGOV, AND AKWUM ONWUNTA +• ν = 1/dD: rate of decease in the IC1 state. +• ρ = (ρ1, . . . , ρ5)⊤ ∈ R5: correction coefficients of the Exposed → SubClinical 1 tran- +sition rate for different age bands. +• ρ′ = (ρ′ +1, . . . , ρ′ +5)⊤ ∈ R5: correction coefficients of the SubClinical → Clinical 1 tran- +sition. +• N = (N1, . . . , N5)⊤ ∈ R5: total number of individuals in each age group. +• N 0: total number of infected individuals on day 0. +• N in = (0.1, 0.4, 0.35, 0.1, 0.05)⊤N 0: age partition of the initial number of infected +individuals. +The ODE (6.14) is initialized by setting +E(0) = N in +3 , +ISC1(0) = 2 +3diag(ρ)N in, +ISC2(0) = 2 +3diag(1−ρ)N in, +IC1(0) = IC2(0) = 0. +The population sizes S = N are taken from the Office for National Statistics, mid 2018 +estimate. +However, none of the model parameters above are known beforehand. In [11], those were +treated as random variables, and their distributions were estimated from observed numbers +of infections and hospitalizations during the first 90 days using Approximate Bayesian Com- +putation (ABC). In general, these variables are correlated through the posterior distribution, +sampling from which is a daunting problem. Here, we replace the joint ABC posterior dis- +tribution by independent uniform distributions with a scaled posterior standard deviation +centered around the posterior mean: +χ ∼ U(0.13 − 0.03σ, 0.13 + 0.03σ), +dL ∼ U(1.57 − 0.42σ, 1.57 + 0.42σ), +(6.17) +dC ∼ U(2.12 − 0.80σ, 2.12 + 0.80σ), +dR ∼ U(1.54 − 0.40σ, 1.54 + 0.40σ), +dR,C ∼ U(12.08 − 1.51σ, 12.08 + 1.51σ), +dD ∼ U(5.54 − 2.19σ, 5.54 + 2.19σ), +ρ1 ∼ U(0.06 − 0.03σ, 0.06 + 0.03σ), +ρ2 ∼ U(0.05 − 0.03σ, 0.05 + 0.03σ), +ρ3 ∼ U(0.08 − 0.04σ, 0.08 + 0.04σ), +ρ4 ∼ U(0.54 − 0.22σ, 0.54 + 0.22σ), +ρ5 ∼ U(0.79 − 0.14σ, 0.79 + 0.14σ), +ρ′ +1 ∼ U(0.26 − 0.23σ, 0.26 + 0.23σ), +ρ′ +2 ∼ U(0.28 − 0.25σ, 0.28 + 0.25σ), +ρ′ +3 ∼ U(0.33 − 0.27σ, 0.33 + 0.27σ), +ρ′ +4 ∼ U(0.26 − 0.11σ, 0.26 + 0.11σ), +ρ′ +5 ∼ U(0.80 − 0.13σ, 0.80 + 0.13σ), +N 0 ∼ U(276 − 133σ, 276 + 133σ), +c123 ∼ U(0.63 − 0.21σ, 0.63 + 0.21σ), +c4 ∼ U(0.57 − 0.23σ, 0.57 + 0.23σ), +c5 ∼ U(0.71 − 0.23σ, 0.71 + 0.23σ). +Here, σ is the standard deviation scaling parameter, taken to be 0.03 in our experiment. +This distribution behaves qualitatively similar to the posterior distribution in the vicinity +of the posterior mean. +It provides sufficient randomness to benchmark the constrained +optimization method, while admitting independent sampling and gridding, needed for the +TT approximations. That is, (6.17) form a random vector +ξ = (χ, dL, dC, dR, dR,C, dD, ρ1, ρ2, ρ3, ρ4, ρ5, ρ′ +1, ρ′ +2, ρ′ +3, ρ′ +4, ρ′ +5, N 0, c123, c4, c5) +of d = 20 independent random variables, the state vector is +y(ξ, t) = (E1, . . . , E5, ISC1 +1 +, . . . , ISC1 +5 +, ISC2 +1 +, . . . , ISC2 +5 +, IC1 +1 , . . . , IC1 +5 , IC2 +1 , . . . , IC2 +5 ), + +STATE-CONSTRAINED OPTIMIZATION PROBLEMS UNDER UNCERTAINTY +25 +and the ODE (6.14) constitutes the forward problem. +For the inverse problem, we use the total number of deceased patients as the cost function. +The rate of decease is proportional to the number of Clinical type 1 individuals, so the total +number of deceased individuals can be computed as +D(ξ, t) = ν +ˆ t +0 +IC1(ξ, s)ds. +(6.18) +To regularize the problem, we add also the norm of the control u(t) = (uwork(t), uschool(t), uother(t)). +Thus, the total cost function reads +j(u) = 1 +2E[D(ξ, T)] + α +2 +ˆ 90 +17 +∥u(t)∥2 +2dt, +(6.19) +where T = 100 is the final simulation time, and α is the regularization parameter, which we +set to 100 in our experiment. Note that the norm of the control is taken only over the time +interval [17, 90] where the control varies. +We introduce the following constraints. Firstly, we limit the control components to the +intervals uwork ∈ [0, 0.69], uschool ∈ [0, 0.9] and uother ∈ [0, 0.59]. Next, we constrain the +R number at the end of the variable control interval, R(ξ, 90) ≤ 1. In our model, the R +number can be computed as R(ξ, t) = λmax(K), where +K = − +� +����� +0 +Au +Au +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +� +����� +� +����� +−κI +0 +0 +0 +0 +κ · diag(ρ) +−ηCI +0 +0 +0 +κ · diag(1 − ρ) +0 +−ηRI +0 +0 +0 +ηC · diag(ρ′) +0 +−νI +0 +0 +ηC · diag(1 − ρ′) +0 +0 +−ηR,CI +� +����� +−1 +, +and λmax denotes the maximal in modulus eigenvalue. Recall that R < 1 implies that the +epidemic decays, while R > 1 corresponds to an expanding epidemic. The full smoothed +Moreau-Yosida cost function becomes +jγ,ε(u) = 1 +2E[D(ξ, T)] + α +2 +ˆ 90 +17 +∥u(t)∥2 +2dt + γ +2E +���gε(R(ξ, 90) − 1) +��2� +. +(6.20) +Since the control is applied nonlinearly in the model, computation of derivatives of the cost +function (6.20) is complicated. Thus, instead of the Newton method, we use the projected +gradient descent method, where the gradient of (6.20) is calculated using finite differencing +with anisotropic step sizes 10−6 · max(|u|, 0.1). The ODE (6.14) is solved using an implicit +Euler method with a time step 0.1. +In this experiment, we use a fixed Moreau-Yosida +parameter γ = 5 · 105 in all iterations, and the smoothing width is chosen as ε = 50/√γ. +The iteration is stopped when the cost value does not decrease in two consecutive iterations. +Each random variable (6.17) is discretized with n = 3 Gauss-Legendre quadrature nodes, and +the TT approximations are carried out with a relative error tolerance of 10−2. The control +u(t) is discretized using 7 Gauss-Legendre nodes on [17, 90] with a Lagrangian interpolation +in between. +In Figure 6, we compare optimizations without constraining R(ξ, 90) (left), and with the +a.s. constraint (right) as described above. We plot the time evolution of the mean and +confidence interval of the total number of hospitalized individuals, IC(t) = IC1(t) + IC2(t). + +26 +HARBIR ANTIL, SERGEY DOLGOV, AND AKWUM ONWUNTA +0 +20 +40 +60 +80 +100 +0 +5 +10 +15 +20 +25 +t (days) +hospitalizations (thousands) +0 +20 +40 +60 +80 +100 +0 +5 +10 +15 +20 +25 +t (days) +hospitalizations (thousands) +0 +20 +40 +60 +80 +100 +0 +0.2 +0.4 +0.6 +0.8 +1 +t (days) +u +work +school +other +0 +20 +40 +60 +80 +100 +0 +0.2 +0.4 +0.6 +0.8 +1 +t (days) +u +work +school +other +Figure 6. Top: optimized IC = IC1 + IC2, mean (blue circles) and 95% +confidence interval (shaded area). Bottom: optimized control signals. Left: +unconstrained optimization, Right: optimization constrained with R(ξ, 90) ≤ +1 a.s. approximated with γ = 5 · 105. Black dashed lines indicate the end of +the optimization time horizon t = 90. +The unconstrained scenario is a finite horizon optimization problem, which drives the control +to near zero values at the end of the controllable time interval, t = 90, due to the zero terminal +condition on the adjoint state. Naturally, this leads to infection growing again for t > 90, +since we extrapolate these small values of the control from t = 90 onwards. +In contrast, if we constrain the R number at the end of the optimization interval to be +below 1 almost surely, this drives the control to higher values again. If we extrapolate these + +STATE-CONSTRAINED OPTIMIZATION PROBLEMS UNDER UNCERTAINTY +27 +control values beyond the optimization window, the epidemic continues decaying, albeit with +a slightly larger uncertainty. This indicates that almost sure constraints can suggest a more +resilient control in risk-critical applications. +Appendix A. Proof of Lemma 3.11 +Introduce a new variable t = exp(s/ε), then +ˆ 0 +−∞ +s log(1 + exp(s/ε))ds = +ˆ 1 +0 +ε log(t) log(1 + t) +t/ε +dt += ε2 +ˆ 1 +0 +log(t) log(t + 1)d log(t) += ε2 +2 (log(t))2 log(t + 1) +��1 +0 − ε2 +2 +ˆ 1 +0 +(log(t))2d log(t + 1). +The first term is zero at t = 1, and at t = 0 we can use that 0 ≤ log(t + 1) ≤ t for 0 ≤ t < 1 +and limt→0(log(t))2 log(t + 1) ≤ limt→0(log(t))2t = 0. For the second term, we proceed as +follows, +ˆ 0 +−∞ +s log(1 + exp(s/ε))ds = −ε2 +2 +ˆ 1 +0 +(log(t))2 +t + 1 +dt +≥ −ε2 +2 +ˆ 1 +0 +(log(t))2dt += −ε2 +2 t(log(t))2��1 +0 +� +�� +� +0 ++ε2 +ˆ 1 +0 +log(t)dt += ε2 t log t|1 +0 − ε2 +ˆ 1 +0 +dt = −ε2. +The proof is completed by recalling that sgε(s) = ε · s log(1 + exp(s/ε)). +References +[1] A. Alphonse, C. Geiersbach, M. Hinterm¨uller, and T. M. Surowiec. 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Optimal Control of Partial Differential Equations: Theory, Methods and Applications. +American Mathematical Society, 2010. +Harbir Antil, The Center for Mathematics and Artificial Intelligence (CMAI) and De- +partment of Mathematical Sciences, George Mason University, Fairfax, VA 22030, USA. +Email address: hantil@gmu.edu +Sergey Dolgov, Department of Mathematical Sciences, University of Bath, Bath, BA2 +7AY, UK. +Email address: s.dolgov@bath.ac.uk +Akwum Onwunta, Department of Industrial and Systems Engineering, Lehigh University, +Bethlehem, PA 18015, USA. +Email address: ako221@lehigh.edu + diff --git a/L9FAT4oBgHgl3EQfxB4e/content/tmp_files/load_file.txt b/L9FAT4oBgHgl3EQfxB4e/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..37d7f56974797565a665135860b098388d65bc53 --- /dev/null +++ b/L9FAT4oBgHgl3EQfxB4e/content/tmp_files/load_file.txt @@ -0,0 +1,1459 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf,len=1458 +page_content='STATE-CONSTRAINED OPTIMIZATION PROBLEMS UNDER UNCERTAINTY: A TENSOR TRAIN APPROACH HARBIR ANTIL, SERGEY DOLGOV, AND AKWUM ONWUNTA Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' We propose an algorithm to solve optimization problems constrained by partial (ordinary) differential equations under uncertainty, with almost sure constraints on the state variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' To alleviate the computational burden of high-dimensional random variables, we approximate all random fields by the tensor-train decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' To enable efficient tensor-train approximation of the state constraints, the latter are handled using the Moreau- Yosida penalty, with an additional smoothing of the positive part (plus/ReLU) function by a softplus function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' We derive theoretical bounds on the constraint violation in terms of the Moreau-Yosida regularization parameter and smoothing width of the softplus function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' This result also proposes a practical recipe for selecting these two parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' When the optimization problem is strongly convex, we establish strong convergence of the regularized solution to the optimal control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' We develop a second order Newton type method with a fast matrix-free action of the approximate Hessian to solve the smoothed Moreau-Yosida problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' This algorithm is tested on benchmark elliptic problems with random coefficients, optimization problems constrained by random elliptic variational inequalities, and a real- world epidemiological model with 20 random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' These examples demonstrate mild (at most polynomial) scaling with respect to the dimension and regularization parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Introduction Over last two decades optimization problems constrained by physical laws, such as partial (ordinary) differential equations (PDEs/ODEs), have emerged as a prominent research area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' This is fueled by many applications in science and engineering, such as controlling pathogen propagation in built environment [26, 25], shape and topology optimization [36, 28], optimal strategies to predict shutdowns due to pandemics [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' The optimization variables consist of state (y) and control/design (u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' However, often due to noisy measurements and ambiguous models due to incomplete physics, the underlying physical laws contain uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' This has led to significant theoretical and algorithmic developments in the area of optimization problems constrained by physical laws under uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' See for instance [23, 4, 14, 3] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' These papers focus on problems with control constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' The literature on state-constrained optimization problems under uncertainty is scarce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' For instance, [12, 17] use probability constraints, and [15, 13, 16] consider almost surely Date: 20 January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' 49J55, 93E20, 49K20, 49K45, 90C15, 65D15, 15A69, 15A23 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' almost surely constraints, state constraints, risk neutral, tensor train, reduced space, preconditioner, variational inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' HA is partially supported by NSF grant DMS-2110263 and the AirForce Office of Scientific Research under Award NO: FA9550-22-1-0248.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' SD is thankful for the support from Engineering and Physical Sciences Re- search Council (EPSRC) New Investigator Award EP/T031255/1 and New Horizons grant EP/V04771X/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='08684v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='OC] 20 Jan 2023 2 HARBIR ANTIL, SERGEY DOLGOV, AND AKWUM ONWUNTA type constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' It is well-known that even in the deterministic setting, the state con- strained problems are highly challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' One of the fundamental difficulties is that the state constraints are imposed in the sense of continuous functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' As a result, the Lagrange multipliers corresponding to those constraints are Radon measures that exhibit low regu- larity [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' The situation is much more delicate in the stochastic setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' We refer to the aforementioned references for a detailed discussion on this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Motivated by the deter- ministic setting, [13] introduces a Moreau-Yosida based approximation scheme to solve the state-constrained optimization problems when the PDE constraints are given by an elliptic equation with random coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Further extensions of this work are considered in [1, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' However, all of these papers approximate expectations of random fields by Monte-Carlo-type methods, which may converge slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' In [3], we introduced an algorithm (TTRISK) based on the tensor train (TT) decom- position [30] to solve risk-averse optimization problems with control constraints, and the conditional value-at-risk (CVaR) [32] risk measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' We demonstrated that the extra com- putational cost due to the uncertainty can scale proportionally to error−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='5 when the TT approximation is used, in contrast to a error−2 scaling of Monte Carlo quadratures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' In the current paper, we continue this program and develop a TT based algorithm for state- constrained optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' For simplicity of presentation, we only consider the risk-neutral setting, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=', the objective function is given by the expected value of a quantity of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Similarly to [13, 16], we tackle the state constrains using Moreau-Yosida based relaxation with a softplus smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' The main contributions of this paper are listed next: (i) We consider an ε-softplus regularization of the positive part function (·)+ = max{·, 0} and derive a probabilistic estimate of state constraint violation in terms of Moreau- Yosida regularization parameter γ and ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' In particular, we show that selecting ε ∝ γ−1/2 ensures the convergence of the constraint violation with a rate γ−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' This result is motivated by [13, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Notice that the ε-smoothing is carried out because the irregular function (·)+ may lack an efficient TT decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' (ii) When the optimization problem is strongly convex, we establish strong convergence of the regularized solution to the optimal control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Our final results can be seen as generalizations of the results in deterministic setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' (iii) We derive a second order Newton type method to solve the regularized problem with a fast matrix-free action of the approximate Hessian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' (iv) We test the proposed method on elliptic equations in one and two physical dimensions and random coefficients, as well as an ODE example (motivated by a realistic applica- tion) with 20 random variables, and show that the algorithm is free from the curse of dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' (v) The proposed approach has been also successfully applied to an example where the PDE constraint is given by an elliptic variational inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Outline: The remainder of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' In Section 2, we provide a rigorous mathematical formulation of the problem under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Section 3 is devoted to the Moreau-Yosida approximation, derivation of the second order Newton method and approximation error estimates due to the Moreau-Yosida approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' In Section 4, we provide a brief description of the TT format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' This is followed by practical aspects of Moreau- Yosida approximations in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Finally, in Section 6, we provide a series of numerical STATE-CONSTRAINED OPTIMIZATION PROBLEMS UNDER UNCERTAINTY 3 experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' At first, we consider an optimization problem with an elliptic PDE in one spatial dimension as constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' This is followed by a two-dimensional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' After these benchmarks, an optimization problem with an elliptic variational inequality as constraint is considered in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' The numerical experiments conclude with a realistic ODE example for designing optimal lockdown strategies in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Problem Formulation Let (Ω, F, P) denote a complete probability space, where Ω represents the sample space, F is the Borel σ-algebra of events on the power set of Ω, and P : Ω → [0, 1] is an appropriate probability measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' We denote by E[·] the expectation with respect to P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Let U be a real deterministic reflexive Banach space of optimization variables (control or design) defined on an open, bounded and connected set D ⊂ Rn with Lipschitz boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' We denote by ∥ · ∥U the norm on U, and the duality pairing between U and U ∗ as ⟨·, ·⟩U∗,U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Let Y = L2(Ω, F, P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' ˆY) and Z = L2(Ω, F, P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' ˆZ) be Bochner spaces of random fields, based on deterministic Banach spaces ˆY �→ L2(D) �→ ˆY∗ and ˆZ, with corresponding norms and duality pairings ∥y∥2 Y = E[∥y(ω)∥2 ˆY], ⟨y, v⟩Y∗,Y = E � ⟨y(ω), v(ω)⟩ ˆY∗, ˆY � , and similarly for Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Let Uad ⊆ U be a closed convex nonempty subset and let c : Y×Uad×Ω → Z denote, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=', a partial differential operator, then consider the equality constraint c(y, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' ω) = 0, in Z, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' ω ∈ Ω, where a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' indicates “almost surely” with respect to the probability measure P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' In this paper, we consider the optimization problems of the form min y,u R[J(y, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' ω)] (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='1) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='t c(y, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' ω) = 0, in Z, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' ω ∈ Ω, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='2) where R represents the risk measure and R[J(y, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' ω)] is a deterministic cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' More precisely, we will focus on the so-called risk-neutral formulation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' that is, R is simply the expectation, denoted by E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' We are particularly interested in the case in which the state variable y is constrained by a random variable: y ≤ ymax(ω) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=', (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='3) where we assume that ymax ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' In what follows, we discuss the Moreau-Yosida approximation for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='1)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='3) and derive a Newton type method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Throughout the paper, without explicitly stating, we will make use of the following assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='1 (unique forward solution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' There exists an injective operator S(ω) : Uad → Y (maybe nonlinear) such that c(S(ω)u, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' ω) = 0 ∀u ∈ Uad a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' This allows us to define the reduced-space cost function j(u) := R[J(S(ω)u, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' ω)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='4) 4 HARBIR ANTIL, SERGEY DOLGOV, AND AKWUM ONWUNTA The resulting reduced optimization problem is given by min u∈Uad j(u) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='t y ≤ ymax(ω) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='5) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Smoothed Moreau-Yosida approximation Solving (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='5) with state constraints involve computation of the indicator function of an active set and/or Lagrange multiplier as a random field that is nonnegative on a compli- cated high-dimensional domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' This may be difficult for many function approximation methods, especially for tensor decompositions that are considered in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' We tackle this difficulty by first turning the constrained optimization problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='5) into an uncon- strained optimization problem with the Moreau-Yosida penalty, and further by smoothing the indicator function in the penalty term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' The classical Moreau-Yosida problem reads, with γ ≥ 0 denoting the regularization pa- rameter, min u∈Uad jγ(u), where jγ(u) := j(u) + γ 2E ���(Su − ymax(ξ))+ ��2 L2(D) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='1) where the so-called positive part or ReLU function (·)+ reads (s)+ = s if s ≥ 0 and 0, oth- erwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Here, we have removed the need to optimize the Lagrange multiplier (corresponding to the inequality constraints) over the nonnegative cone, but the function approximation of a nonsmooth high-dimensional random field (Su − ymax(ξ))+ (and derivatives thereof) may be still inefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' For this reason, we replace the ReLU function in the penalty term by a smoothed version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' In this paper, we use the softplus function gε(s) = ε · log(1 + exp(s/ε)) ∈ C∞(R), g0(s) = lim ε→0 gε(x) = (s)+, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='2) although other (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' piecewise polynomial) functions are also possible [24, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Now, the cost function becomes jγ,ε(u) := j(u) + γ 2E ���gε(Su − ymax) ��2 L2(D) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='3) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Discretization and Derivatives of the Cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' In practice, the operator S involves the solution of a differential equation, which needs to be discretized (using e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Finite Element methods and/or time integration schemes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' For a given mesh parameter h > 0, we introduce the discretized (maybe nonlinear) operator Sh(ω) : Uad → Rny, where ny is the total number of degrees of freedom in the discrete solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' We denote the induced Bochner space Yh ∼= L2 h(Ω, D) := L2(Ω, F, P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Rny).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' The L2-norm can be written as an expectation of a vector quadratic form, ∥y∥2 L2 h(Ω,D) = E � y(ω)⊤My(ω) � , ∀y ∈ L2 h(Ω, D), where M = M⊤ > 0 ∈ Rny×ny is a mass matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' The discretized problem cost is denoted by jh(u) ≈ j(u), and the discretized constraint is yh max ∈ Yh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Now, the semi-discretized Moreau-Yosida cost function (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='3) becomes jγ,ε,h(u) := jh(u) + γ 2E � ∥gε(Shu − yh max)∥2 M � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='4) STATE-CONSTRAINED OPTIMIZATION PROBLEMS UNDER UNCERTAINTY 5 To derive a Newton type method, we compute the expressions of gradient and Hessian: ∇ujγ,ε,h = ∇ujh + γE � S∗ h · diag(g′ ε(Shu − yh max)) · Mgε(Shu − yh max) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='5) ∇uujγ,ε,h = ∇2 uujh + γE � S∗ h · diag(g′ ε)Mdiag(g′ ε) · S′ h � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='6) + γE � S∗ h · (tendiag(g′′ ε) ×3 (Mgε)) · S′ h � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='7) + γE � ∇uS∗ h ×3 (diag(g′ ε(Shu − yh max)) · Mgε(Shu − yh max)) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='8) where tendiag(·) is producing a 3-dimensional tensor out of vector by putting the vector elements along the diagonal, and zero elements otherwise, and ×3 is the tensor-vector con- traction product over the 3d mode of the tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' If Sh is a nonlinear operator, S′ h = ∇uSh(u) denotes the gradient of an image of u, and S∗ h is the adjoint of S′ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Matrix-free Fixed Point Gauss-Newton Hessian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' The exact assembly of all terms of the Hessian (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='6)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='8) can be too computationally expensive, since this involves dense tensor-valued random fields (such as ∇uS∗ h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' To simplify the computations, we can firstly omit the terms (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='7) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='8) which contain order-3 tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Secondly, we can replace the exact expectation by a fixed-point evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Rewriting (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='1) using Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='1 we can define J(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' ω) = J(S(ω)u, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' ω) and its discretized version Jh(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' ω) = J(Sh(ω)u, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' The Hessian of jh can then be written as ∇2 uujh = E � ∇2 uuJh(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' ω) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' For practical computations, it is convenient to parametrize all random fields with inde- pendent identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=') random variables with a known probability density function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Those variables can then be sampled independently, and an expectation can be computed simply by quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Therefore, we will use the following assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='1 (finite noise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' There exists a d-dimensional random vector ξ(ω) ∈ Rd with a product probability density function π(ξ) = π(ξ1) · · · π(ξd), such that any random field y ∈ Y can be expressed as a function of ξ, y(ω) = y(ξ(ω)) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=', and E[y] = ˆ Rd y(ξ)π(ξ)dξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' In particular, the vector ξ can often be derived from a parametrization of the forward solution operator Sh(ω) = Sh(ξ(ω)), and/or the constraint yh max(ω) = yh max(ξ(ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Let y = S(ν(ω))u be the resolution of an elliptic PDE −∇(κ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' ν(ω))∇y) = u, where the diffusivity κ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' ν(ω)) = κ0(x) + p � k=1 ψk(x)νk(ω) and the constraint ymax(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' η(ω)) = y0(x) + q � k=1 φk(x)ηk(ω) 6 HARBIR ANTIL, SERGEY DOLGOV, AND AKWUM ONWUNTA are given by Karhunen-Loeve expansions (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=', [27]), where ν and η are independent random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Then, we can define ξ = (ν1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' , νp, η1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' , ηq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Now we can replace ∇2 uujh = E[∇2 uuJh(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' ξ)] by ˜∇2 uujh = ∇2 uuJh(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' E[ξ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' This is exact if ∇2 uuJh is linear in ξ, but we can take it as an approximation in the general case too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Now to apply ˜∇2 uujh to a vector we just need to apply one deterministic ∇2 uuJh(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' E[ξ]), which involves solving one forward, one adjoint, and two linear sensitivity (of state and adjoint) deterministic problems in the most general setting [4, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' 1, Algo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Similarly we approximate the second term in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='6) by γS∗ h(ξ∗)MS′ h(ξ∗), where ξ∗ = E � ξ · 1⊤g′ ε(Shu − yh max(ξ)) � E � 1⊤g′ ε(Shu − yh max(ξ)) � is the mean of the random variable with respect to the probability density πg′ ∝ π · (1⊤g′ ε(Shu − yh max)), and 1 ∈ Rny is the constant vector, averaging the spatial components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Note that 1⊤g′ ε(Shu − yh max) is a nonnegative function bounded by ny, so π1⊤g′ ε(Shu − yh max) is nonnegative and normalizable, and πg′ is indeed a probability density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Finally, we obtain a deterministic approximate Hessian ˜H = ∇2 uuJh(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' E[ξ]) + γS∗ h(ξ∗)MS′ h(ξ∗), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='9) which can be applied to a vector by solving 2 forward, 2 adjoint, and 2 sensitivity problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Probability of the Constraint Violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' In the rest of this section, we prove certain properties about the quality of the solution of the smoothed problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='3) with respect to the constraint, and the exact solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='1)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' This needs a few properties of the softplus smoothing function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' For any ε ≥ 0, the softplus function (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='2) satifies: gε(s) ≥ (s)+ for any s ∈ R, g′ ε(s) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='5 for s ≥ 0, and g′ ε(s) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='5 for s ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Using the monotonicity of the logarithm, gε(s) = ε log � 1 + exp(s/ε) � ≥ � ε log � exp(s/ε) � = s = (s)+, s ≥ 0, 0 = (s)+, s < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' The remaining inequalities follow simply from the monotonicity of the sigmoid function g′ ε(s) = 1/(1 + exp(−s/ε)) and that g′ ε(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' □ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Let uγ,ε be a minimizer of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='3), and assume that j(u) ≥ 0 for any u ∈ Uad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Then for any δ > 0, we have P � ∥(S(ω)uγ,ε − ymax(ω))+∥2 L2(D) > δ � ≤ C1 + C2γε2 γδ , where C1 = 2j(u∗), C2 = log2 2 · ∥1∥2 L2(D), and u∗ is a minimizer of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='1)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' This motivates the condition ε ≲ 1/√γ to overcome the effect of smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' STATE-CONSTRAINED OPTIMIZATION PROBLEMS UNDER UNCERTAINTY 7 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Using Markov’s inequality, we obtain P � ∥(Suγ,ε − ymax(ω))+∥2 L2(D) > δ � ≤ E ���(Suγ,ε − ymax(ω))+ ��2 L2(D) � δ ≤ E ���gε(Suγ,ε − ymax(ω)) ��2 L2(D) � δ , where in the second inequality we used Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Since uγ,ε minimizes (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='3), it holds j(uγ,ε) + γ 2E[∥gε(Suγ,ε − ymax(ω))∥2 L2(D)] ≤ j(u∗) + γ 2E[∥gε(Su∗ − ymax(ω))∥2 L2(D)] for any u∗ ∈ Uad such as the minimizer of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='1) constrained to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Dividing by γ/2 and neglecting j(uγ,ε) ≥ 0, we get E[∥gε(Suγ,ε − ymax(ω))∥2 L2(D)] ≤ C1 γ + E[∥gε(Su∗ − ymax(ω))∥2 L2(D)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' For the latter term, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='3) implies Su∗ − ymax(ω) ≤ 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=', and due to monotonicity of gε, gε(Su∗ − ymax(ω)) ≤ gε(0) = ε · log 2 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Taking this upper bound out of the expectation and norm, we obtain E[∥gε(Suγ,ε−ymax(ω))∥2 L2(D)] ≤ C1 γ +ε2·log2 2·E[∥1∥2 L2(D)] = C1 γ +ε2·log2 2·∥1∥2 L2(D), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='10) and the estimate on probability follows by the Markov’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Strong Convergence with Strongly Convex Cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' To prove the strong conver- gence of the minimizer of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='3) to the minimizer of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='1)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='3) we need further assumptions on the cost and smoothing functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='6 (Bounded derivative of the cost).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' There exists L < ∞ such that ∥j′(u)∥U∗ ≤ L ∀u ∈ Uad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='7 (α-strong convexity of the cost).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' There exists α > 0 such that ⟨j′(u) − j′(v), u − v⟩U∗,U ≥ α∥u − v∥2 U, ∀u, v ∈ Uad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='8 (Smoothing function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' The smoothing function gε possesses the following properties g′ ε(s) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='5, gε(s) ≥ s, for s ≥ 0, g′ ε(s) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='5, for s ≤ 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='11) and either: gε(s)s ≥ −ηmax(ε), for s ≤ 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='12) or, for any random field y(ω) ∈ Y such that y(ω) ≤ 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=', ⟨y, gε(y)⟩Y∗,Y ≥ −ηint(ε), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='13) where ηmax(ε), ηint(ε) ≥ 0, ∀ε > 0, ηmax(ε), ηint(ε) → 0 as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Notice that all the conditions in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='11) are satisfied by the softplus function (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='2) (see Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' We only need to check (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='12) or alternatively (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' 8 HARBIR ANTIL, SERGEY DOLGOV, AND AKWUM ONWUNTA Conjecture 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Our numerical experiments demonstrate that for the softplus function (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='2) it holds ηmax(ε) = O(ε2) and ηint(ε) = O(ε3), although we are only able to prove the latter estimate under specific conditions (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='11 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Now we are able to prove the strong convergence of the smoothed optimal control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Under Assumptions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='6–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='8, linear operator S, and ε = εγ depen- dent on γ in such a way that γ min{ηmax(εγ), ηint(εγ)} → 0, as γ → ∞, and ⟨f, f⟩Y∗,Y = ∥f∥2 L2(Ω,D) for any f ∈ Y, the minimizer uγ of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='3) converges to the solution u∗ of the exact problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='1)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='3), α∥uγ − u∗∥2 U + γ 2∥(Suγ − ymax)+∥2 L2(Ω,D) → 0, γ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' The optimality condition for the smoothed problem, ⟨∇ujγ,ε(uγ), v − uγ⟩U∗,U ≥ 0, ∀v ∈ Uad, can be expanded by introducing an auxiliary variable λγ to match the gradient of the Moreau-Yosida term: ⟨j′(uγ) + S∗λγ, v − uγ⟩U∗,U ≥ 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='14) γg′ ε(Suγ − ymax)gε(Suγ − ymax) = λγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='15) In turn, the KKT conditions for the original problem read ⟨j′(u∗) + S∗λ∗, v − u∗⟩U∗,U ≥ 0 ∀v ∈ Uad (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='16) λ∗ ≥ 0 Su∗ − ymax ≤ 0 ⟨λ∗, Su∗ − ymax⟩Y∗,Y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='17) Adding (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='16) with v = uγ to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='14) with v = u∗, and casting S∗ onto another side of the duality pairing, we get 0 ≥ ⟨j′(uγ) + S∗λγ − j′(u∗) − S∗λ∗, uγ − u∗⟩U∗,U = ⟨j′(uγ) − j′(u∗), uγ − u∗⟩U∗,U + ⟨λγ, Suγ − Su∗⟩Y∗,Y + ⟨j′(u∗), uγ − u∗⟩U∗,U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='18) Due to the strong convexity, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='18), and Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='6 we arrive at α∥uγ − u∗∥2 U + ⟨λγ, Suγ − Su∗⟩Y∗,Y ≤ ⟨j′(u∗), u∗ − uγ⟩U∗,U ≤ ∥j′(u∗)∥U∗∥u∗ − uγ∥U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='19) The second term on the left hand side can be bounded as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Using the fact that ymax − Su∗ ≥ 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' and the definition of λγ, we obtain that ⟨λγ, Suγ − Su∗⟩Y∗,Y = ⟨λγ, (Suγ − ymax) + (ymax − Su∗)⟩Y∗,Y ≥ ⟨λγ, Suγ − ymax⟩Y∗,Y = γ⟨g′ ε(Suγ − ymax)gε(Suγ − ymax), Suγ − ymax⟩Y∗,Y = γ⟨g′ ε(Suγ − ymax)(Suγ − ymax), gε(Suγ − ymax)⟩Y∗,Y = γ⟨g′ ε(Suγ − ymax)(Suγ − ymax)+, gε(Suγ − ymax)⟩Y∗,Y + γ⟨g′ ε(Suγ − ymax)(Suγ − ymax)−, gε(Suγ − ymax)⟩Y∗,Y, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='20) STATE-CONSTRAINED OPTIMIZATION PROBLEMS UNDER UNCERTAINTY 9 where we have split Suγ − ymax into positive and negative parts, with (s)− = min(s, 0) denoting the negative part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Next using Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='8 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='20), we readily obtain that ⟨λγ, Suγ − Su∗⟩Y∗,Y ≥ γ⟨0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='5(Suγ − ymax)+, (Suγ − ymax)+⟩Y∗,Y + γ⟨0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='5(Suγ − ymax)−, gε(Suγ − ymax)⟩Y∗,Y (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='21) ≥ γ � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='5∥(Suγ − ymax)+∥2 L2(Ω,D) − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='5ηint(ε) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='22) Alternatively, we can bound (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='21) using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='12) to arrive at ⟨λγ, Suγ − Su∗⟩Y∗,Y ≥ γ � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='5∥(Suγ − ymax)+∥2 L2(Ω,D) − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='5ηmax(ε)∥1∥2 L2(Ω,D) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' In either case, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='19) implies that uγ is bounded in Uad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Therefore, there exists a weakly converging subsequence uγ ⇀ ˆu in U as γ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Since, Uad is closed convex, therefore ˆu ∈ Uad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' If ε = εγ → 0 as γ → ∞, Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='8 (for both ηmax and ηint) implies that 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='5γ∥(Suγ −ymax)+∥2 L2(Ω,D) is bounded, which means ∥(Suγ −ymax)+∥2 L2(Ω,D) → 0 as γ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Since S is injective and linear, ∥(Suγ − ymax)+∥2 L2(Ω,D) is continuous and convex, hence [38, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='12]: 0 = lim inf γ→∞ ∥(Suγ − ymax)+∥2 L2(Ω,D) ≥ ∥(Sˆu − ymax)+∥2 L2(Ω,D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Since D is a connected domain of positive measure, this yields |(Sˆu − ymax)+| = 0, that is, Sˆu ≤ ymax a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Adding again (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='16) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='14) and using strong convexity of j, but keeping both λγ and λ∗, we get α∥uγ − u∗∥2 U ≤ ⟨λ∗ − λγ, Suγ − Su∗⟩Y∗,Y (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='23) ≤ ⟨λ∗, (Suγ − ymax) + (ymax − Su∗)⟩Y∗,Y (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='24) − γ 2∥(Suγ − ymax)+∥2 L2(Ω,D) + γ 2 min{∥1∥2 L2(Ω,D)ηmax(εγ), ηint(εγ)}, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='25) where we used (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='22) with the negative sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' If γηmax(εγ) → 0 or γηint(εγ) → 0, then 0 ≤ lim γ→∞[α∥uγ − u∗∥2 U] ≤ lim γ→∞⟨λ∗, Suγ − ymax⟩Y∗,Y = ⟨ λ∗ ���� ≥0 , Sˆu − ymax � �� � ≤0 ⟩Y∗,Y ≤ 0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='26) due to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='17), so uγ → u∗, thereby completing the proof of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' For the softplus function (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='2) it holds for any ε ≥ 0: ˆ 0 −∞ sgε(s)ds ≥ −ε3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' The proof uses elementary calculus and is given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' □ In order to search for a rate of convergence, we establish the following result: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Suppose Assumptions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='6–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='8 hold, ˆY is a space of scalar functions, the operator S is linear, and |∂(Su − ymax)/∂ξ1| ≥ c > 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' ∀u ∈ Uad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Suppose that ⟨f, g⟩ ˆY∗, ˆY = ´ D f(x)g(x)dx ∀f, g ∈ ˆY, and maxξ1∈R π(ξ1) = P < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Let ε = ε0/√γ 10 HARBIR ANTIL, SERGEY DOLGOV, AND AKWUM ONWUNTA with any ε0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Then the minimizer uγ of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='3) converges to the solution u∗ of the exact problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='1)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='3), and ∥uγ − u∗∥2 U ≤ Cε3 0γ−1/2 + 1 α⟨λ∗, Suγ − ymax⟩Y∗,Y → 0, γ → ∞, where C > 0 is independent of γ and ε0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' For the classical Moreau-Yosida penalty with ε0 = 0, we recover existing convergence estimates [21, 2] that depend only on ⟨λ∗, Suγ − ymax⟩Y∗,Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' This term converges to 0 as shown in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='26), but the rate of this convergence can be estimated only if bounds on ∥λ∗∥L2(Ω,D) or ∥Suγ −ymax∥Y can be established from other sources, such as the discretization of Y [21, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' We aim at refining the estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Specifically, we need to lower-bound ⟨(Suγ − ymax)−, gε(Suγ − ymax)⟩Y∗,Y, where (y)− = min(y, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' For brevity, let f(x, ξ) = Suγ − ymax(x, ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Using the particular form of duality pairing and Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='1, we can write ⟨(Suγ − ymax)−, gε(Suγ − ymax)⟩Y∗,Y = ˆ Rd ˆ D (f)−gε(f)dxπ(ξ1) · · · π(ξd)dξ = ˆ D ˆ f(x,ξ)≤0 fgε(f)π(ξ1) · · · π(ξd)dξdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='27) Introduce a change of variables � ���� ξ1 ξ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' ξd � ���� → � ���� f(x, ξ) ξ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' ξd � ���� with the Jacobian J := ��������� det � ���� ∂f ∂ξ1 ∂f ∂ξ2 · · ∂f ∂ξd 0 1 · · 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' 0 · · 0 1 � ���� ��������� = ���� ∂f ∂ξ1 ���� ≥ c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Now we can express (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='27) using univariate integration, ⟨(Suγ − ymax)−, gε(Suγ − ymax)⟩Y∗,Y = ˆ D ˆ 0 min f ˆ Rd−1 fgε(f)J−1π(ξ1(f)) · · · π(ξd)dξ2 · · · dξddfdx ≥ ˆ D ˆ 0 −∞ fgε(f)1 cPdfdx ≥ −|D|P 1 cε3, where in the second line we used that the expression under the integral is nonpositive, and ´ π(x2)dx2 = · · · = ´ π(xd)dxd = 1, and in the third line we used Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' STATE-CONSTRAINED OPTIMIZATION PROBLEMS UNDER UNCERTAINTY 11 Now we can replace (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='22) as follows: ⟨λγ, Suγ − Su∗⟩Y∗,Y ≥ γ � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='5∥(Suγ − ymax)+∥2 L2(Ω,D) − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='5|D|P 1 cε3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Proceeding as in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='10, we replace (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='25) by α∥uγ − u∗∥2 U ≤ ⟨λ∗, Suγ − ymax⟩Y∗,Y + γ 2|D|P 1 cε3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Setting ε = ε0/√γ, we obtain that ∥uγ − u∗∥2 U ≤ 1 α⟨λ∗, Suγ − ymax⟩Y∗,Y + |D|P 2cα � �� � C ε3 0 γ1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Thus the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' This theorem can be generalized to vector-valued functions straightforwardly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Indeed, if fi(x, ξ) denotes the ith component of a vector function, the duality pairing (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='27) reads ⟨(f)−, gε(f)⟩Y∗,Y = ˆ Rd ˆ D � i (fi)−gε(fi)dxπ(ξ)dξ = � i ˆ D ˆ fi(x,ξ)≤0 figε(fi)π(ξ)dξdx, and ξ1 can be changed to fi for each term of the sum over i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' The assumption of a lower bound of the Jacobian is practical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' The Karhunen-Loeve expansion as in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='2 is normally derived as the eigenvalue expansion of the covariance function of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' By the Perron-Frobenius theorem, ψ1(x) = ∂κ/∂ξ1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Further, ∂y/∂κ ̸= 0 due to ellipticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Hence ∂(Su)/∂ξ1 ̸= 0 whenever either u or boundary conditions or source term are nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' The remaining assumptions of Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='12 are also reasonable for practical solutions of regularized optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' A convenient observation is that ε = ε0/√γ is the sufficient condition on the law of decay of the smoothing parameter for both Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='4 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Tensor-Train decomposition Throughout this section, we use Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Recall that the bottleneck is the com- putation of the expectation in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' gradient (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' While it may be possible to use a Monte Carlo quadrature, its convergence is usually slow, which may make estimates of small values of the gradient near the optimum particularly inaccurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' In this section, we describe the Tensor-Train (TT) decomposition as a function approximation technique that allows fast computation of the expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' The original TT decomposition [30] was proposed for ten- sors (such as tensors of expansion coefficients), and the functional TT (FTT) decomposition [5, 19] has extended this idea to multivariate functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Let us introduce a basis {ℓi(ξk)} nξ i=1 in each random variable ξk, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' , d, and a quadrature with nodes Z = {zj} and weights {wj} which is exact on this basis, E[ℓi] = nξ � j=1 wjℓi(zj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' 12 HARBIR ANTIL, SERGEY DOLGOV, AND AKWUM ONWUNTA For example, we can take Lagrange interpolation polynomials built upon a Gaussian quadra- ture, or orthogonal polynomials up to degree nξ − 1 together with the roots of the degree-nξ polynomial, or Fourier modes and the rectangular quadrature with the number of nodes corresponding to the highest frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Then we can approximate any random field y ∈ Y in the tensor product basis, y(ξ) ≈ nξ � i1=1 · · nξ � id=1 Yi1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=',idℓi1(ξ1) · · · ℓid(ξd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Note that the expansion coefficients Y form a tensor of nd ξ entries, which is impossible to store directly if d is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' The TT decomposition aims to factorize this tensor further to a product of tensors of manageable size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' A tensor Y ∈ Rnξ×···×nξ is said to be approximated by the TT decomposition with a relative approximation error ϵ if there exist 3-dimensional tensors Y(k) ∈ Rrk−1×nξ×rk, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' , d, such that ˜Yi1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=',id := r0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=',rd � s0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=',sd=1 Y(1) s0,i1,s1Y(2) s1,i2,s2 · · · Y(d) sd−1,id,sd, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='1) and ∥Y− ˜Y∥F = ϵ∥Y∥F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' The factors Y(k) are called TT cores, and the ranges of summation indices r0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' , rd ∈ N are called TT ranks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Note that without loss of generality we can let r0 = rd = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Plugging in the basis and redistributing the summations we obtain the FTT approximation ˜y(ξ) := r0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=',rd � s0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=',sd=1 y(1) s0,s1(ξ1)y(2) s1,s2(ξ2) · · · y(d) sd−1,sd(ξd), where y(k) sk−1,sk(ξk) = nξ � i=1 Y(k) sk−1,i,skℓi(ξk), k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Smooth [35], weakly correlated [33] or certainly structured [20] functions have been shown to induce rapidly converging TT approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Given the TT decomposition, its expectation can be computed by first integrating each TT core, and then multiplying the TT cores one by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Let V(k) sk−1,sk = nξ � j=1 wjy(k) sk−1,sk(zj) = nξ � i,j=1 wjLi,jY(k) sk−1,i,sk, where Li,j = ℓi(zj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='2) Now we multiply the matrices V(k) ∈ Rrk−1×rk in order: E[˜y] = ��� V(1)V(2)� V(3) � · · V(d) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='3) Note that each step in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='3) is a product of 1 × rk−1 vector by rk−1 × rk matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' In turn, the univariate quadrature (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='2) requires n2 ξrk−1rk floating point operations if the Vandermonde matrix L is dense, and nξrk−1rk if it’s sparse, for example, if Lagrange polynomials are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' STATE-CONSTRAINED OPTIMIZATION PROBLEMS UNDER UNCERTAINTY 13 Introducing r := maxk rk, we conclude that the expectation of a TT decomposition can be computed with a complexity O(dr2) which is linear in the dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' To compute a TT approximation, we employ the TT-Cross algorithm [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' We start with an empirical risk minimization problem min Y(1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=',Y(d) N � j=1 � ˜y(ξj) − y(ξj) �2 , where Ξ = {ξj} is a certain set of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' To avoid minimization over all Y(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' , Y(d) simultaneously (which is non-convex), we switch to an alternating direction approach: iterate over k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' , d, solving in each step min Y(k) N � j=1 � ˜y(ξj) − y(ξj) �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='4) This problem can be solved by linear normal equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Indeed, introduce a matrix Y̸=k ∈ RN×(rk−1nξrk) with elements (Y̸=k)j,t = � s0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=',sk−2 y(1) s0,s1(ξj 1) · · · y(k−1) sk−2,sk−1(ξj k−1)ℓi(ξj k) � sk+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=',sd y(k+1) sk,sk+1(ξj k+1) · · · y(d) sd−1,sd(ξj d), where t = (sk−1 − 1)nξrk + (i − 1)rk + sk, and a vector y(k) ∈ Rrk−1nξrk with elements y(k) t = Y(k) sk−1,i,sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' Now ˜y(Ξ) = Y̸=ky(k), and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='4) is minimized by y(k) = (Y⊤ ̸=kY̸=k)−1(Y⊤ ̸=ky(Ξ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9FAT4oBgHgl3EQfxB4e/content/2301.08684v1.pdf'} +page_content='5) To both select “good” sample set Ξ and simplify the assembly of Y̸=k, we restrict the set to have the Cartesian form Ξ = Ξk, where Ξ ( +8 +rect(83, 27, 64, 64); +ellipse(177, 172, 103, 103); +9 +10 +rect(213, 38, 70, 64); +11 +bezier(63, 240, 49, 335, 187， 370, 338, 245); +})open ; +12 +13 +7 +function draw() ( +background"pink"); +reset +Editor.shapeToolbox() open +save +Click the open s +7 +widgetA Study of Editor Features in a Creative Coding Classroom +CHI ’23, April 23–28, 2023, Hamburg, Germany +3 +4 +5 +Canvas Ruler +Time Travel Slider +In-context Docs +p5 State Displays +Interactive Value Inspector +Linked Copy-and-Paste +Code Snippet Templates +Drag-and-Drop Refactoring +Very +useful +Useful +Neither useful +nor unuseful +Hypothetical Features +Year 1 +Year 2 +Survey
 +Responses +Implemented Features +p5/y1 +p5/y2 +Very +useful +Useful +Neither useful +nor unuseful +3 +4 +5 +Coding by +Drawing Tools +Directly +Manipulate +Shapes +These Y1 features
 +were merged in Y2 as +Shape Toolbox +mean +standard +error +† +Linters +Color Picker +Tidy Code +Autocomplete +Shape Toolbox +Auto-refresh +Code Folding +Number Sliders +Number Picker* +† was present in +, but +not asked about in the Y2 survey +* not asked about in the Y1 surveys +p5/y2 +* +Figure 6: The features surveyed across both years and how “Useful” they were deemed to be on a 5-point Likert scale. +Through this process we collected ∼1.2 million events across +∼6730 sessions (defined as before). Due to a configuration error +(present only in wi22), events were not collected with unique ses- +sion identifiers, although we were able to reconstruct 75.4% of the +sessions—the remainder are excluded from analyses requiring spe- +cific session information. While the incomplete data is unfortunate, +it still provides a more detailed picture of activity than in Year 1, +which saw 68% log study participation across sp21 and su21. Within +this reduced sample, sessions lasted 𝜇=24.1±38.0 minutes. +4.2.3 +(Short-Format) Feature Survey. Near the end of wi22, students +were invited to take an abbreviated version of the Year 1 survey +(Sec. 4.1.3), containing only features that were added or improved +upon in p5/y2. Following the structure of the previous survey, we +asked about frequency of use and Usefulness for features one at +a time, followed by a summary table asking about Interest and a +suite of reflection questions. Participants were compensated with +extra credit roughly equivalent to 1% of the final course grade. +A total of 23 students participated in the survey. Our survey +provider did not measure time taken to respond, but based on pi- +loting we believe that the survey took 10-15 minutes to complete. +Presentation order was not correlated with any of our metrics +(𝑝=0.779-0.939). Again, the ratings exhibited reasonable agreement: +𝑟=0.727 (Useful/Often), 𝑟=0.607 (Useful/Interested), 𝑟=0.693 (Of- +ten/Interested) with 𝑝<0.001. As with Year 1, we focus only on +Usefulness in the body of the text (see the appendix for the others). +We found prior experience to be statistically significantly (𝑝<0.01) +correlated with only a single feature, auto-refresh, for which there +was a somewhat negative correlation (𝑟=-0.487). +5 +ANALYSIS +We now reflect on the features, connecting them to the themes +summarized in Sec. 1, denoted T1 through T4. We consider features +implemented in both p5/y1 and p5/y2 (Sec. 5.1), followed by those +added in p5/y2 (Sec. 5.2), and then introduce concerns that cut +across multiple features (Sec. 5.3 and Sec. 5.4). Our analysis draws +on data from the survey studies (summarized in Fig. 6) and the +log studies as appropriate. Participants from the sp21, su21, wi22, +and su22 surveys are referred to as A1-16, B1-9, C1-19, and D1-4, +respectively, and are colored by year. +5.1 +Features in Both p5/y1 and p5/y2 +We begin by considering features present in both editor versions. +5.1.1 +Linting. This static analysis tool eagerly executes after small +code edits, checking simple syntactic assertions akin to spell check +for code. It was well received in both years and was mostly seen as +helpful, although sometimes impolite. +Students found linting to be “very helpful” (A5,7,17,20, C10,16, +D1, 2) and “very useful” (A19, C2,4,6,12,18, D4), because it “saves +time and energy” (A4) and shows “where I needed to go to fix simple +bugs” (A16). C10 believed that debugging “would be way more +annoying without it” because “it’s not always obvious what you did +wrong” (D4). Whereas 86.0% of executions in Year 1 passed lint, in +Year 2 (where we had visibility into all lint runs) code passed 13.5% +of lint runs (which happened after most small text edits). This may +indicate that students address lint errors before running code as a +simple integrity check, or that the analyses are executed too early; +however, student comments seem to indicate the former. Unlike +other features, students were incentivized to attend to it, as the +absence of lint errors was a small part of homework grades (98.7% +and 97.2% of submissions in Years 1 and 2, respectively, passed lint). +Beyond code style, linting can provide opportunities to expose +novices to other best practices. For example, CSSLint [26] (used +in p5/y2) explained that the * selector is considered bad practice +because it is inefficient. Indeed, C3 felt that linting “trained me to +think and type in a certain way”, and A5 observed that it could be +“a nice way to point out when I am making stylistic errors (instead of +[Tidy Code] just magically fixing all of them for me).” Utilizing this +well-received channel for introducing programming features and +practices is an opportunity for future IDE design. T1 +Participants also offered ideas to improve the feature. Because the +editor eagerly ran the linter, “the yellow line warning[s] often exist +all the time. It annoys me” (B4). Instead, some students would have +preferred not to see lint errors “until I finish typing” (A13) or “before +finishing a line of code” (B5)—the mechanics of exactly when and +how to display errors for incomplete code will require careful design +(as considered, e.g. in Hazel [79, 80]). Others expressed a desire for +more nuance—“acknowledging the difference between ‘This Must Be +Changed To Have Nice Code™’ and ‘hey, maybe consider changing +this thing!”’ (A5)—and control—being able to “ignore/exit out of +a warning” (A3). Poorly-received default choices and persistent +errors can repel users. As an extreme example, one wi22 student +decided to use Replit [89], rather than p5/y2, for their final project +because too many (CSSLint) errors seemed irrelevant or unclear +how to fix. Linters integrated with editors in this way do not offer +mechanisms to override general advice or to indicate that the user +knows what they are doing. This is impolite computing [104]: it + +CHI ’23, April 23–28, 2023, Hamburg, Germany +McNutt et al. +forgoes user agency and generally is perceived as a pest. Avoiding +these pitfalls is important to leverage the instructive opportunities +offered by the well-received, static analysis-informed tools. T1 +5.1.2 +Tidy Code. Auto-formatters provide on-demand code restyling +without semantic modification, and are common in professional cod- +ing workflows [86]. We often encouraged the use of this tool—called +Tidy Code in the p5 editor—in lectures, but we did not incentivize +its usage in grading. It was invoked manually (from the top menu +bar or keyboard shortcut) rather than being executed on every save. +Like linting, this feature was generally well received. Students +found auto-formatting to be “super useful” (A15) and “very satis- +fying” (A2). The formatting choices were not always appreciated, +however. Whereas C15 “only rarely preferred my own organization”, +A12 felt the results “appeared less organized, such as having irregu- +lar line breaks” and A10 “worried it would mess up my organization.” +We observed that students in Year 1 often (needlessly) invoked +auto-formatting twice in a row. In particular there was a probability +of 16.15% and 8.65% (in sp21 and su21) of auto-formatted code being +auto-formatted again right away—with similar behavior observed +for saves (see appendix for details). This suggests that providing +clear code-state signals (analogous to linting’s visual indicators) +may reduce needless anxiety-motivated saves and tidyings. The +presence of this behavior in Year 1 suggests it was likely repeated in +Year 2; however, the aforementioned configuration error prevented +us from collecting auto-formatting usage. While simple indicators +may seem to be trivial UI modifications, we suggest that it will +impact the perception and understanding of such features. +Several students would have liked the feature to be customizable, +rather than enforcing a fixed set of “preferences that should not be +forced by tidy code” (C16). Indeed, some students would have liked +auto-formatting better “if it was a little configurable” (A16); for +example, “if there [were] multiple common/standard rulesets there +could be a way to choose which you want to follow” (A5). Further- +more, it “would be helpful to be able to specif[y] which block of code +to tidy” (A13). Thus, extending well-chosen defaults with ways to +selectively customize style preferences—a notion which has been +referred to as “code style sheets” [69]—could further increase the +politeness of this feature and thus its utility. T1 +Like the teachable moments offered by linting, B8 felt they +“Learnt a lot about code organization using this feature!” As imple- +mented, however, the results of auto-formatting are updated in the +code box without explanation. Better would be for the editor to +“show you what you are doing ‘incorrectly”’ (C19), for example, using +visual highlights and annotations to explain the differences—which +could also serve as scaffolding to introduce version control tools. +5.1.3 +Auto-refresh. This feature re-executes code upon text edits— +a workflow demonstrating “level-3 liveness” [96, 97]. Auto-refresh +was present in both p5/y1 (inherited from the original editor) and +in p5/y2 (where it was modified). In principle, live feedback would +seem particularly helpful in a creative coding context as programs +are often updated with small graphical adjustments, and thus well +matched with a short edit-run cycle. It was also well matched with +our setting: the Normalized Programming State model [23] sug- +gests that spending longer periods of time in syntactically unknown +states (such as when the code has not been executed in a while) is +Year 1 +Year 2 +Fraction of sessions using auto-refresh by students +0% +10% +20% +30% +40% +50% +60% +70% +0 +10 +20 +2 +28 +1 +2 +2 +3 +1 +80% +90% +0 +10 +20 +4 +7 +6 +5 +21 +4 +1 +4 +2 +2 students used auto-refresh +in 40-50% of their sessions +Figure 7: Histograms of the fraction of sessions where a stu- +dent used auto-refresh any amount. 58.7% and 5.1% of stu- +dents never used auto-refresh in Years 1 and 2 respectively. +negatively correlated with program success. This needs to be bal- +anced with the cognitive load [54] caused by repeated executions. +Auto-refresh in p5/y1 did not achieve a fruitful balance. As +indicated in Fig. 7, only a handful of participants regularly used +it and most students used it rarely, if at all. The survey responses +color this imbalance. Whereas A9 “used this all the time and loved +it”, finding it “way easier than clicking the ‘play’ button all the time”, +others felt that the keyboard hotkey was sufficient (A16, B1,8, D1). +More important than convenience were differing views on the +fundamental interaction model itself. B7 appreciated the ability “to +see what I was creating as I coded”, finding it useful even though +“error messages that kept popping up got in the way a little”, while +others found the errors “very distracting” (A5,6). Participants felt +the feature was “running incomplete code unintentionally” (B6) and +“when you don’t want it to” (B1). Instead, some students felt robbed +of their agency over their code, desiring “to be the boss of when my +code reran” (A5) and in “control my own pace” (B2), only running +the code when “I know I have something that I want to see” (A13). +This suggests that, while spending too much time in syntactically +invalid states may be detrimental [23], spending too little time is also +problematic. Developing a careful understanding of the tradeoffs is +an important avenue for future live programming work. T2 +For the purposes of this work, we made only simple changes +to auto-refresh in p5/y2 based on our observations from the first +year. We increased the refresh delay from 400ms to 1s, and, more +importantly, in the event that executed code had lint errors—a proxy +for run-time errors—the editor did not refresh the canvas, instead +indicating that it was “stale.” Thus, in the (many) cases when edits +are incomplete or erroneous, the canvas remains visually stable. +The modified auto-refresh was modestly better received, with its +Usefulness increasing from 𝜇=3.1 to 𝜇=3.7. In addition, per Fig. 7, +it was used more often—although we note that auto-refresh was +demonstrated more at the beginning of wi22 and su22 than in prior +editions. A one-sided t-test indicates that students in Year 2 used +auto-refresh significantly (𝑝<0.001) more often. Yet, the overall bal- +ance remained far from perfect. Some participants were “stressed” +at “all the errors that pop up as I implement new things” (C15) and +“before I got to fix them” (C4). These negative views seemed more +likely to come from those with prior experience (𝑟=-0.487, 𝑝<0.001), +which may suggest that expectations are set by experience with + +A Study of Editor Features in a Creative Coding Classroom +CHI ’23, April 23–28, 2023, Hamburg, Germany +tools exhibiting a different execution cadence. Others, however, +found it “very useful for certain exercises that needed lots of small ad- +justments” (C3) and “very helpful when using trial and error” (C16). +Overall, we observed no significant changes in user behavior after +modifying auto-refresh, despite the improved perception of the +feature. This again underscores that designing UIs to be polite (or +at least not irritating) is critical to their usage. +5.2 +Features Only in p5/y2 +Next, we consider the features that were added in p5/y2. While +Year 2 survey responses are based on hands-on experience with +the features in p5/y2, Year 1 responses are based on descriptions +in the survey and experience with other tools. Feature use in wi22 +is shown in Fig. 8. +5.2.1 +Shape Toolbox. The most significant addition to p5/y2 was +the Shape Toolbox feature that allowed GUI-based specification +of primitive shapes using direct manipulation which generated +matching code (Fig. 5). The constituent parts of this feature were +highly perceived in Year 1: 𝜇=4.8±0.44 for Coding by Drawing +Tools, and 𝜇=4.2±1.0 for Directly Manipulate Shapes. Some students +believed it would be “very beginner friendly” (A3) and would make +work “a lot easier and faster” (B7). Others believed it would also +reduce errors (A9) and help with debugging (B6). +Help programming curved shapes—such as the trees in Fig. 4— +was particularly enticing: “for bezier curves, changing the input +values rarely produced an expected result” (A12), highlighting a +gulf of execution [78]. The process usually involved “lots of trial +and error” (A3), sometimes resulting in student disengagement: +“Coding the bezier curves manually turned me off of them, and I did +not attempt them in my work” (A14). However, that same student +noted “If I had had a tool like this, I certainly would have used them.” +Several students in Year 2 embraced the feature. For example, +C18 found it “EXTREMELY helpful, especially when it came to draw- +ing Bezier curves. Every time I had to draw a curve, I used the shape +toolbox. I probably would have cried without it.” C10 mentioned that +it “Was very nice to use it to get approximate coordinates then fine +tune them after.” +Although the feature was “very useful for beginner projects” (C2), +several students, including C6, “used them less as time progressed.” +Shape Toolbox was used often for the tree homework (see HW3 +in Fig. 8), and use per execution by week was minimal after that +assignment, being used in only 2.12% of all (available) sessions. +Perhaps because the feature did not have a stable visual presence +(as with the auto-refresh button), some students “completely forgot +this existed, but I think it would have been really really useful if I had +remembered” (C4). In addition, although we expected the feature to +be used extensively for HW 2, in wi22 Editor.shapeToolbox was +announced but not demonstrated in class until after the assignment +was released. Bezier curves accounted for the majority of invoca- +tions (see Fig. 5.5). Toolbox sessions (from open to save) lasted +𝜇=22±30 seconds, indicating that it may have been used relatively +often to make small graphical adjustments, as opposed to building +larger compositions. +Students may have continued to use them later in the course “if it +allowed for some of the shapes that are more complicated” (C16). Fur- +ther limiting the utility of the feature, within an invocation shape +Jan 09, 2022 +Jan 17 +Jan 25 +Feb 01 +Feb 09 +Feb 17 +Feb 25 +Mar 05 +Mar 13 +0 +2k +4k +6k +8k +10k +12k +14k +Executions Per Day +HW 1: Color Wheel +HW 2: Freeze Frame +HW 3: Trees +HW 4: Book of Patterns +HW 5: Deck of Cards +HW 6: Snake +HW 7: Wordle +Project: Proposal +HW 8: Blackout Poetry +Project: Progress Report +Project: Final +0 +0.2k +0.4k +0.6k +0.8k +1k +1.2k +Feature Use Per Day +HW 1: Color Wheel +HW 2: Freeze Frame +HW 3: Trees +HW 4: Book of Patterns +HW 5: Deck of Cards +HW 6: Snake +HW 7: Wordle +Project: Proposal +HW 8: Blackout Poetry +Project: Progress Report +Project: Final +Auto +Autocomplete +Color Picker +Manual +Number Widgets +Shape Toolbox +Slider +Figure 8: Feature use in wi22 was guided by course content. +For instance, autocomplete was demonstrated prior to HW3 +and sliders were included in the starter code for HW5. +drawing functions allowed only literals—once a student wanted +to use variables and arithmetic expressions, the Toolbox would no +longer open. “[C]reating an object without this feature would be bet- +ter because of the precision” (B5) afforded by variables, expressions, +and so on. Thus, the feature ultimately fell short of what students +imagined: 𝜇=3.8±1.2. +Bidirectional updates are being explored in a growing number of +systems (as in Sketch-n-Sketch [49]), but there remain significant +technical and UI design challenges to explore, before even consider- +ing their value to novices. As predicted by a couple students, more +feature-rich bidirectional synchronization would need to reconcile +ambiguous graphical interactions (“There are many parameters and +it would be hard to make it so it manipulates them one at a time” (B5)) +and their effect on other parts of the program (“My only but major +concern would be that it doesn’t confuse the other lines of code, and +that it may not run the way the programmer wants to use it” (B6)). +Nevertheless, the experience suggests that even a simple imple- +mentation of this very desirable feature was promising. However, +as we discuss in Sec. 5.3, many students were skeptical about the +effect of this feature on learning. +T4 +5.2.2 +Autocomplete. We enabled a simple autocomplete menu [45] +and populated it with p5-specific identifiers (variables and function +names), syntax templates (common patterns, like for-loops with +holes), and commands for invoking the Shape Toolbox and Number +Sliders. Passively supporting learning in this way would seem to +be a natural fit for our setting, but some students were leery of it. +Year 1 survey respondents anticipated autocomplete positively +(𝜇=4.6±0.5), believing it would help in several ways. For example, +to “increase speed and productivity when coding” (B9) and “make it +faster to get debugging done” (B6). In addition, A13 believed auto- +complete would encourage better code style: “not having dynamic +autocomplete incentivizes me to write non-descriptive function names +and variables for the sake of efficiency.” Participants also believed +autocomplete would help “discover new features” (B5), “expose us +to new things we didn’t know existed” (B7), and provide “an idea +of what to write or what could be written” (B4). These beliefs are +in line with how professional programmers use autocomplete to +debug and explore APIs [71]. + +CHI ’23, April 23–28, 2023, Hamburg, Germany +McNutt et al. +However, the experienced reality of p5/y2 fell short (𝜇=3.7±1.0) +of anticipation. Autocomplete was used in only 12% of sessions (with +33.6% selections being templates), although it was used progres- +sively less as wi22 and su22 went on. While this relative infrequency +of use may be related to the simple implementation (which did not +include embedded documentation or other common guidance fea- +tures) or the emphasis later in the course on web programming (the +DOM was not thoroughly reflected in the autocomplete sugges- +tions), this trend appears to agree with how Vihavainen et al. [101] +observed novice usage of autocomplete. They note that 27.3% of +novices initially used autocomplete to create a particular command +(Java’s system print), which decreased to 1.64% after a week of use. +This appears to suggest that autocomplete can serve as a vehicle +for teaching: it is “a useful guide until I was able to type certain +things in by memory” (C13). Some perceived the ability to “stop +memorizing certain code” (A9) as a benefit, while others thought +“it’s a give and take” (C7) and might hinder “programmers’ knowledge +about commands and their forms in the long run” (B8). We return to +this concern about the effect on learning in Sec. 5.3. T4 Beyond these +hesitancies, it is unclear why more students did not engage with +the feature, although some noted that it can be “annoying when +you already know what you want” (C15)—which suggests that the +clutter T3 or cognitive noise T2 may be a factor. Given this diversity +of opinion, we suggest that configurability is important to designing +such features politely, as some students (such as D4) wanted to be +able to turn off autocomplete (to limit its disturbances). +5.2.3 +Color Pickers. Integrating a color picker into an editor for +creative coding was perceived as very useful in the Year 1 surveys +(𝜇=4.7±0.56). Whereas A4, probably like many students, did not +pick colors as much in the second half of the course, A6 said “I had +a color picker tab open for every single assignment.” Based on this +enthusiasm, we implemented a modal color picker dialog box in +p5/y2 (Fig. 1), following a sentiment from A13 that such a design +“would probably be more helpful than in the code to prevent clutter.” +T3 +Note that a similar color picker was recently added to the p5 editor, +highlighting the value of this feature. However, ours supports more +color formats, namely, web color names—driven by a participant +suggestion (B9)—and RGB values as numeric lists. +After experiencing color pickers, students viewed them posi- +tively (𝜇=4.4±0.65), with the wi22 students using them frequently +in the first half of the course, as depicted in Fig. 8. They helped +make the process “more efficient” (A9), “[taking] out the hassle” (C6) +of “open[ing] up another program” (A2). Color pickers may foster +creativity, as they could “let me pick some irregular colors” (B2). +Several participants also voiced support for the idea, suggested +in the survey prompt, for an eyedropper tool. Others suggested +additional features inspired by drawing programs like Illustrator, +such as grid (D1), zoom (C15), “better proportions” (A3), or a way +to “group lines and shapes and move them all at once” (B7). Such +lightweight and familiar tools from creativity domains are natural +enhancements—as long as they are not impolitely imposed T1—that +we intend to investigate in the future. +5.2.4 +Number Pickers and Sliders. “Scrubbers” [100], which allow +direct manipulation of numeric values by dragging, are often touted +in live programming systems and interactive documents [99] as +being representative of the value of those environments. Despite +the overlap between live programming’s close connection to the +visual domain and the interests of creative coding, the clutter T3 and +lack of control T2 brought on by these features impeded adoption. +Some Year 1 students were positive about hover-based Number +Sliders, believing they would allow them to “experiment with the +code more quickly” (B6) and “more efficiently” (B9). However, some +worried that “it could make the editor look more crowded” (A3), while +A5 noted “I would rather just do it myself.” +Nevertheless, we added Number Sliders to p5/y2, which appear +(per Sec. 4.2.1) inline via Editor.slider(min, max, value), as +well as Number Pickers (Fig. 1), which are buttons surrounding each +number literal that allow it to be incremented and decremented +( +). (Such small modifications explain the large absolute number +of Number Picker events in Fig. 8.) Students found these additions +could be a “quick, helpful way to make sure my assignments didn’t +break at a larger scale” (C6), as was the case for HW5 (cf. Fig. 8). Al- +though scrubbers were perhaps most useful toward the latter stages +of a task, “when I’m playing around with my final result” (C17), C13 +felt they “allowed me to tap into my creativity.” +Yet, per Fig. 6, the feature was not so highly rated. A recurring +theme is that scrubbers—in various configurations—felt “messy” (B4, +C9), “disrupted the look of the code” (C4) +T3 or were just generally +unnecessary (B10). A14 felt that the transitory changes would be +confusing, and hard to maintain a model of different parameter +configurations. T2 Others wanted more refinement in the numeric +type, such as limiting it to numbers “divisible by five” (A5). While +these features are typically well-used in graphical applications like +Figma, it seems that this type of feature is “trying to solve or better a +process that needs no help” (B10). While there is evident overlap be- +tween our domain with other artistic settings, not every translated +feature will match the interests of learners. +5.3 +On Skepticism +Next, we grapple with the perception that some tools take away +learning opportunities that may be needed to “become a good pro- +grammer.” +T4 Several features were perceived as making things too +easy for novices. Fig. 9 summarizes how “skeptics” worried about +different features. These perceptions are valuable: as the technol- +ogy acceptance model [27] and related theories highlight, perceived +usefulness is a central part of whether a system is ultimately used. +Several students worried how syntax templates (see appendix) +and autocomplete balanced the tradeoff between augmenting their +abilities and enfeebling their development of skills. Whereas A5 +was “not sure if it actually matters” to practice memorizing names +and function signatures, C17 weighed the tradeoff according to +the goals of the student: “I wouldn’t consider it a horrible thing for +those who don’t want to go into coding professionally/too much”— +implying that a more serious programmer might indeed miss out +on practicing an important skill. A16 thought such features “might +reduce some of the learning by doing that you get when coding, so I’m +not sure if it’s great for a class. I learn through my coding mistakes +and this would reduce the number of mistakes, so a mixed bag.” +There were similar concerns about in-editor documentation (also +discussed in the appendix). For example, A15 said that “new coders +need to learn the process of going into the manual.” A4 reconciled +the aforementioned tradeoff as follows: “However, depending on + +600 ++A Study of Editor Features in a Creative Coding Classroom +CHI ’23, April 23–28, 2023, Hamburg, Germany +No +Some +Prior Experience +Yes +C1 +C14 +Linting +A5 +Canvas +Ruler +B8 +Number +Sliders +A4 +p5 State +Displays +Tidy +Code +C1 +C14 +C17 +B8 +Auto- +complete +C1 +C7 +C17 +A2 +Shape +Toolbox +A5 +A6 +A15 +A13 +A16 +B8 +Code +Snippets +A7 +A15 +A4 +A16 +In-Context +Docs +A4 +B5 +Year 1 +Year 2 +Survey Responses: +Total skeptics in all surveys: 14/48 +Skepticism by Feature +Participant + in +wi22 worried that +linting would prevent +them from learning +C14 +Figure 9: Some skeptical survey respondents worried that +some features would deleteriously affect learning. +the specific goal of this course, if it is to focus more on the creative +coding aspect and not necessarily ‘become a good programmer’ then +in-context docs would be awesssommeee.” +Some of these concerns might be ameliorated by introducing a +notion of documentation or autocomplete levels (in a similar style +as DrRacket’s language levels [73]), which gradually adjust what +information is available as new concepts are introduced. However, +without sufficient signaling, students might construct mental mod- +els of the information present in the feature and then dismiss all +subsequent configurations. +Conflicting views over the Shape Toolbox in Year 1 were most +striking. “This feature would be so useful and allow for more creative +opportunities especially for beginner coders” (A10). It would also +be a “useful learning tool” (A3) by allowing students to “see how +the code changes in order to learn how certain parts of the code are +working” (B9). But many students were skeptical. This “feels like +cheating!” (A6) and “saves way too much work for the new learn- +ers” (A15). “It’s way too useful but can hinder with the learning +process of basics of coding. As a student, I won’t want this but as a +programmer who knows the basics, it’s a nice feature” (B8). “I think +some of these features while helpful would have discouraged learning. +Some of the most rewarding parts was sweating through inconve- +nient parts” (A2). As shown in Fig. 9, some of this hesitation was +self-censorship by students with little or no prior experience. +However, among Year 2 participants (all of whom had access to +the feature), there were no skeptics of the feature. Perhaps the idea +of others having improved tools is jarring, while students who are +given improved tools simply worry about the plenty of challenging +learning left to do. We view this situation as akin to giving students +calculators in a math class: they help with specific classes of tasks +that, once simplified, enable learning about richer topics. +Students seem to construct a naive model of what makes a good +programmer, suggested above as being someone who has memo- +rized the entire language and does not depend on digital assistants +or developer-experience tools, thereby dismissing behaviors besides +this as being inauthentic. We suggest that reorganizing and reform- +ing this model is part of the value that classroom-based computing +education offers, as it can help to offer a thicker model of what is +authentic [91]. Enhancements to novice-oriented IDEs such may +also help to dispel these notions if they are perceived as realistic +tools rather than as something akin to training wheels. +5.4 +On Creativity +Finally, we consider the role of creativity in our editor. While +there exists little agreement on what creativity means in HCI re- +search [35], we found that students espoused two clear views on +how tools might help them creatively: automating tasks that impede +of creativity and helping explore unknown functionality. +For students, creativity often appeared to be something which +typical coding tasks stood in the way of; obstacles that some techni- +cal interventions could ameliorate. Shape Toolbox was emblematic +of this style of reduction. For instance, A10 believed that such a tool +would “allow for more creative opportunities especially for beginner +coders,” and B9 believed that it “could help with planning ideas for +art projects and increase creativity.” As noted in Sec. 5.2.1 students +in p5/y2 embraced this feature and appeared to use it to reduce the +tedium required to precisely locate shapes, thereby making greater +room for artistic expression. Others highlighted the value that tools +that reduced tedious tasks, such as picking individual coordinates +through a ruler (e.g. A3 and A9) or identifying which lines corre- +sponded to which components of the image (B9). Summarizing this +view, C16 observed that “I think what this editor did well as an art +tool was streamlining certain common processes.” +Beyond reducing tedium were opportunities for exploration, +which were manifested both as moments of play (A8) or fun (A4, +C6), as well as discovering new functionality. For instance A12 +believed that “comprehensive documentation would have allowed me +to be both more creative,” a view which was confirmed by C17, who +believed that such features help do “things I don’t yet know how to do +by myself” and thereby “help me be more creative.” A9 believed that +surfacing program state might encourage reflection and discovery, +such as by seeing that “circle has a round stroke cap, so it might +make me wonder what other shapes the stroke cap could have.” A14 +believed that an assistant that made artistic suggestions might be +well received, “[f]or instance, if I’m editing text, and I was given +suggestions for font, color, etc.” Similarly, A4 noted that it would be +useful receive suggestions to help inspire their designs, for example, +through “videos on youtube, images, articles.” Some features, such as +number sliders, were highlighted as being only valuable “when I’m +playing around with my final result” (C17), but that they “encourage +a lot of experimentation and creativity” (A4). These observations +align with prior work, which highlighted the value of providing +assistance in exploring the space of possible designs in creative +coding contexts [77] and in creativity support tools generally [93]. +Whether a student’s primary purpose was closer to coding or +to making art was an additional source of skepticism to those in +Sec. 5.3. Again, regarding Shape Toolbox: “This would be great, but +would reduce the amount of time figuring out the code. This would +make it more an explicit art tool, and less a ‘make art with code’ +tool” (A16). A5 similarly noted that “feels a little too much like +draw-ing for my taste” and took the class “with the primary goal +of getting better at programming so I’d want to do things the code-y +way”. Similarly, C17 felt that it “hinder[ed] the process of creative +discovery– including trial and error”, however this was mostly not an +issue as they sought to be “to be more accurate than creative” in this +course. While it is natural to want tools to be familiar, we believe +that new authoring paradigms (e.g. bidirectional programming) +should be viewed as complementary rather than antagonistic. + +CHI ’23, April 23–28, 2023, Hamburg, Germany +McNutt et al. +6 +DISCUSSION +This paper explored the observed behaviors and surveyed percep- +tions of novice programmers in a creative coding course. To wrap +up, we recap the main themes, reflect on the connection between +our work and other domains, describe limitations of our studies, +and offer avenues for future work. +6.1 +Recap: Themes +In our analysis, we chose four recurring themes to highlight. +T1 Static Analyses. We observed that simple static analyses were +seen as supportive of a variety of types of work—notable given +that error messages sometimes are obstacles in introductory set- +tings [16]. Polite lightweight assistants that respect user agency, like +those expressed through linting or auto-formatting, can be a helpful +platform on which to learn and test new skills with confidence. On +the other hand, A5 noted that they “would also probably prefer to +do things by hand” rather than use advanced features because there +lacked visual indicators of a particular action’s effect—highlighting +the importance of clear effect-forecasting for feature trust. +T2 Liveness. We saw that overeager evaluation can overwhelm +and stress users through distracting updates that are unsynchro- +nized with their expected edit-run cadence. Live programming of- +fers enticing benefits for novice and creative contexts (e.g. feedback +immediacy or a closeness of mapping between code and graphics). +Yet, these interaction challenges for non-expert settings are not +yet thoroughly understood, leaving open questions about how to +blend user control with system eagerness in a profitable way that +maintains an experience level-attuned sense of agency. +T3 Clutter. We noted that amateurs are mindful of how the edit- +ing space can become overwhelming if too much visual noise or +unfamiliar forms of interaction are introduced. For instance, stu- +dents are aware that individual features (e.g. Number Scrubbers, lint +errors, and autocomplete menus) can break their flow. We highlight +the difficulty and importance of developing design guidelines that +can aid the development of novel features within these constraints. +T4 Skepticism. Finally, we discussed how user perceptions of a +feature can inspire skepticism about its propriety in learning envi- +ronments. Year 1 students believed that the Shape Toolbox would +impede learning; however, those who used it in Year 2 did not +share that concern, instead viewing it as a convenience. Year 2 +students also saw knowledge assistants such as autocomplete as +detrimental to their development as programmers. We believe it is +valuable future work to better understand what types of features +and knowledge assistants are likely to be viewed as detrimental. +6.2 +Connections to Other Domains +Next we reflect on how our findings may apply more broadly. +6.2.1 +Programming Pedagogy. Our work is merely situated within +a classroom; we do not seek to make claims about the learning +effects of the features we studied—this is an important, separable +direction for future work. Yet, some of our themes may carry over +to pedagogically-minded editors in more general learning contexts. +We suggest that skepticism T4 about features perceived as being +too useful, such as autocomplete, may continue to be prevalent +in learning contexts. Such concerns might be circumvented by +emphasizing tools that help correct, rather than help complete, +such as how linting T1 can identify an error while also providing +justification and explanation for that error. We also note the value +of having a programming environment that is perceived as being +approachable (A3,5, B1). Furthermore, tools having not “got in +the way” (C16) or otherwise cluttering T3 the display in unhelpful +ways seem intuitively valuable, particularly in learning contexts. +Similarly, live execution T2 may be beneficial in non-visual contexts +as it promotes immediate feedback, such as by rerunning a test +suite dynamically, as in Jest’s watch mode [32] or Huang et al.’s use +of projection boxes [52] in a classroom to expose live values. +Finally, like others before us [36, 70, 105], we found that a cur- +riculum centered around media-art topics—as opposed to more +abstract content often found in intro CS courses—invited a broad +range of students who might not otherwise study CS in a formal +setting (the appendix lists majors represented in the courses). +6.2.2 +Other Domains. Editors specialized to a given domain can +make adaptations that aid that context. In this work we focused on +creative coding and designed affordances specific to this domain, +however our findings might be applied in related contexts. We +highlight the value of bidirectional editing, linting, and designing +editors with their effects on creativity in mind. +Bidirectional synchronization of code and effect (such as in our +Shape Toolbox) seems to be an especially valuable approach in +domains that have a prominent visual component. This has been +explored by Asai et al. [10] as a mechanism to clean and synthesize +data for statistical modeling, as well by DeLine [28] and Wu et al. +[107] for data science tasks such as modeling and analysis. We +suggest such synchronization might be usefully applied to other +visualization contexts (like preparing charts for presentation), as +well as other creative coding contexts. Such interfaces may poten- +tially reduce tedium in certain tasks and, more fundamentally, may +provide opportunities for learning about the domain, for example, +demonstrating how to achieve a particular effect using code. +Next, we highlight that linters (or other static analysis tools T1) +can provide a straightforward channel for introducing newcom- +ers to basic principles and best practices of a particular domain. +While they have already been explored in some contexts—such as +for spreadsheets [13] and visualizations [51, 74]—additional fields +such as data science [75] and music editors might integrate these +concepts as well in order to surface best practices, such as high- +lighting statistical fallacies, helping guide usage with unusual tools +(e.g. Orca [55]), or surfacing accidental discordance or inaudible +components in music editors. As discussed, such ambient assis- +tants should be designed in a polite manner (e.g., through granular +dismissal of advice) to avoid being irritating and then dismissed. +While most technical tasks require some amount of creativity, +we argue that features in editors in creativity centered-domains +should be constructed in order to align specifically with goals of +either reducing tedium or aiding in exploration. Barke et al. [12] +observe a similar pattern of exploration vs. acceleration in use of the +AI-powered code assistant Copilot for traditional, non-creative soft- +ware development tasks, suggesting overlap in editor features that + +A Study of Editor Features in a Creative Coding Classroom +CHI ’23, April 23–28, 2023, Hamburg, Germany +support creative coding and coding more generally. Compton [24] +argues for IDEs with features that are valuable unto themselves— +for example, for being playful or thought-provoking—rather than +their use as a means to end. Non-productivity focused techniques +may be useful in creative coding contexts more generally, perhaps +as a design advisor as A4 suggested. These additions may drive +unexpected patterns of usage, leading to new types of discovery +through play—which might even valuable in technical domains like +data science or visualization [102]. At the same time, such inter- +ventions may inspire skepticism +T4 about their authenticity if they +are perceived as too whimsical or unrealistic. +6.3 +Limitations and Future Work +As described throughout, our study had a number of limitations. +These included data collection errors (such as the configuration +error in wi22) and the relative simplicity of the survey. For in- +stance, our use of static images—as opposed to videos or interactive +prototypes—limited our ability to accurately explore reactions to +proposed features. However, the use of non-interactive stimuli (fol- +lowing Kery et al. [60]) allowed respondents to project their own +beliefs about the features and ignore potentially distracting low- +level bugs or stylistic issues. Further, we only implemented a subset +all designs we identified, so we cannot make inferences about what +features would be most valuable in general. Instead, we focus only +on the observed themes and interactions with implemented fea- +tures. This approach was a coarse and inexpensive way to identify +and explore some potentially fruitful features, however not all such +features were necessarily identified nor considered. Future work +could implement more of the identified features—and also augment +our observations with lab studies—to better understand the effects +of particular features. Furthermore, whereas our work investigated +how novices perceive the utility of various editor features, subse- +quent work should also investigate their pedagogical effects on +learning outcomes—one notable point for comparison is that Oviatt +et al. [82] found novel interfaces can hinder learning. +The biases of our particular student populations may not be re- +flective of a more general student population, however the views of +the college-aged (sp21, wi22) students seem aligned with those of the +high-school students (su22). In addition, they are in agreement with +those of su21 students who, because of pandemic era-distancing, at- +tended from around the world and thus drawn from a substantially +different population. Our own biases were likely projected onto the +students in teaching this material, and different instructors may +have inspired different responses in students. To this end, student +perceptions are likely reflective of the context and content of the +work they were asked to do. For instance, the open-ended nature of +many assignments likely shaped student opinions of the features we +asked about, which may have been different under more structured +programming tasks. In future work we would like to reexamine +our findings by teaching the course to and soliciting feedback from +students from other institutions, age groups, and backgrounds. +Students were generally positive about the editor being online +and the way in which our feedback and submission systems were in- +tegrated (A3,5), with B1 noting that they were especially beginner- +friendly. Nevertheless, the choice to use a web tool had limitations. +Students with inconsistent internet connections struggled with the +online environment (prompting B6 to suggest an offline mode), +while others had computers that were unable to handle the com- +putational weight of a larger web application (which made some +students hesitant to explore some editor features). For instance, A2 +noted that they hesitated to use auto-refresh because “my computer +was already very slow and I didn’t want my code to crash while it +was running.” These concerns were particularly prominent during +the fully online sp21 and su21 editions. While in-person teaching +has resumed (as in wi22 and su22), that consideration of how to +build novice-oriented tools that support those with limited internet +connectivity or less powerful computers should not cease. +While our target population in this work was students, in future +work we wish to understand what features instructors see as valu- +able or concerning in such a setting. Similarly, it would be useful +to consider whether these user interface patterns are applicable to +professional artists working in creative coding spaces—questions +which are closely connected to Li et al.’s [67] study of the tools +that artists make for themselves. Of particular relevance are artist- +designed custom coding environments used for teaching and artistic +practice (such as Field [30]). +In sum, creative coding has been, and continues to be, fertile soil +for HCI research. We believe that studying the problems users in +these creative domains face is valuable unto itself, and is ever more +relevant as creative coding becomes an increasingly common way +to introduce computing and to make art. +ACKNOWLEDGMENTS +We are grateful to those who made our courses possible, includ- +ing the course staff (Brian Hempel, Angela Liu, and Bhakti Shah), +Kevin Workman for allowing us to incorporate his Happy Coding +tutorials, and the p5 community and developers for building such +useful tools. We thank Lilian Huang, Shriram Krishnamurthi, Elsie +Lee-Robbins, Justin Lubin, and the anonymous reviewers for their +helpful commentary. Finally, we thank our students, without whom +this work could not have taken place. 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IEEE, +223–233. https://doi.org/10.1109/ICSE.2015.43 + +CHI ’23, April 23–28, 2023, Hamburg, Germany +McNutt et al. +APPENDIX +In this appendix we provide supplementary material that fell outside the scope of the main content of the paper. +• In Sec. A we make several notes about the course design and other ancillary details. +• In Sec. B we provide additional details about several of our studies. +A +THE CREATIVE CODING COURSE +Here we provide additional context for our course. In Fig. 11, we show the schedule for the 3-week su21 edition (link to course site). This +edition featured only a sequence of homeworks and exercises, and did not include the self-guided project found in the “full” editions of +course (sp21, wi22). A similar version of the course was also taught as su22. There were six individual homework assignments in su21, the +first five of which appeared in all course editions: +1. Color Wheel. Recreate a given red-yellow-blue color wheel. (function calls, color and shape-drawing APIs, trigonometric expressions) +2. Freeze Frame. Pick a static frame from the “StoryBots: Shapes” video and recreate it. (function calls, color and shape-drawing APIs) +3. Trees. Use variables and arithmetic expressions to implement a symmetric tree drawing. (variables, arithmetic, curves) +4. Book of Patterns. Implement several 2-dimensional grid patterns—stripes, polka dots, checks, plaid, chevron, harlequin, argyle, +and honeycomb—inspired by the designs in My First Book of Patterns. (nested loops) +5. Snake. Make a simple version of the classic snake game. Starter code was provided with function stubs for different aspects of a +simple model-view-controller architecture (mutable variables, arrays, objects, animation, mouse and keyboard events) +6. Subway Font. Rewrite a webpage using a “font” that resembles the signage of the New York City subway. As shown on the cover +of Subway, some letters are rendered white-on-black and others are set atop colored circles. (HTML, CSS, DOM API, dictionaries) +As highlighted in the main text, we designed our course primarily for college students with little-to-no programming experience who +were not planning to major in computer science. In sp21, 4 out of the 31 students were undeclared, and among the remaining 27 students 14 +different programs of study were represented. In wi22, 10 out of the 27 students were undeclared, and among the remaining 17 students 12 +different programs of study were represented. All told, students from 23 different departments participated in the course (Fig. 12). +Based on both our study and pre-course on-boarding surveys, students self-reported high levels of prior experience (as highlighted in +Fig. 3) in each edition of the course. In sp21, students who had previously completed computer science courses at the university—in a couple +cases many such courses—were mistakenly allowed to enroll. This enrollment issue was fixed for wi22, but still nearly half of the students +(who completed the course) self-reported prior experience through self-study, courses in high school, and from other university departments +or institutions. In su21 and su22, enrollment was unrestricted (the high-school students were not already associated with the university), and +a large majority of these students reported prior experience. In any case, the different levels among our student populations helps color some +of the observations in the main text. +B +ADDITIONAL STUDY DETAILS +In this section we present aspects of our studies that did not fit in the main text: the ethics statement for our studies, additional results, +followed by descriptions of the hypothetical features from our Year 1 survey that were not implemented in Year 2. +B.1 +Ethics Statement +All studies were reviewed and determined to be exempt by our university’s institutional review board. We did not collect demographic data, +because it was not a core aspect of our investigation. Although we designed and taught these courses with an eye towards the associated +studies, we believe the course materials we developed and delivered (through lectures, office hours, and online discussions) were minimally +affected by the presence of these studies and our use of custom versions of the p5 editor. +B.2 +Additional Results +Here we list a series of one-off results that were observed. Then in Sec. B.2.1 include an analysis of the code folding feature (original part of +the analysis in Sec. 5.1). Finally we provide an additional analysis of the the auto-refresh feature in Sec. B.2.2. +• In Fig. 13, we show an alternative depiction of the results from both years of our survey which includes metrics other than the one +used in the main text (namely Usefulness). +• In Fig. 14 we provide a simple summary of the volume of code executions across all three editions along with assignment due dates, +which highlights that execution volume tended to be higher for earlier graphic-only assignments (compared to later assignments +which involved interactivity or HTML/CSS). +• In Fig. 15 we provide a related graphic showing execution history for sp21 and su21, along with the relative error rates by day. +In Year 1, our logging scheme did not include a mechanism for explicitly collecting run-time errors, so they were reconstructed +post-course by running each logged sketch for 10 seconds and collecting all errors generated during that period. This approach may +exclude errors students saw, such as those generated through interaction with the sketch or through randomness. On average each +session had 𝜇=7.27±32.8 errors, with outliers excluded. Within our reduced sample from Year 2, sessions exhibited 𝜇=30.7±95.8 +errors, again with outliers excluded. A one-sided t-test indicated that there were significantly more mean errors per session in Year + +A Study of Editor Features in a Creative Coding Classroom +CHI ’23, April 23–28, 2023, Hamburg, Germany +2 (𝑝<0.001). This increase is likely due to the new collection method, which captured errors witnessed by the user rather than just +reconstructed errors. +• In Fig. 16 we provide a figure showing the bi-gram action sequence probability of actions in Year 1. +• We then provide tables in Fig. 6 for Fig. 17 and Fig. 18. +B.2.1 +Code Folding. This simple feature, common to most modern editors [33], allows functions and other blocks of code to be collapsed +and later expanded. This feature was generally well liked as it made code “feel more organized” (A9), while helping users avoid “being +overwhelmed” (A2) and making things “look neater and less intimidating” (A1). This has the organizational benefit that it is “easier to find +specific chunks of code” (A1), which, as noted by A13 and A16, reduces the amount of scrolling—these are well-understood benefits of this +feature [50]. Being able to organize and navigate code are important concerns for novice creative coders. T3 +However, the feature was not universally appreciated. Whereas B8 found that code folding “Helped a lot while debugging and re- +organization!”, A9 asked “when debugging, what if the problem is in one of the lines of code that are hidden?” A number of participants noted +that they simply did not use it or did not find it helpful. Some students only invoked it accidentally (A10), while others found it confusing +(A5,14) because it did not clearly communicate what code was to be folded. Code folding, or other interface-based code organization tools, +seem especially valuable in this context as most sketches typically involved only a single file (e.g. sp21 and wi22 final projects had a median +of 1 JavaScript file). As the file structure abstraction for code organization is underused, there is opportunity for interface-based abstraction. +Debugging +Programming +Year 1/2 Outliers were dropped. Year 2 has missing data due to a data collection error. +Cycle Frequency (Count) +Average number of executions that it takes to switch +from one episode type to another +0 +5 +10 +15 +Cycle Length (Minutes) +Average length of time between code executions. +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +Episode Length (Minutes) +Alaboudi & LaToza +Auto-refresh +Manual +Year 1 +Year 2 +Year 1 +Year 2 +Period of time that a programmer is either in a +debugging or programming state. +0 +2 +4 +6 +8 +10 +12 +Q3 +mean +Q1 +Figure 10: Comparing how students shifted between debugging and programming states (using different execution styles) +against a baseline of professional programmers [7]. +B.2.2 +Live Coding. In addition to the analysis of the auto-refresh feature considered in the main text (Sec. 5.1.3), we also sought to understand +how student edit-run behavior compared to that of professional programmers. Fig. 10 shows how auto-refresh usage affected the length, +size, and frequency of edit-run cycles with regard to debugging versus programming states adopting the metrics used by Alaboudi and +LaToza [7], who studied how professional programmers shift between debugging and programming states during edit-run cycles in their +own work. A salient difference from the baseline was that the number of executions to transition from a programming to a debugging +state (and vice versa) was shorter for our students. This is likely informed by the domain: the professionals were working on projects such +as Firefox and Curl, which likely have a different execution cadence than the graphic-oriented work conducted in creative coding. The +programming episode length was similar for the professionals and those using manual execution—although debugging episodes for the latter +group were much shorter, suggesting that the errors were much less complex for our students. However, given the differences in expertise +and domain between these groups, it is difficult to identify a primary cause of the changes. +The key observation is that the usage patterns exhibited by our students were not the same as those of professionals, but were not +fundamentally dissimilar. This suggests the potential transferability of our observations about novices to more experienced users. +Using auto-refresh does not appear to have an effect on cycle frequency, although it seems to be associated with shorter episodes and cycle +lengths. This coheres with our expectation of auto-refresh, as it triggers executions more quickly than one might with manual execution, but +suggests a certain consistency related to task. + +CHI ’23, April 23–28, 2023, Hamburg, Germany +McNutt et al. +Figure 11: Schedule for su21 edition of the course. Readings and their corresponding images were adapted from Workman’s +HappyCoding tutorials [106]. +sp21 +wi22 +Anthropology +Chemistry +Undeclared +Economics +English +Global Studies +Humanities +Mathematics +Neuroscience +Physics +Political Science +Psychology +Visual Arts +Exchange +Undeclared +Economics +English +Env. and Urban +General Studies +Law +Humanities +Media Arts +Music +Physics +Public Policy +Russia, East Europe +1 +1 +1 +2 +2 +2 +1 +2 +4 +10 +Total 27 +Total: 31 +10 +5 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +2 +1 +1 +1 +1 +Comparative +Human +Dev. +Comp + Science +Figure 12: Home departments of students who completed the sp21 (top) and wi22 (bottom) courses. + +WEEK 2 +WEEK 3 +Tu Jul 13 +W Jul 14 +Th Jul 15 +F Jul 16 +M Jul 19 +Tu Jul 20 +W Jul 21 +Th Jul 22 +F Jul 23 +M Jul 26 +Tu Jul 27 +W Jul 28 +Th Jul 29 +Class +Class +Class +Random Topics +Loops +Class +Ftomagesit +Class +Class +Class +Recursion +classes +Class +stiders +(Unidentified Flying) +aries +L-systems +Functions +Objects +Recursione +Exercise 5 +Abstract Art +oops +(classified) +Class +1，2，3 +Exercise 11 +Exercise 4 +Objects +Class +(nothing) +Exercise 1 +SPRAY +Exercise_ 9 +Class +Class +Coding Mondrian +Counting Tags +Fractals +Exercise 2 +If statements +Arrays +PAINT +Abstracting Mondrian +Exercise _7 +taths//Br +Reading +count[tag]++; +Input +(nothing) +Exercise 8 +Images +Exercise 12 + Pong +Exercise 6 + Pixelate +Class +Reading +Reading +Reading +Exercise 10 +Living Line +L-systems +Exercise 3 +Random +Reading +For Loops +Interactive HTML +Reading +Walk NYC +Bouncy Bali +Using Objects +Project +From p5.js to Web Dev +:: +Walk ANYC +Description +Reading +html +Reading +circle Conjecture +Reading +Creating Functions +If statements +Array Functions +Bonus Reading +Creating Classes +Reading +Homework 3 +HTML, Tags, CSs +Bonus Reading +Homework 5 +Counting Pancakes +Welcome to Coding + Just okay. +Trees +Reading +Local Storage + +Debugging +Snake +p5Js +Animation +Arrays +p5Js +Homework 4 +Calling Functions +Homework 2 +Book of Patterns +SubwayFont +Input +Freeze Frame +cloud Storage +Homework 1 +SHATES +Post-Processing +Using Variables +Color wheel +1 +Creating Variables +Post-Processing +8A Study of Editor Features in a Creative Coding Classroom +CHI ’23, April 23–28, 2023, Hamburg, Germany +Year 1 +Year 2 +Implemented +Implemented +Implemented +Implemented +Standard +Standard +All features implemented +Coding by Drawing Tools +Color Picker +Linters +Autocomplete +Canvas Ruler +Tidy Code +Directly Manipulate Shapes +Time Travel Slider +Code Folding +In-context Docs +p5 State Displays +Interactive Value Inspector +Linked Copy-and-Paste +Code Snippet Templates +Number Sliders +Auto-refresh +Drag-and-Drop Refactoring +Interested +Often +Useful +1 +2 +3 +4 +5 +Linters +Color Picker +Tidy Code +Number Sliders +Coding by Drawing Tools +Auto-refresh +Autocomplete +Number Picker +Figure 13: Participants in both the long (Year 1) and short (Year 2) survey were asked about a variety of features and rated each +of them on how Interested they were in it, how Often they would use it, and how Useful they thought it was. Most features +considered are non-standard, however several were implemented in our editor or standard (but not implemented). It is notable +that although some of the most commonly instrumented features are not necessarily predictive of perceived utility. + +CHI ’23, April 23–28, 2023, Hamburg, Germany +McNutt et al. +Auto-Refresh +Manual Execution +Average Executions per Student +Mar 29, 2021 +Apr 05, 2021 +Apr 13, 2021 +Apr 21, 2021 +Apr 29, 2021 +May 07, 2021 +May 15, 2021 +May 23, 2021 +May 31, 2021 +0 +100 +200 +300 +400 +500 +600 +hw-1-mondrian-nyc +hw-2-color-wheel +hw-3-freeze-frame +hw-4-trees +hw-5-sleepy-face +hw-6-patterns +hw-7-snake +hw-8-pancakes +hw-9-subway-font +hw-10-typoglycemia +proposal +progress-report +final +Jul 12, 2021 +Jul 14, 2021 +Jul 16, 2021 +Jul 18, 2021 +Jul 20, 2021 +Jul 22, 2021 +Jul 24, 2021 +Jul 26, 2021 +Jul 28, 2021 +0 +100 +200 +300 +400 +500 +600 +hw-04-book-of-patterns +hw-01-color-wheel +hw-03-trees +hw-06-subway-font +hw-05-snake +hw-02-freeze-frame +Jan 09, 2022 +Jan 17, 2022 +Jan 25, 2022 +Feb 01, 2022 +Feb 09, 2022 +Feb 17, 2022 +Feb 25, 2022 +Mar 05, 2022 +Mar 13, 2022 +0 +100 +200 +300 +400 +500 +600 +hw-1-color-wheel +hw-2-freeze-frame +hw-3-trees +hw-4-book-of-patterns +hw-5-deck-of-cards +hw-6-snake +hw-7-wordle +proposal +hw-8-blackout-poetry +progress-report +final +Jun 12, 2022 +Jun 14, 2022 +Jun 16, 2022 +Jun 18, 2022 +Jun 20, 2022 +Jun 22, 2022 +Jun 24, 2022 +Jun 26, 2022 +Jun 28, 2022 +0 +100 +200 +300 +400 +500 +600 +hw-1-color-wheel +hw-2-freeze-frame +hw-3-trees +hw-4-book-of-patterns +hw-5-snake +hw-6-wordle +hw-7-typoglycemia +sp21 +su21 +wi22 +su22 +Figure 14: A summary of executions for each of the course editions show auto-refresh versus manual execution. + +A Study of Editor Features in a Creative Coding Classroom +CHI ’23, April 23–28, 2023, Hamburg, Germany +No Error +DOMException +Error +FirebaseError +RangeError +ReferenceError +SyntaxError +TypeError +Live +Programming +Executions +Manual +Executions +Normalized +Error Type +sp21 +su21 +0 +50k +100k +0 +2k +4k +6k +Week 1 +Week 2 +Week 3 +Week 4 +Week 5 +Week 6 +Week 7 +Week 8 +Week 9 +Week 10 +0% +50% +100% +0 +1k +2k +0 +1k +2k +3k +4k +Week 1 +Week 2 +Week 3 +0% +50% +100% +Figure 15: Errors over time in the first year of courses. Outliers have not been dropped. Our research questions are generally +not motivated by analysis of error types, as they were typically driven by course material, rather than by interface design. +78.7% +19.6% +46.1% +48.9% +0.2% +45.7% +43.2% +1.9% +16.1% +39.7% +0.9% +40.6% +0.2% +1.4% +0.0% +81.3% +12.5% +0.6% +0.6% +95.9% +2.7% +2.9% +1.2% +0.2% +2.9% +3.9% +0.0% +0.0% +3.1% +5.4% +0.4% +3.1% +Auto +refresh +Auto-refresh +Manual +Execution +Manual Execution +Save +Save +Find and +Replace +Find and Replace +Find and +Replace All +Find and Replace All +Tidy +Code +Tidy +Code +Tidy Code +Auto +refresh +Manual +Execution +Find and +Replace +Find and +Replace All +Tidy +Code +To Action +From Action +1.1% +71.8% +0.7% +94.5% +4.2% +5.0% +26.5% +61.5% +0.2% +62.7% +30.7% +3.2% +2.4% +0.9% +8.6% +24.8% +0.1% +34.6% +19.2% +46.2% +0.0% +0.2% +0.3% +85.1% +7.3% +0.1% +7.6% +0.4% +0% +100% +sp21 +su21 +Figure 16: The bi-gram action sequence probability in Year 1 shows the rate at which a given action is followed by another +particular action. We do not show wi22 because we mistakenly did not collect Tidy Code executions. +Feature Name +rating +mean +𝜎 +Q1 +Q3 +Auto-refresh +3.08 +1.26 +3 +4.00 +Autocomplete +4.56 +0.51 +4 +5.00 +Canvas Ruler +4.48 +0.71 +4 +5.00 +Code Folding +4.04 +0.84 +3 +5.00 +Code Snippet Templates +3.64 +1.29 +3 +5.00 +Coding by Drawing Tools +4.76 +0.44 +5 +5.00 +Color Picker +4.68 +0.56 +4 +5.00 +Directly Manipulate Shapes +4.20 +1.00 +4 +5.00 +Drag-and-Drop Refactoring +3.04 +1.34 +2 +4.00 +In-context Docs +4.04 +1.24 +4 +5.00 +Interactive Value Inspector +3.92 +1.08 +4 +4.00 +Linked Copy-and-Paste +3.68 +1.25 +3 +5.00 +Linters +4.60 +0.58 +4 +5.00 +Number Sliders +3.56 +1.26 +3 +5.00 +Tidy Code +4.48 +0.71 +4 +5.00 +Time Travel Slider +4.12 +0.73 +4 +5.00 +p5 State Displays +4.00 +1.12 +3 +5.00 +Figure 17: The computed values for Fig. 13, the survey results +relating to Usefulness from Year 1. +Feature Name +rating +mean +𝜎 +Q1 +Q3 +Auto-refresh +3.71 +1.23 +3 +5.00 +Autocomplete +3.71 +0.95 +3 +4.00 +Coding by Drawing Tools +3.79 +1.14 +3 +5.00 +Color Picker +4.38 +0.65 +4 +5.00 +Linters +4.50 +1.02 +4 +5.00 +Number Picker +2.79 +0.98 +2 +3.25 +Number Sliders +3.83 +0.70 +3 +4.00 +Tidy Code +4.29 +1.08 +4 +5.00 +Figure 18: The computed values for Fig. 13, the survey results +relating to Usefulness from Year 2. + +CHI ’23, April 23–28, 2023, Hamburg, Germany +McNutt et al. +B.3 +Hypothetical Features +Here we return to the hypothetical features asked about in Year 1 surveys but not implemented in p5/y2. Because responses were based only +on a brief description and static image, we limit our discussion of each feature. +Canvas Ruler. As mentioned earlier, the Canvas Ruler was widely viewed as a useful tool to add for creative coding—however, B8 felt +“it would take away the fun of mouseX and mouseY!” Several additional suggestions were made, such as being able to “measure angles, so +a ruler and compass.”(A16) In future editions of the course we intend to return to this feature, as it seems like a natural next step. The +primary concern in implementing such an addition would be that it does not clutter the interface T3, and perhaps, per commentary on Syntax +Templates, be controllable with a keyboard. +In-context Docs. Many full-featured editors (e.g. VS Code) include relevant documentation about language features and user-defined +variables as a tooltip. As with autocomplete, B4 believed In-context Docs would be helpful because “gives me an idea of what to [write].” +Several participants echoed this sentiment, believing that it “would have drastically widened my skill set” (A14). On the other hand, A4 was +“actually a little torn by [it] because I think googling and traveling to the reference is really important. It may start off as inconvenient but just +becomes more natural with practice”—which is in line with our observations about student skepticism. T4 Among the quantitative ratings +from the surveys in the first year, this feature was the only one that had a statistically significant relationship (𝑝<0.01) with self reported +experience was in-context docs, in particular exhibiting a negative correlation (𝑟=-0.308). It is possible that an alternative presentation of this +feature (perhaps in the search-based style of Blueprint [19]) might elicit more positive responses, however, based on these results we believe +that users might be similarly skeptical, although exploration of such responses could be usefully explored in future work. + +sketch.js +Saved: 2 minutes ago +Preview +const size=5o; +2 +const gap = 10; +37 +functionsetup()( +4 +createCanvas(400,400); +5 +9 +7vfunctiondraw()( +8 +background(220); +9 +10V +for（leti=0:i<10i++)( +11V +for（letj=o;j<10;j++) +12 +fill((i+j)%2？'black' +'white') +13 +square(i *(size + gap),j *(size + gap), size); +14 +15 +16 +17 +Left ruleredge (0,170)Right ruler edge (400,170)background(colorstring, [a]) +background(gray,[a]) +background(v1,v2,v3,[a]) +background(values) +background(image,[a]) +1function se +The background() function sets the color used for the +background of the p5js canvas. The default background is +"t +transparent. This function is typically used within draw() to +5V +function dr +clear the display window at the beginning of each frame +can be usedlinside setup( to set the backaround o +6 +background(250); +7 +rotateY(frameCount*0.01); +8 +for +letj=;j<5;j++)( +10 +push(); +11V +for(leti=0;i<80;i++)( +12 +translate( +13 +sin(frameCount*0.001+j)*100 +14 +sin(frameCount*0.001+j)*100 +15 +i*0.1 +16A Study of Editor Features in a Creative Coding Classroom +CHI ’23, April 23–28, 2023, Hamburg, Germany +Syntax Templates. The syntax templates we implement in our autocomplete were similar to our proposed Code Snippet Templates, +however the latter feature was docked (in the manner of Google Colab’s Code Snippet library), and thus required mouse clicks, which +may have dampened enthusiasm for the feature: “I think if there were keyboard shortcuts for these then I would use them extensively” (A13). +Some thought these features would be an “easy way to get students started with no experience” (A24), but as discussed in Sec. 6 others were +skeptical. We still believe this feature would be valuable to implement in the future, possibly integrated into the autocomplete, in order to +keep the interface tidy and unencumbered. T3 +Time Travel Slider. This proposed feature would allow the state of the code execution to be paused and rewound in order to support +debugging tasks—which a majority of respondents either understood as a GUI-based shortcut for p5’s frameRate setting (which specifies +how many times per second the draw loop is called) or as a mechanism for version control, both of which, while interesting, are not the +feature we intended. While several students expressed enthusiasm for this latter idea (indicating the potential utility of a Variolite-style [59] +or other selective undos, such as that of Yoon and Myers [108] or Mikami et al.’s [76] Micro-Versioning), this did not yield coherent feedback, +beyond confusion about unfamiliar features. + + Auto-refresh +Sketch name +Wood cushion +SUBMIT +Sketch Files +V +sketch.js +Preview + index.html + sketch.js +1functionsetup() +D style.css +2 +createCanvas(400,400); +3 +t +4 +5V +function draw()( +9 +background(220); +7 +bezier(x1, y1, x2, y2, x3, y3, x4, y4); +8 +7 +9 +Code Snippets +10 +Filter code snippets +11 +Add color +Add colored rectangle +Add beziercurve +4 +Insertedlineofcode +Add mouseDrags +/ent +Add mouseClicked event +> +Load image +Access web cam +Adid custom snippet + +Consolesketch.js +Saved: 2 minutes ago +Preview +constsize=50; +const gap =10; +3V +function setup()( +createCanvas(400,400); +function draw() +8 +background(220); +9 +10V +for(leti=0;i<10;i++)( +11V +for(letj=o;j<1o;j++) +12 +fill((i+j)%2?black' +"white'; +13 +square(i * (size + gap),j * (size + gap), size); +14 +15 +16 +17 +Frame +2.2kCHI ’23, April 23–28, 2023, Hamburg, Germany +McNutt et al. +p5 State Displays. A display of current values for common library variables, such as strokeWidth and fill color, at particular lines of +code—similar in spirit to object value displays in creativity tools like Illustrator. Some students were enthusiastic about this feature, noting +that it “would be extremely useful to be able to see all this information in one place” (B7), while others felt it might enrich creativity by showing +what options are available (A9). Others were less enthusiastic, noting that it would be “a little redundant” (A1) with running the code, or +that it would be tedious (A7) compared to simply writing code. +Interactive Value Inspector. A growing thread of research allows users to inspect the current value of variables at various lines of code +on demand, such as in Lerner’s Project Boxes [65] or Kang and Guo’s [58] “DISPLAY ALL THE VALUES!” approach to novice coding in +Omnicode, as well through live probes [85]. In this feature we proposed a Projection Box style feature that included a customizable inspector. +Students were generally enthusiastic about this feature, noting that it would be helpful for beginners (B3,8) as it would make “loop +definitions” (A16) and debugging (A1,10). Yet some worried that the implementation might be overwhelming (A5) or distracting (A6), or +would not substantially improve over console.log-based debugging (A14). T2 In addition one skeptical student believed that it might “make +the coder (especially early learner) to be lazy” (A15), and prevent them from learning good debugging skills +T4. + + Auto-refresh +Sketch name +Woodcushion +SUBMIT +8 +Sketch Files +< +sketch.js` +Preview + index.html + sketch.js +1vfunctionsetup()( +D style.css +2 +createCanvas(360,280); +3 +nostroke(); +4 +noLoop(); +5 +6 +77 +functiondraw()( +8 +drawcircle(width/2,280/2,6); +9 +10 +P5 State Values +11 +function drawcircle(x,radius,level)( +12 +consttt=（126*leve1)/4.0; +Stroke +None +13 +fill(tt); +Stroke Weiqht +1px +14 +ellipse( +height/2,radius*2,radius*2); +Stroke Cap +Round +15V +if +(leve) +1) ( +Fill +Multiple +16 +level =Ievel - 1; +Translate +0, 0 +17 +drawcircle(x-radius /2,radius /2,level) +18 +Rotation +drawcircle(x + radius/2,radius/2,level) +19 +20 +21 +22 +Watch additional values + +Console< +sketch.js +Saved: 9 days ago +Preview +175 +//Acounterto help safeguardagainstinfiniteloops +176 +//during development.Youmaytry adjusting this +177 +//valueif there is toolittleflooding. +178 +let fuel = 100000; +179 +180vwhile(worklist.length>0&&fuel>0)( +181 +//Gettheposition atthefront oftheworklist +182 +const pos=worklist.shift(); +183 +const x = pos[o]; +184 +const y= pos[1]; +185 +186 +const withinBounds=!(x<0Ilx>img.width Ilyimg.height); +187 +constnotAlreadyFilled=!alreadyFilled['$(x}-$(y}'J; +188 +189V +if(withinBounds&¬AlreadyFilled)( +190 +const c = img.get(x,y); +191 +const [r,g,b,a］ = c; +C: +[255,0,0,0] +192 +constiswhitish=r>240&&g>240&&b>240; +r: +255 +193 +constisTransparent=a<5; +g: +0 +194V +if(iswhitis +isTransparent) +195 +['$(x}-$ty=true; +b: +0 +alreadyFil +196 +img.set(x, +，color(fiilcolor)); +a: +0 +197 +iswhitish: +true +198 +//Addneighboringpixelstoendofworklist +isTransparent: +false +199 +worklist.push([x +1,yJ);A Study of Editor Features in a Creative Coding Classroom +CHI ’23, April 23–28, 2023, Hamburg, Germany +Linked Copy-and-Paste. Vihavainenet al. [101] note that novices tend to make heavy use of copy-paste, so a natural point of enhancement +then would be to embed variable-style abstraction into copy-paste itself. This idea has been discussed in research works previously [31, 98], +however is not typically seen in this style of editor. Some students thought this would be helpful, by “facilitat[ing] better organizational +practices” (A12) or in niche situations (A7,13). However, most others were apprehensive about the feature’s value. Some noted that it seemed +to be a more oblique version of creating a variable (A1,16). Some thought that what was already in the editor sufficiently addressed any +tasks linked copy-and-paste might accomplish, through regular copy-paste (A9) and Find & Replace (B8). A13 argued that a Sublime-style +multi-cursor selection would be more flexible and preferable. We note that multi-cursor support was enabled in our editor (as part of +CodeMirror), although students were not explicitly made aware of this functionality. Others still simply thought it would not be useful, and +would “creat[e] mess for me” (B4) or otherwise be confusing (A10, B5). T3 +Drag-and-Drop Refactoring. Clicking and dragging values to create arguments, variables, and other functions in a technique that has +been previously explored to useful effect [64]. In this feature we proposed a simple version of this feature, however our presentation lacked +the nuance of the presentations used by Lee et al. [64], which may have led feature being rated lowest. Although a few respondents were +intrigued, some said they would prefer copy and paste (A12, B1,5,6,8), most were disinterested. For example: “I personally don’t like dragging +and dropping things because there is room for dragging and dropping into the wrong section especially if your computer is slow. I don’t think copy +paste was too time consuming and encourages greater accuracy” (A2); several others shared these views about efficiency and accuracy. In +addition, there were concerns about the usability of the feature: “Clicking and dragging is not an ergonomic motion on a laptop touchpad” (A6). +We highlight this as an especially valuable concern, as dragging may not be an accessible motion for some users, although something like +Kobayashi and Igarashi’s suspendable drag-and-drop interactions [62] may usefully address these concerns. +Other Suggested Features. Beyond the hypothetical features we presented, some respondents suggested ideas like scratchpads or selective +execution contexts similar to some of the ideas expressed in Code Bubbles [18] or Jupyter notebooks. Others suggested course-specific +affordances, such as hints relevant to the assignment or integration of the assignment directly into the editor. This has a similar flavor as +DrRacket’s language levels, and Marceau et al. [72] briefly sketched out learner-attuned error messaging levels. This is similar to Interactive +Tutor Systems [53] which integrate curriculum and course work into a single environment. While this level of integration can be helpful, it +may undermine the utility of an in-class instruction model because such interfaces are naturally self- rather than group-paced, although that +should be investigated in future work. + +sketch.js +Saved: 13 minutes ago +Preview +1functionsetup()( +2 +createCanvas(360,280); +nostroke(); +4 +noLoop(); +5 +6 +7vfunction draw()( +8 +drawcircle(width /2,280/2,6); +6 +10 +11V +functiondrawcircle(x,radius,level)( +12 +consttt=(126*level)/4.0; +13 +fill(tt); +14 +ellipse(x, +height/2,radius*2,radius*2); +15V +if(level>1)( +16 +level = level -1 +17 +drawcircle(x - +radius / 2， +radius / 2, +level); +18 +drawcircle(x+radius/2, +radius/ +level); +19 +20 +21 +23 +Thepurplevalueswerecopy+pastedfrom +thegreenvalue,linkingthosevalues.Achange +Console +to any linked value will change all linked values.sketchjs +Saved: 24 minutes ago +Preview +17 +function setup()( +2 +createCanvas(360,280); +3 +nostroke(); +4 +noLoop(); +5 +7 +6 +77 +functiondraw()( +8 +drawcircle(width/2,280/2,6); +9 +10 +11V +function drawcircle(x,radius,level)( +12 +consttt=（126*level)/4.0; +13 +fill(tt); +14 +ellipse(x,height/2,radius*2,radius *2); +15V +if(level>1) +16 +level = level - 1; +17 +drawCircle(x + radius / 2, radius / 2,level) +18 +drawcircle(x -radius /2,radius / +2,level); +Drag a line of code to move it +19 +drawCircle(x +radius +radius +Level +20 +21 +22 +23 +ConsoleCHI ’23, April 23–28, 2023, Hamburg, Germany +McNutt et al. +C +SURVEY INSTRUMENT FOR INITIAL SURVEY — YEAR 1 (sp21, su21) +C.1 +Page 1: Consent to Participate in a Research Study +Research Project Title: Post-Course Survey of Students in Creative Coding (2021) +Principal Investigator: Ravi Chugh +Graduate Student: Andrew McNutt +IRB Protocol: IRB21-1062 +This form is designed for students younger than 18 years of age who took the Creative Coding Pre-College Immersion class in Summer +2021 and their parents, respectively referred to as “you” and “your child” below. You (or your child) is being asked to take part in a research +study. This form has important information about the reason for doing this study, what we will ask you (or your child) to do, and the way we +would like to use information about you (or your child) if you choose to allow yourself (or your child) to be in the study. +Purpose of Research Study: You are (or your child is) being asked to participate in a research study regarding the usability of editors +for creative coding. In our recently completed course we used an in-browser editor that was slightly modified from the publicly available p5 +editor. We are interested in understanding what editor features might be useful to someone learning to code (particularly in the context +of a creative coding course) or otherwise making digital art works. Ultimately, this research may be published and presented at scientific +conferences to improve the community’s knowledge about editors for creative coding, and may be used to improve the editor used in future +iterations of our course. +Participation Procedures and Activities: The full extent of the procedure will involve completing this survey. We anticipate that +completion of this survey will take up to 60 minutes. Due to the difficulty of determining credit for partial completion, no compensation will +be provided for partial completion. At the end of the form you (or your child) will provide a student id and preferred email address, and you +(or your child) will receive a $30 Amazon gift card for participating. +Consent and Assent Process: If you are (or your child is) 18 years or older, you (or your child) can provide the consent required to +opt-in to the study. If you are (or your child is) under 18 years of age, you can give your assent (or you can give your parental consent) to +join the study. For students under 18 years of age, participation in this study requires both consent from a parent as well as assent from the +student. +Risks/Discomforts of Being in this Study: The risks to your participation in the survey are those associated with basic computer +tasks, including boredom, fatigue, or mild stress. Benefits of Being in this Study The only benefit to you (or your child) is the learning +experience from participating in a research study. The benefit to society is the contribution to scientific knowledge. +Confidentiality of Data and Limits to Confidentiality: Any reports and presentations about the findings from this study will not +include your (or your child’s) name or any other identifying information. +Use of Your Research Data: We will never share the data beyond the University of Chicago research team. However, an analysis of the +data may be analyzed and published in scientific conference proceedings or journal articles. The free-text responses provided to any portion +of this survey may be quoted in part or in whole in this publication. We will remove any information from the analysis that could identify +you (or your child) before providing the analysis for publication. +Voluntary Participation and Right to Refuse or Withdraw: Participation in this study is voluntary. The decision to participate in +this study is entirely up to you and your child. You (or your child) may refuse to take part in the study at any time without prejudice or +penalties and will not result in any loss of benefits to which you (or your child) are otherwise entitled. +Mandatory Reporting of Child Abuse or Neglect: The research study staff are mandated reporters and are required to report suspected +child abuse or neglect to the Illinois Department of Child and Family Services. For more information, please see the University policy: +https://tinyurl.com/mr26uazn +Contact Information for Research Questions and Participation: If you have questions or concerns about the study, you can contact +the researchers at: +Principal Investigator +Ravi Chugh, +Associate Professor +John Crerar Library +University of Chicago +5730 S Ellis Ave +Chicago, IL 60637 +Email: rchugh@uchicago.edu +Graduate Student +Andrew McNutt, +PhD student +John Crerar Library + +A Study of Editor Features in a Creative Coding Classroom +CHI ’23, April 23–28, 2023, Hamburg, Germany +University of Chicago +5730 S Ellis Ave +Chicago, IL 60637 +Email: mcnutt@uchicago.edu +If you have any questions about your rights as a participant in this research, feel you have been harmed, or wish to discuss other +study-related concerns with someone who is not part of the research team, you can contact the University of Chicago Social and Behavioral +Sciences Institutional Review Board (IRB) Office by phone at (773) 702-2915, or by email at sbs-irb@uchicago.edu. +Parental Consent +(1) Parent full name (Last, First) +(2) Parent Email address +(3) I have read and understood this consent form. Yes ⃝ no ⃝ +(4) I am a parent and give consent for my child, under 18 years of age, to participate in this study. Yes ⃝ no ⃝ +Student Assent +(1) Student full name (Last, First) +(2) Student Email address +(3) Student GitHub username (same as used for homework submission in this class) +(4) Student CNetID (the username before your @uchicago.edu email address) +(5) I have read and understood this consent form. Yes ⃝ no ⃝ +(6) I am a student, under 18 years of age, and give assent to participate in this study. Yes ⃝ no ⃝ + +CHI ’23, April 23–28, 2023, Hamburg, Germany +McNutt et al. +C.2 +Page 2: Introduction and Reflection +In this section, we’ll ask you some questions about your programming background, and to reflect on your experience during the course. +(1) Pre-Course Experience. How much programming experience did you have prior to taking the course? +(2) Post-Course Confidence. How confident do you feel in your programming skills after taking this course? Have they improved? +(3) Challenges. What aspect of coding or learning to program gave you the most trouble? As a way to help organize your thinking, +consider the assignment that you had the most difficulty with. Could the editor have done anything to help you with that? +(4) Debugging. Think about the experience of debugging. How did you go about doing it? If you used console.log to help debug, did +you find it helpful? Did you use any other strategies? Is there anything about it or the debugging process that you wish could have +been different? +(5) Error Messages. Think about the error messages you encountered (inline in the code box, in the console area under the code +box, in the browser console, or elsewhere). Were they useful? How did you deal with them? Do you wish they were presented +differently? +(6) Code Organization. How did you go about organizing your code? For instance, how did you decide where to place variables, +create functions? Was there ever a point when your organizational scheme ran into problems, if so how did you handle it? Is there +anything the editor could have done to help you during these organizational tasks? +(7) Freeze Frame Homework. Think about the freeze frame assignment (or any other time during the course when you needed to +repeatedly edit and re-run the code in order to get particular positions or other values to achieve a desired effect). Is there anything +the editor could have done to help you get your image to be just right? +(8) External Tools. It’s natural to use other tools as part of the programming process, such as color eye droppers or p5’s online +documentation. Do you think it would be useful to integrate these tools as part of the editor? What other tools can you imagine +wanting to be part of your in-editor coding workflow? +(9) Desired Features. What sorts of editor features might have allowed you to be more effective in your coding? What sorts of editor +features might have allowed you to be more creative? + +A Study of Editor Features in a Creative Coding Classroom +CHI ’23, April 23–28, 2023, Hamburg, Germany +C.3 +Pages 3-18: Editor Features +In this section, we’ll ask for your thoughts and opinions about some features that appeared in the editor as well as some hypothetical features +that we may implement for future iterations of the course. +(1) Autocomplete Imagine an editor feature which provides autocomplete suggestions as you type. This would be akin to the predictive +text feature found in many messaging applications, but would be sensitive to variables you’ve created and functions available from +imported libraries. While this feature appears in some other editors it did not appear in our editor. +(a) Do you think this would be useful? (1) Not Very Useful ⃝, (2) ⃝, (3) Neither useful nor unuseful ⃝, (4) ⃝, Very Useful ⃝ +(b) How often do you think you would use this feature? (1) Never ⃝, (2) ⃝, (3)Occasionally ⃝, (4) ⃝, All the time ⃝ +(c) Why or why not? Is there any way you would like to modify this feature to make it more useful? +(2) Linters Our editor featured a tool called a “linter” that surfaced stylistic or coding errors through on-screen alerts, as in the image +below. This tool is analogous to spell- and grammar-checkers in standard word processors. The particular linter used in our editor, +called JSHint, tends not to give many warnings for stylistic errors. Other available linters give many more warnings for stylistic +errors. +(a) Do you think this is useful? (1) Not Very Useful ⃝, (2) ⃝, (3) Neither useful nor unuseful ⃝, (4) ⃝, Very Useful ⃝ +(b) How often do you think you used this feature? (1) Never ⃝, (2) ⃝, (3)Occasionally ⃝, (4) ⃝, All the time ⃝ +(c) Why or why not? Is there any way you would like to modify this feature to make it more useful? +(3) Tidy Code Our editor featured a button called “Tidy Code” which automatically reorganized your code. This feature is sometimes +seen in other editors and is more commonly known as an “auto-formatter”. These can typically be configured to enforce a particular +coding style. +(a) Do you think this is useful? (1) Not Very Useful ⃝, (2) ⃝, (3) Neither useful nor unuseful ⃝, (4) ⃝, Very Useful ⃝ +(b) How often do you think you used this feature? (1) Never ⃝, (2) ⃝, (3)Occasionally ⃝, (4) ⃝, All the time ⃝ +(c) Why or why not? Is there any way you would like to modify this feature to make it more useful? + +1 +function hello() +2 +alert('Hello world!') +3 +let counter = 0; +4 +for(let idx = ; +idx < 10; idx++) ( +5 +counter += +idx; +6 +7 +console.l +8 +log +method)Console.log(...data:any):.. +timeLog1vfunctionsetup()( +2 +createCanvas(710,40o,WEBGL); +3 +4 +5vfunctiondraw()( +6 +background(250) +× Missing semicolon +rotateY(frameCount * 0.01)s +Expected an assignment or function call and instead saw an expression. +8 +9V +for(letj=o;j<5;j++)( +10 +push(); +11 +for(leti=0;i<80;i++) +12 +translate( +1.2 +100 +Ap5 * cs11 +File +Edit +Help +0 +Tidy Code +++Tab +duct. +Find +3+FCHI ’23, April 23–28, 2023, Hamburg, Germany +McNutt et al. +(4) Auto-refresh There is a feature in our editor called “Auto-refresh.” When selected, it re-runs your code every time you finish +typing (or sometimes before). This enables small update cycles as you code. +(a) Do you think this is useful? (1) Not Very Useful ⃝, (2) ⃝, (3) Neither useful nor unuseful ⃝, (4) ⃝, Very Useful ⃝ +(b) How often do you think you used this feature? (1) Never ⃝, (2) ⃝, (3)Occasionally ⃝, (4) ⃝, All the time ⃝ +(c) Why or why not? Is there any way you would like to modify this feature to make it more useful? +(5) Code Folding There is a feature in our editor, and many other editors, called “code folding” as in the screenshot below. This allows +you to collapse certain sections of code, such as functions and loops. The “folded” code is still there and can be referenced from +other places, but it’s temporarily hidden and replaced with “...”. +(a) Do you think this is useful? (1) Not Very Useful ⃝, (2) ⃝, (3) Neither useful nor unuseful ⃝, (4) ⃝, Very Useful ⃝ +(b) How often do you think you used this feature? (1) Never ⃝, (2) ⃝, (3)Occasionally ⃝, (4) ⃝, All the time ⃝ +(c) Why or why not? Is there any way you would like to modify this feature to make it more useful? +(6) Canvas Ruler Imagine a feature which allows you to place a draggable ruler into the drawing side of the editor. You can use it as a +way to visually identify screen coordinates. This feature might involve a way to display the current direction and placement of the +coordinate origin, especially with regard to translation and rotation functions. +(a) Do you think this would be useful? (1) Not Very Useful ⃝, (2) ⃝, (3) Neither useful nor unuseful ⃝, (4) ⃝, Very Useful ⃝ +(b) How often do you think you would use this feature? (1) Never ⃝, (2) ⃝, (3)Occasionally ⃝, (4) ⃝, All the time ⃝ +(c) Why or why not? Is there any way you would like to modify this feature to make it more useful? + +sketch.js +Saved: 2 minutes ago +Preview +const size=5o; +2 +const gap = 10; +37 +functionsetup()( +4 +createCanvas(400,400); +5 +9 +7vfunctiondraw()( +8 +background(220); +9 +10V +for（leti=0:i<10i++)( +11V +for（letj=o;j<10;j++) +12 +fill((i+j)%2？'black' +'white') +13 +square(i *(size + gap),j *(size + gap), size); +14 +15 +16 +17 +Left ruleredge (0,170)Right ruler edge (400,170)p5 * cs111 +File v +Edit +Help +0 +Auto-refresh +Sketch name +Small mapusaurus functionsetupO...) +4 +5V +function draw()( +6 +background(25o); +rotateY(frameCount * 0.01); +8 +9V +for (let j = 0; j< 5; j++)( +10 +push(); +11V +for(leti=0;i<80;i++)( +12 +translate( +13 +sin(frameCount *0.001 +j)* 100 +14 +sin(frameCount * 0.001 +j)* 100 +15 +i*0.1 +16A Study of Editor Features in a Creative Coding Classroom +CHI ’23, April 23–28, 2023, Hamburg, Germany +(7) Number Sliders Imagine a feature which allows you to modify the numeric values in the code without typing or re-running the +program (such as in the image below). With this feature, you click a value of interest and then drag a slider that appears above it to +change it. The canvas is continuously re-rendered as you drag the slider. This would be similar to using p5’s slider function, but, +rather than just changing the value in the running code, it would also modify the text of the code. +(a) Do you think this would be useful? (1) Not Very Useful ⃝, (2) ⃝, (3) Neither useful nor unuseful ⃝, (4) ⃝, Very Useful ⃝ +(b) How often do you think you would use this feature? (1) Never ⃝, (2) ⃝, (3)Occasionally ⃝, (4) ⃝, All the time ⃝ +(c) Why or why not? Is there any way you would like to modify this feature to make it more useful? +(8) Color Picker Imagine having a color picker integrated into the editor. When selecting color values in the code, the color picker +could appear on hover (as in the image below) to modify the value, or the tool could be docked into the bottom of the editor +(allowing it to be always on). This could include pre-configured or document-based palettes, as in Illustrator or Photoshop. +(a) Do you think this would be useful? (1) Not Very Useful ⃝, (2) ⃝, (3) Neither useful nor unuseful ⃝, (4) ⃝, Very Useful ⃝ +(b) How often do you think you would use this feature? (1) Never ⃝, (2) ⃝, (3)Occasionally ⃝, (4) ⃝, All the time ⃝ +(c) Why or why not? Is there any way you would like to modify this feature to make it more useful? + +> +sketch.js +Saved: 2 minutes ago +Preview +1 +1 +const size = 50; +2 +const gap = 10; +3V+ +functionsetup +4 +createCanvas(400,400); +5 +口 +6 +7vfunctiondraw() +8 +background(220); +10V +for（leti=0;i<10:i++)( +11V +for(letj=o;j<10;j++)( +12 +fill((i+j)%2？'black' +'white'); +13 +square(i * (size + gap),j * (size + gap), size); +14 +t +15 +16 +17sketch.js* +esago +Preview +const size = 5o; +const gap = 10; +3V +functionsetup(） +4 +createCanvas(400,400); +5 +6 +7vfunction draw()( +8 +background(220); +#000000 +100 +10V +for(leti=0;i<10; +HEX +H +S +Alpha +11V +for(letj=0;j<10 +12 +fill((i+j)%2？'black' +"white'); +13 +square(i * (size + gap),j * (size + gap), size) +14 +15 +7 +16 +17CHI ’23, April 23–28, 2023, Hamburg, Germany +McNutt et al. +(9) p5 State Displays In p5 it is common to set values for variables such as strokeWidth (which describes the width of subsequently +drawn lines), or fill (which describes the interior color of subsequently drawn shape). These are examples of ""state variables"". +There are a variety of such variables in p5, however (in contrast with digital drawing tools like Photoshop), these variables are not +displayed anywhere in the editor.

Imagine a feature where all of the relevant state values are shown, such that when +you move the text cursor to a line in your code, the display shows the state values at that point in time. This would allow you to +evaluate if your drawing tools are configured as you want them to be. +(a) Do you think this would be useful? (1) Not Very Useful ⃝, (2) ⃝, (3) Neither useful nor unuseful ⃝, (4) ⃝, Very Useful ⃝ +(b) How often do you think you would use this feature? (1) Never ⃝, (2) ⃝, (3)Occasionally ⃝, (4) ⃝, All the time ⃝ +(c) Why or why not? Is there any way you would like to modify this feature to make it more useful? +(10) Interactive Value Inspector Imagine a feature which allows you to see the value of the current program execution by hovering +over chunks of the code. It would provide similar information as when inserting console.log statements into your code, but instead +you would extract that same information through hovering. In contrast with ""p5 State Displays"" (which only shows p5 state +variables like fill and strokeWidth) this feature would allow you to see both state variables as well as the value of all variables, +including ones you’ve defined. This information could be presented through a tooltip (as in the below image) or through a docked +panel. +(a) Do you think this would be useful? (1) Not Very Useful ⃝, (2) ⃝, (3) Neither useful nor unuseful ⃝, (4) ⃝, Very Useful ⃝ +(b) How often do you think you would use this feature? (1) Never ⃝, (2) ⃝, (3)Occasionally ⃝, (4) ⃝, All the time ⃝ +(c) Why or why not? Is there any way you would like to modify this feature to make it more useful? + + Auto-refresh +Sketch name +Woodcushion +SUBMIT +8 +Sketch Files +< +sketch.js` +Preview + index.html + sketch.js +1vfunctionsetup()( +D style.css +2 +createCanvas(360,280); +3 +nostroke(); +4 +noLoop(); +5 +6 +77 +functiondraw()( +8 +drawcircle(width/2,280/2,6); +9 +10 +P5 State Values +11 +function drawcircle(x,radius,level)( +12 +consttt=（126*leve1)/4.0; +Stroke +None +13 +fill(tt); +Stroke Weiqht +1px +14 +ellipse( +height/2,radius*2,radius*2); +Stroke Cap +Round +15V +if +(leve) +1) ( +Fill +Multiple +16 +level =Ievel - 1; +Translate +0, 0 +17 +drawcircle(x-radius /2,radius /2,level) +18 +Rotation +drawcircle(x + radius/2,radius/2,level) +19 +20 +21 +22 +Watch additional values + +Console< +sketch.js +Saved: 9 days ago +Preview +175 +//Acounterto help safeguardagainstinfiniteloops +176 +//during development.Youmaytry adjusting this +177 +//valueif there is toolittleflooding. +178 +let fuel = 100000; +179 +180vwhile(worklist.length>0&&fuel>0)( +181 +//Gettheposition atthefront oftheworklist +182 +const pos=worklist.shift(); +183 +const x = pos[o]; +184 +const y= pos[1]; +185 +186 +const withinBounds=!(x<0Ilx>img.width Ilyimg.height); +187 +constnotAlreadyFilled=!alreadyFilled['$(x}-$(y}'J; +188 +189V +if(withinBounds&¬AlreadyFilled)( +190 +const c = img.get(x,y); +191 +const [r,g,b,a］ = c; +C: +[255,0,0,0] +192 +constiswhitish=r>240&&g>240&&b>240; +r: +255 +193 +constisTransparent=a<5; +g: +0 +194V +if(iswhitis +isTransparent) +195 +['$(x}-$ty=true; +b: +0 +alreadyFil +196 +img.set(x, +，color(fiilcolor)); +a: +0 +197 +iswhitish: +true +198 +//Addneighboringpixelstoendofworklist +isTransparent: +false +199 +worklist.push([x +1,yJ);A Study of Editor Features in a Creative Coding Classroom +CHI ’23, April 23–28, 2023, Hamburg, Germany +(11) In-context Docs Imagine an editor feature which gives you access to the documentation while you are writing code. This might +involve a tooltip that appears on hover (as in the image below) which describes the usage of a particular function. It could also +involve showing the description in a dedicated pane on the side. While such features appear in some other editors it did not appear +in our editor. +(a) Do you think this would be useful? (1) Not Very Useful ⃝, (2) ⃝, (3) Neither useful nor unuseful ⃝, (4) ⃝, Very Useful ⃝ +(b) How often do you think you would use this feature? (1) Never ⃝, (2) ⃝, (3)Occasionally ⃝, (4) ⃝, All the time ⃝ +(c) Why or why not? Is there any way you would like to modify this feature to make it more useful? +(12) Code Snippet Templates Imagine a feature which allows you to paste in common code snippets from a list. After clicking one of +the desired options (such as in the image below) a piece of code achieving that functionality will be added to your code. These +snippets could include small structures, such as for-loops, or larger structures, such as particular API uses or classes. This feature +sometimes appears in other coding systems, but was not implemented in our system. +(a) Do you think this would be useful? (1) Not Very Useful ⃝, (2) ⃝, (3) Neither useful nor unuseful ⃝, (4) ⃝, Very Useful ⃝ +(b) How often do you think you would use this feature? (1) Never ⃝, (2) ⃝, (3)Occasionally ⃝, (4) ⃝, All the time ⃝ +(c) Why or why not? Is there any way you would like to modify this feature to make it more useful? +(13) Coding by Drawing Tools Imagine an editor feature which allows you to fill out the arguments to particular functions graphically. +For instance, you might indicate to the editor that you are interested in drawing a bezier curve, and then draw each of the vertices +in the curve directly on the editor, just as you would in a GUI-based tool like Illustrator, which in turn inserts a corresponding line +of bezier command in your code. Unlike in the previous feature, which just inserted code templates, this feature allows you to +specify the values of the inserted code with your mouse on the output canvas. +(a) Do you think this would be useful? (1) Not Very Useful ⃝, (2) ⃝, (3) Neither useful nor unuseful ⃝, (4) ⃝, Very Useful ⃝ +(b) How often do you think you would use this feature? (1) Never ⃝, (2) ⃝, (3)Occasionally ⃝, (4) ⃝, All the time ⃝ +(c) Why or why not? Is there any way you would like to modify this feature to make it more useful? + +background(colorstring, [a]) +background(gray,[a]) +background(v1,v2,v3,[a]) +background(values) +background(image,[a]) +1function se +The background() function sets the color used for the +background of the p5js canvas. The default background is +"t +transparent. This function is typically used within draw() to +5V +function dr +clear the display window at the beginning of each frame +can be usedlinside setup( to set the backaround o +6 +background(250); +7 +rotateY(frameCount*0.01); +8 +for +letj=;j<5;j++)( +10 +push(); +11V +for(leti=0;i<80;i++)( +12 +translate( +13 +sin(frameCount*0.001+j)*100 +14 +sin(frameCount*0.001+j)*100 +15 +i*0.1 +16 Auto-refresh +Sketch name +Wood cushion +SUBMIT +Sketch Files +V +sketch.js +Preview + index.html + sketch.js +1functionsetup() +D style.css +2 +createCanvas(400,400); +3 +t +4 +5V +function draw()( +9 +background(220); +7 +bezier(x1, y1, x2, y2, x3, y3, x4, y4); +8 +7 +9 +Code Snippets +10 +Filter code snippets +11 +Add color +Add colored rectangle +Add beziercurve +4 +Insertedlineofcode +Add mouseDrags +/ent +Add mouseClicked event +> +Load image +Access web cam +Adid custom snippet + +Console> +sketch.js +Saved: 2 minutes ago +Preview +Drawing tools +NTHE+ +Additional drawing tools +1function setup()( +click +Add Slider ++ +2 +createCanvas(400,400); +Add Button ++ +7 +click +4 +Add Radio ++ +5function draw()( +Adid Vector Shane +6 +background(220); +7 +noFiil(); +8 +stroke(255,102,0); +9 +10 +stroke(0,0,); +11 +bezier(285,20,10,10,90,90,15,80); +12 +13 +click +click +Insertedlineofcode +ConsoleCHI ’23, April 23–28, 2023, Hamburg, Germany +McNutt et al. +(14) Linked Copy-and-Paste A common abstraction mechanism that we used in class is to create variables or functions rather than +copy-pasting chunks of code. While variables and functions are a useful form of computational thinking, there are other ways to +approach this task.

Imagine a feature which keeps track of your copy-and-pastes: whenever you edit a value you’ve +copied and pasted, all pieces of code which were copied are also changed. This special linked copy-paste can be selectively turned +on and off so that you can make edits without changing all copies. +(a) Do you think this would be useful? (1) Not Very Useful ⃝, (2) ⃝, (3) Neither useful nor unuseful ⃝, (4) ⃝, Very Useful ⃝ +(b) How often do you think you would use this feature? (1) Never ⃝, (2) ⃝, (3)Occasionally ⃝, (4) ⃝, All the time ⃝ +(c) Why or why not? Is there any way you would like to modify this feature to make it more useful? +(15) Drag-and-Drop Refactoring Refactoring is the process of changing the way a piece of code is organized such that the functionality +remains the same, but the code is easier to work with. You probably did this during the course by making a variable to capture +repeated code or by creating a function to represent some repeated functionality.

Imagine a feature which allows you to +click and drag values to create arguments, variables, and functions. This might allow you to reorder lines of code by clicking and +dragging them, or to highlight a series of repeated values and drag them to automatically create a new variable. +(a) Do you think this would be useful? (1) Not Very Useful ⃝, (2) ⃝, (3) Neither useful nor unuseful ⃝, (4) ⃝, Very Useful ⃝ +(b) How often do you think you would use this feature? (1) Never ⃝, (2) ⃝, (3)Occasionally ⃝, (4) ⃝, All the time ⃝ +(c) Why or why not? Is there any way you would like to modify this feature to make it more useful? +(16) Time Travel Slider Imagine an editor feature which allows you to go back to earlier points in time of your code’s execution. With +such a feature you’d press Play, as normal, and watch your code execute. If there was an intermediate state you were curious about +you can pause the execution and go back (by dragging a slider) to an earlier state of the canvas. Once you are finished inspecting +you can resume execution without rerunning the code. This would allow you to inspect how your code was adding shapes to the +canvas over time. +(a) Do you think this would be useful? (1) Not Very Useful ⃝, (2) ⃝, (3) Neither useful nor unuseful ⃝, (4) ⃝, Very Useful ⃝ +(b) How often do you think you would use this feature? (1) Never ⃝, (2) ⃝, (3)Occasionally ⃝, (4) ⃝, All the time ⃝ +(c) Why or why not? Is there any way you would like to modify this feature to make it more useful? + +sketch.js +Saved: 2 minutes ago +Preview +constsize=50; +const gap =10; +3V +function setup()( +createCanvas(400,400); +function draw() +8 +background(220); +9 +10V +for(leti=0;i<10;i++)( +11V +for(letj=o;j<1o;j++) +12 +fill((i+j)%2?black' +"white'; +13 +square(i * (size + gap),j * (size + gap), size); +14 +15 +16 +17 +Frame +2.2ksketch.js +Saved: 13 minutes ago +Preview +1functionsetup()( +2 +createCanvas(360,280); +nostroke(); +4 +noLoop(); +5 +6 +7vfunction draw()( +8 +drawcircle(width /2,280/2,6); +6 +10 +11V +functiondrawcircle(x,radius,level)( +12 +consttt=(126*level)/4.0; +13 +fill(tt); +14 +ellipse(x, +height/2,radius*2,radius*2); +15V +if(level>1)( +16 +level = level -1 +17 +drawcircle(x - +radius / 2， +radius / 2, +level); +18 +drawcircle(x+radius/2, +radius/ +level); +19 +20 +21 +23 +Thepurplevalueswerecopy+pastedfrom +thegreenvalue,linkingthosevalues.Achange +Console +to any linked value will change all linked values.sketchjs +Saved: 24 minutes ago +Preview +17 +function setup()( +2 +createCanvas(360,280); +3 +nostroke(); +4 +noLoop(); +5 +7 +6 +77 +functiondraw()( +8 +drawcircle(width/2,280/2,6); +9 +10 +11V +function drawcircle(x,radius,level)( +12 +consttt=（126*level)/4.0; +13 +fill(tt); +14 +ellipse(x,height/2,radius*2,radius *2); +15V +if(level>1) +16 +level = level - 1; +17 +drawCircle(x + radius / 2, radius / 2,level) +18 +drawcircle(x -radius /2,radius / +2,level); +Drag a line of code to move it +19 +drawCircle(x +radius +radius +Level +20 +21 +22 +23 +ConsoleA Study of Editor Features in a Creative Coding Classroom +CHI ’23, April 23–28, 2023, Hamburg, Germany +(17) Directly Manipulate Shape Attributes on Canvas Imagine being able to edit the output canvas and have that change the +corresponding JavaScript code. This would involve making changes to specific values graphically, such as changing the size of +circle or end points of lines by dragging them to a desired position. This differs from the functionality of the previously described +""Code by Drawing Tools"" feature; that one allowed clicking and dragging to add new shapes to the code, whereas this one allows +clicking and dragging to dynamically update and modify existing shapes in the code. +(a) Do you think this would be useful? (1) Not Very Useful ⃝, (2) ⃝, (3) Neither useful nor unuseful ⃝, (4) ⃝, Very Useful ⃝ +(b) How often do you think you would use this feature? (1) Never ⃝, (2) ⃝, (3)Occasionally ⃝, (4) ⃝, All the time ⃝ +(c) Why or why not? Is there any way you would like to modify this feature to make it more useful? + +> +sketch.js +Saved: 15 seconds ago +Preview +5functiondraw()( +6 +background(102); +7 +8 +push(); +9 +translate(width*0.2,height*0.5): +10 +rotate(30); +11 +star(0,0,5 +Wethinkyoumeanforthis +12 +pop(); +shape to be rotated, but +13 +perhapsyou meanforittobe +14 +push(); +translated? If so, click here +15 +translate(wi +16 +star(0,0,80,100,40); +17 +pop(); +18 +19 +push(); +20 +transiate(width * 0.8, height * 0.5); +21 +star(0,0,30,70,5); +22 +pop(); +23 +24 +25vfunction star(x,y,radius1,radius2,npoints)( +26 +letangle=Two_PI/npoints; +27 +let halfAngle = angle / 2.0; +28 +beginShape（); +29V +for(leta=o;a +sketch.js +1 v +function setup() ( +2 +const canvasSize +Editor.slider(0, 800, 500) +3 +createCanvas(canvassize, canvasSize); +4 +5 +function draw()( +7 +background("pink"); +8 +KCHI ’23, April 23–28, 2023, Hamburg, Germany +McNutt et al. +(4) Feature: Linting +(a) How often did you use Linting? ⃝ Never, ⃝ Once in a while, ⃝ Occasionally, ⃝ Frequently, ⃝ All the time +(b) Do you think Linting is useful? ⃝ Not very useful, ⃝ Not useful, ⃝ Neither useful nor unuseful, ⃝ Useful, ⃝ Very Useful +(c) Do you have any comments about Linting? For instance: How did you feel it affected your learning? Is there any way you +would modify it to make it more useful? +(5) Feature: Tidy Code +(a) How often did you use Tidy Code? ⃝ Never, ⃝ Once in a while, ⃝ Occasionally, ⃝ Frequently, ⃝ All the time +(b) Do you think Tidy Code is useful? ⃝ Not very useful, ⃝ Not useful, ⃝ Neither useful nor unuseful, ⃝ Useful, ⃝ Very +Useful +(c) Do you have any comments about Tidy Code? For instance: How did you feel it affected your learning? Is there any way +you would modify it to make it more useful? + +> +sketch.js +1 +const +canvasSize; +2 +function setup() ( +4 +createCanvas(400, 600) +5 +6 +function draw() ( +7 +8 +background("pink"); +9 +if (true ==.false) ( +10 +console.log("WAT?!?") +11 +Missing semicolon. +12p5 +cs111 +File +Edit +Help +Tidy Code ++M +Find ++F +> +sketch.js‘ +Find Next ++# +1 +Find Previous +↑+#+G +2 +3 +// bad style +4 +function setup( ( createCanvas(400, 600); } +5 +function draw() + background("pink" +）；}A Study of Editor Features in a Creative Coding Classroom +CHI ’23, April 23–28, 2023, Hamburg, Germany +(6) Feature: Auto-Refresh +(a) How often did you use Auto-Refresh? ⃝ Never, ⃝ Once in a while, ⃝ Occasionally, ⃝ Frequently, ⃝ All the time +(b) Do you think Auto-Refresh is useful? ⃝ Not very useful, ⃝ Not useful, ⃝ Neither useful nor unuseful, ⃝ Useful, ⃝ Very +Useful +(c) Do you have any comments about Auto-Refresh? For instance: How did you feel it affected your learning? Is there any +way you would modify it to make it more useful? +(7) Feature: Shape Toolbox +(a) How often did you use Shape Toolbox? ⃝ Never, ⃝ Once in a while, ⃝ Occasionally, ⃝ Frequently, ⃝ All the time +(b) Do you think Shape Toolbox is useful? ⃝ Not very useful, ⃝ Not useful, ⃝ Neither useful nor unuseful, ⃝ Useful, ⃝ Very +Useful +(c) Do you have any comments about Shape Toolbox? For instance: How did you feel it affected your learning? Is there any +way you would modify it to make it more useful? +(8) Feature: Autocomplete +(a) How often did you use Autocomplete? ⃝ Never, ⃝ Once in a while, ⃝ Occasionally, ⃝ Frequently, ⃝ All the time +(b) Do you think Autocomplete is useful? ⃝ Not very useful, ⃝ Not useful, ⃝ Neither useful nor unuseful, ⃝ Useful, ⃝ Very +Useful +(c) Do you have any comments about Autocomplete? For instance: How did you feel it affected your learning? Is there any +way you would modify it to make it more useful? + +OnSketch name +SUBMIT +Shape toolbox +sketch.js' +Canvas +1 v +function setup() ( +2 +createCanvas(400, 400); +3 +4 +5 v +function draw() +6 +background("pink"); +7v +Editor.shapeToolbox(() => ( +8 +rect(50， 71， 225， 95); +9 +})open ; +10 +reset +Hide Color Pickers Show Number Pickers +save +Consolefunction draw() ( +background("pink"); +if +if + statement +if else statement +if else-if else statement +SHIFTCHI ’23, April 23–28, 2023, Hamburg, Germany +McNutt et al. +D.3 +Page: Reflection Questions +Next, we’d like you to reflect on a few additional aspects of the course, how they might be improved, and ways in which the editor might be +modified to meet those challenges. +(1) Feature Review Reflect on the features we’ve just been considering. Which of the features we considered are you most excited +about? +• Color Pickers Very Disinterested ⃝, Disinterested ⃝, Neutral ⃝, Interested ⃝, Very Interested ⃝ +• Number Pickers Very Disinterested ⃝, Disinterested ⃝, Neutral ⃝, Interested ⃝, Very Interested ⃝ +• Number Sliders Very Disinterested ⃝, Disinterested ⃝, Neutral ⃝, Interested ⃝, Very Interested ⃝ +• Linting Very Disinterested ⃝, Disinterested ⃝, Neutral ⃝, Interested ⃝, Very Interested ⃝ +• Tidy Code Very Disinterested ⃝, Disinterested ⃝, Neutral ⃝, Interested ⃝, Very Interested ⃝ +• Auto-Refresh Very Disinterested ⃝, Disinterested ⃝, Neutral ⃝, Interested ⃝, Very Interested ⃝ +• Shape Toolbox Very Disinterested ⃝, Disinterested ⃝, Neutral ⃝, Interested ⃝, Very Interested ⃝ +• Autocomplete Very Disinterested ⃝, Disinterested ⃝, Neutral ⃝, Interested ⃝, Very Interested ⃝ +(2) Effect on Learning While it is hard to compare with something you didn’t do, how do you think your experience in the course +would have been different had these features not been part of the editor? Do you feel you would have learned more or less than you +did? +(3) Art Tools Now that you’ve learned some programming for creative coding, how does that affect your perspective of art making? +How might a code editor help or hinder the art making process? +(4) Challenges What aspect of coding or learning to program gave you the most trouble? As a way to help organize your thinking, +consider the assignment that you had the most difficulty with. Could the editor have done anything to help you with that? +(5) External Tools It’s natural to use other tools as part of the programming process, such as color eye droppers or p5’s online +documentation. Do you think it would be useful to integrate these tools as part of the editor? What other tools can you imagine +wanting to be part of your in-editor coding workflow? +(6) Desired Features What sorts of editor features might have allowed you to be more effective in your coding? What sorts of editor +features might have allowed you to be more creative? +(7) Miscellaneous Is there anything else you would like us to know? Any additional feedback you’d like to share about the editor, or +any other technical aspect of the course? + diff --git a/M9FQT4oBgHgl3EQfVza8/content/tmp_files/load_file.txt b/M9FQT4oBgHgl3EQfVza8/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a4c5653b2facd9023a57d76a51f99719ea6f9e0f --- /dev/null +++ b/M9FQT4oBgHgl3EQfVza8/content/tmp_files/load_file.txt @@ -0,0 +1,2923 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf,len=2922 +page_content='A Study of Editor Features in a Creative Coding Classroom Andrew McNutt University of Chicago Chicago,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' IL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' USA Anton Outkine University of Chicago Chicago,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' IL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' USA Ravi Chugh University of Chicago Chicago,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' IL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' USA Figure 1: This modified p5 editor (dubbed p5/y2) was used in a creative coding course to study how students use and perceive various editor features including standard ones,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' such as linting and auto-formatting (“Tidy Code”),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' as well as more experimen- tal features,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' such as live reloading (“Auto-refresh”) and a toolbox for bidirectionally manipulating shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' ABSTRACT Creative coding is a rapidly expanding domain for both artistic expression and computational education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Numerous libraries and IDEs support creative coding, however there has been little con- sideration of how the environments themselves might be designed to serve these twin goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' To investigate this gap, we implemented and used an experimental editor to teach a sequence of college and high-school creative coding courses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' In the first year, we con- ducted a log analysis of student work (𝑛=39) and surveys regarding prospective features (𝑛=25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' These guided our implementation of common enhancements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' color pickers) as well as uncommon ones (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' bidirectional shape editing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' In the second year, we stud- ied the effects of these features through logging (𝑛=39+) and survey (𝑛=23) studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Reflecting on the results, we identify opportunities to improve creativity- and novice-focused IDEs and highlight ten- sions in their design—as in tools that augment artistry or efficiency but may be perceived as hindering learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Copyrights for components of this work owned by others than the author(s) must be honored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Abstracting with credit is permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' To copy otherwise, or republish, to post on servers or to redistribute to 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='$15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='00 https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='1145/3544548.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='3580683 CCS CONCEPTS Human-centered computing → Human computer interaction (HCI);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' • Software and its engineering → Integrated and visual development environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' KEYWORDS Creative coding, Code editors, p5, Introductory programming ACM Reference Format: Andrew McNutt, Anton Outkine, and Ravi Chugh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' A Study of Editor Features in a Creative Coding Classroom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' In Proceedings of the 2023 CHI Conference on Human Factors in Computing Systems (CHI ’23), April 23– 28, 2023, Hamburg, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' ACM, New York, NY, USA, 42 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' https: //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='1145/3544548.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='3580683 1 INTRODUCTION Creative coding is a rapidly expanding computational domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' It generally refers to programming work that “blur(s) the distinc- tion between art and design and science and engineering” [66], encompassing pursuits such as generative art, embedded comput- ing, audio editing, performative live programming, and countless others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Many libraries and languages have arisen to support this programming genre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Some are tuned to domain-specific purposes— such as Orca [55] or Tracery [25] which support creating procedural music and Twitter bots, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Others simplify the process of many common artistic tasks (such as drawing and interactivity) without specializing in a specific area—as in openFrameworks [68] or Processing [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Among these general-purpose tools, those in arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='13302v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='HC] 30 Jan 2023 File v Helpv Edit Menu Containing Tidy Code C carbemo Auto-refresh Toggle Sketch Files sketchjs Number Pickers index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='html 58 let t = -0+;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='js 59 v function draw() ( B style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='css 60 Editor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='shapeToolbox() open Number Slider 61v if (keyCode =--- -70+) ( Shape 62 noStroke();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Toolbox Color Picker 63 fill("black");' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 64 10 circle(mouseX, _mouseY,Editor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='slider(30,200,80) I Missing semicolon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Linting fill("honeydew" 67 v for (let idx Pink Purple 68 const point1 = +++++++ 69 const newAcc = 1 vOrange Yellow 0000 70 Green 0000000000000000000 71 v for (let jdx 72 v if (idx jcBlue 00000000000000 73 continue;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Brown White 74 Gray Hide Color Pickers Hide Number Pickers Close Convert to hex and close Console 16 16 Show/Hide Widgets 91 Controls to toggle Color Pickers 16 and Number Pickers 91 53CHI ’23, April 23–28, 2023, Hamburg, Germany McNutt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' sp21 Log Study 26 13 27+ Students 31 26 27 Survey Study 16 9 19 Study Overlap 14 7 19 wi22 12 12 4 4 su22 su21 + Auto-refresh Improvements + Autocomplete + Number Scrubbers + Color Pickers + Shape Toolbox + GitHub Classroom Integration + Open-Code URLs Unused Features Students refers to those who completed the course,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' as some students dropped or withdrew p5/y2 p5 p5/y1 p5/y1 Year 1 Year 2 Figure 2: We modified the p5 editor before each year of a creative coding course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' We conducted studies to observe stu- dent usage and perception of existing and modified features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' the Processing family—such as Processing itself and p5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='js [1]—are particularly well known, having attracted large and active commu- nities, exemplified by the prevalence of artist- and novice-focused educational media, like the Coding Train [92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Beyond the potential for creative or artistic expression, this genre of work has long been embraced as means by which to teach intro- ductory programming [42, 66, 83, 105]—an approach often referred to as media computation [40] within CS departments—as it may be easier for students to engage with material that interests them [8], and creative or artistic tasks may be more engaging to students [70] not invested in the more common CS Ed topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Greenberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' [37] argue that creative coding-based introductions to computer science are more appealing to women, and create a more inclusive environment than traditional introductory CS curricula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Despite the potential utility for both artistic expression and learning to code, there has been relatively little consideration of how to enhance creative coding environments to facilitate these goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Following a trend exemplified by the development environment bundled with the Processing library, a number of creative coding toolchains come with their own environments, which are often tailored specifically for artists in their domain, as in Orca [55] or Tweakable [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' For example, the p5 editor—a browser-based editor maintained by the p5 community that acts as a gateway to coding for often non-technical users—is intentionally simple, and has limited “features and frills” to make it easier to jump right into coding [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' By definition, however, this guiding principle forgoes potential benefits of many standard IDE features (such as autocomplete), standard GUI features (such as color pickers), and more experimental features explored in research communities (such as bidirectional editing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' To shed light on the gap between creative coding tools and their goals for users, this work considers the following questions: How might we refine and enhance standard tools to extend the creative reach of novices?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' What sorts of features do novices perceive to be beneficial in a creative coding environment?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' We consider these ques- tions in the setting of the p5 editor because of its ubiquity [66] in creative coding contexts, as well as for its simple and mostly standard form, which may inform the design of enhancements to more general-purpose programming tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Paper Summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' We conducted a series of studies that considered how students use a modified version of the p5 code editor in an introductory programming and creative coding course at a private research university in the US.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2 displays an overview of our work, involving four course offerings spanning two academic years taught to college students (sp21, wi22) and to high-school students (su21, su22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' While our studies are situated in a classroom, our work is not about pedagogy per se—rather, we focus on understanding the needs and perceptions of creative coding novices as exhibited across the length of a full programming course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' We used a modified version of the p5 editor (referred to as p5/y1) in the first-year courses (sp21 and su21) and ran two studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' The first was a log analysis based on capturing code executions during the course (𝑛=39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' The second was a long-form survey that sought to understand student opinions and expectations about existing and hypothetical editor features (𝑛=25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' The results of the first-year studies revealed opportunities to improve existing editor features as well as interest in several hypo- thetical features—including direct manipulation widgets for modi- fying colors and a bidirectional shape drawing system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Thus, we further augmented the p5 editor (p5/y2) in the second-year offer- ings (wi22 and su22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Analogous to the studies in the first year, we monitored student behavior (𝑛=39+) through an anonymized track- ing scheme,1 and solicited their opinions through an abbreviated version of our previous survey (𝑛=23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' We identify several key themes based on the results of the studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' T1 Simple static analysis seen as supportive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Tools supporting basic automated formatting and analysis—such as code “tidying” or linting—are well received by our novices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' However, impolite [104] designs (which are those that do not respect user agency or act in an otherwise irritating manner) can lead to frustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' T2 Overeager evaluation can overwhelm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Live programming can give immediate feedback on code changes—potentially ben- eficial for tightening art-making cycles—but it often does so too quickly or in an irritating manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' T3 Students appreciate avoiding clutter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Like all programmers, novice creative coders are sensitive to inherent tradeoffs between minimal and feature-rich coding environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' T4 Useful features may be “too useful.” Students were recep- tive to integrating art-specific and other sophisticated tools into their programming environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Yet such features can inspire skepticism—even by novices—about their effect on learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Next, we situate our study within related work (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2), and then we describe our creative coding course (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' After describing our methodology (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 4), we analyze the results and consider our primary themes (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Through these studies we identify design implications for subsequent creativity- and novice-focused IDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2 RELATED WORK This paper investigates how to integrate advanced editor techniques into tools focused on novices and creative purposes based on ob- servation and analysis of novice programmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Given the broad range of related works, we frame the discussion around our primary design decisions: to start with the p5 editor and its existing feature set (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='1), to add a suite of more advanced features (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='2), and to evaluate these adaptations in a classroom setting (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 1 We did not collect user identifiers in the Year 2 log study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Thus, “39+” indicates that the log data includes students who did not complete the course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' A Study of Editor Features in a Creative Coding Classroom CHI ’23, April 23–28, 2023, Hamburg, Germany 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='1 Creative Coding Environments Creative coding is a multifarious category of work encompassing diverse approaches and topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' One common element is the use of editing environments that have been customized to address the particular domain of consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' One prominent example is p5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='js [1] which (like its predecessor Processing [87]) can be used as a standalone library, but is made substantially more approachable by novices though the availability of a simple development environment specific to doing work with that library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' The p5 editor [3] simulates a simple web server in the browser by combining each of the files in a “sketch” (synonymous with program in this context) and executing them as a standalone web page in an isolated component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' The existing feature set in this editor is a particularly intriguing object for study for several reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' First, it is lightweight, web-based, and supports cloud-based saves and shares—a good fit for an introductory programming class, as it does not have potentially intimidating baggage of a heavyweight IDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Second, the text editor (based on CodeMirror [45]) supports a number of contemporary IDE features—such as linting [57] and auto-formatting [15]—that make our findings potentially general- izable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Furthermore, it contains a live-reloading (“auto-refresh”) feature being actively researched in programming-language user- interface communities [88, 90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' By considering (versions of) the existing, relatively standard p5 editor, our formative study aimed to understand which features were important before pursuing more drastic changes within the scope of this work and beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' In addition to these more general editors, there are a variety of tools that focus on more limited domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' For instance, Shader- toy [84] provides a browser-based editor for creating and sharing shaders and prominently features procedural and generative visual art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Like creative coding in general, these editing environments are not limited to the graphical domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Tweakable [5] and Orca [55] provide environments for creating programmatically generated music, based on node-and-wire composition and 2D livecoding, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' HappyBrackets [34] more closely reflects our approach to enhancing creativity by augmenting a standard IDE, although it is centered on using IoT devices for musical composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Make- Code [11] has an editing environment that contains synchronized block and text representations of code with a focus on creating games for microcontroller-based devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Most similar to our work, p5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='fab [95] modifies the p5 editor to support digital fabrication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Mitchell and Bown [77] studied the needs of creative coders through a lab-based study, highlighting the value of visualizing program state, supporting best practices and short iteration cycles, and assisting exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Our findings are closely related to theirs, but located within a classroom and conducted on a longer time- scale—following Frich et al.’s [35] call for more studies to evaluate extant tools in their in-vivo usage context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' This scale and scope informs our different, but complementary, set of themes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Creative coding IDEs, and other such tools, target users at an intriguing intersection: many are relatively inexperienced but are strongly motivated to use these systems effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Lessons learned from studying users of these systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' students in a creative coding classroom) may translate to other venues with non-technical high-engagement users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Such populations occur widely and include spreadsheet users [14], tinkerers [17, 21], and artists more generally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='2 Advanced Editor Features Many works have experimented with new ways to augment con- ventional text-based code editors with more interactive capabil- ities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Among the plethora of such features, we chose several to consider in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' In the first year of our study, students had access to a live-reloading feature in the existing p5 editor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' In the second year, we implemented domain-specific graphical widgets (namely, color picker and number sliders), and bidirectional shape drawing—among many other advanced features being actively researched—because they are closely aligned with the concerns of creative coding, were well-received in the first-year survey, and were feasible to implement given finite resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' We discuss these features below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Tanimoto [96, 97] describes programming affordances on a live- ness spectrum, relating to the degree of agency that users express in the execution of code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' These range from the familiar edit-run cycle to predictive execution, with the always-executing style of live-reloading in the p5 editor falling in the middle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Immediate feed- back evidently has rich educational utility, as in Python Tutor [39], and the sprawling number of systems its design informs [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Omni- code [58] takes a “Display all the values” approach to help novices understand and debug code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' They find that the always-on strategy is useful for these purposes, which agrees with Kramer et al.’s [63] findings that live programming helps users fix bugs more quickly than a traditional edit-run cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' [52] also found that live programming helps students perform some tasks more quickly, but in their study learning outcomes remained unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Augmenting text with graphical representations can provide a more natural way to specify code than textual input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' The complexi- ties of these vary from simple inline widgets, such as sliders and color pickers, to more complex designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Graphite [81] explores a notion of palettes which allow for domain-specific editors, such as for color and regular expressions, surfaced through autocomplete- style menus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Barista [61] integrates interactive structured visual representations inline with code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Andersen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' [9] build on this premise by formalizing how GUIs might be integrated directly into Racket code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Several features we added to the p5 editor follow the hybrid textual-plus-visual approach found in these works, targeting our specific domain and audience, novice creative coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Some systems bidirectionally synchronize code and GUI manip- ulations: changes made to either the source text or corresponding graphical output are reflected in the other [4, 47, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' While this has been most prominently used to create parametric drawings [44, 49], both ours and previous works suggest potential value for novices as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' For instance, Hundhausen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' [53] find that this type of bidirectional development has educational utility and promotes skill transfer to text-based languages and environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Contrast- ingly, Do et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' [29] utilize a mixed text-and-direct manipulation approach to teach an Hour of Code course to 5th and 6th graders using a JavaScript-like language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Yet, they did not find as rich an educational benefit, but argue that further development is neces- sary to situate this UI paradigm in creative-educational contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Our results tentatively suggest that this approach can be useful—in terms of student usage and perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' However, further study is needed to understand the effect it has on learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' CHI ’23, April 23–28, 2023, Hamburg, Germany McNutt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='3 Classroom Studies Computer science education researchers have studied the poten- tial benefits—regarding gender diversity, retention, and learning outcomes—of emphasizing computing with media in introductory programming courses [41, 42, 94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Our work provides a step to- ward understanding the role that programming tools—as opposed to curricular design—might play in creative-educational settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Despite this work taking place in a classroom, however, our aims in this paper are not focused on measuring the pedagogical impact of individual editor features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Instead, we pursue an understand- ing of the needs and perceptions of novice creative coders, and our classroom setting allows us to engage with such users on the time scale of an introductory programming course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' As Weintrop and Wilensky [103] argue, “it is critical that we conduct studies .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' analyzing tools not from the perspective of those who have al- ready mastered the content, but instead from the perspective of the learners who the tools is designed for.” Our work embraces this approach, deriving guidelines for IDE design based on perceived utility of different features “to better inform educators on how to best utilize them in their classrooms .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' [and] provide a roadmap for the improvement of these tools moving forward” [103].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' While we have specifically opted for a text-based environment, block-based environments are notable for their frequent use with younger learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' In a study with high school students in a formal classroom setting, Weintrop and Wilensky [103] found that students (i) considered it easier to read programs as blocks rather than as text, (ii) liked the visual cues offered by blocks (though this preference diminished over time), (iii) found blocks easier to compose (via drag-and-drop), and (iv) liked how the interface organized blocks into related functionality and helped serve as memory aid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Their study also identified several perceived limitations of blocks: that they are potentially less powerful, slower and more verbose, and inauthentic—in the sense of not “doing the same kinds of things they will do in ‘real life’ outside of the environment in which learning takes place” [91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Notably, even though novice high school students appreciate the pedagogic value of blocks, they still perceive them as inauthentic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' This comports with our findings about skepticism about unfamiliar or advanced features among novices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' A variety of works have employed similar logging studies to ours (often referred to as learning analytics [56]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Some such works use in-course logging studies as a means by which to analyze student progression through assignments, which are described in multiple surveys [54, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Helminen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' [46] used a similar environment as our own to understand the types of errors students encountered in an introductory Python course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Vihavainen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' [101] conduct a key-level logging study of novice coding behavior, although they seek to understand student behavior rather than IDE design for novices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Our work is related to these, but we are less interested in understanding issues like student progress through assignments than the hindrances they encounter in the UI generally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 3 COURSE DESCRIPTION Our course aimed to teach basic computing skills (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' variables, iteration, and function decomposition) to students with little-to-no- programming experience in the context of creative coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Learn- ing to program typically also requires learning many surrounding skills, such as facility with command-line interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' To eliminate such possibly intimidating setup difficulties—and allow tighter in- tegration between our web-based instructional texts and the venue where work was to be done—we decided to centralize all student work within the online p5 editor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Following common practices in creative coding courses [66] and tutorials (such as from Khan Academy [6] and Happy Coding [106]), we used JavaScript and the p5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='js library as the primary learning mediums, although the basics of web programming with HTML and CSS were also introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' p5 exposes a variety of drawing and interaction methods as primitive functions (such as rect and circle) which the programmer combines either to make static or dynamic compositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' While p5 can be used to fully manipulate the native DOM, the majority of coding occurs inside a simplified environment focused on HTML-canvas manipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' We taught four editions of the course, referred to chronologically as sp21, su21, wi22, and su22 (summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' The sp21 and wi22 editions were “full” 10-week college courses, offered from within a computer science department but cross-listed with media arts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Students had broad academic interests: more than 20 different degree programs were represented by the 58 students (see the ap- pendix for a breakdown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' The su21 and su22 editions were intensive 3-week versions taught over the summer to high school students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' A fifth version of the course was taught during the summer of 2021, but was dropped from our analysis because participation was too small to meaningfully analyze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Required coursework consisted of graded individual homeworks, collected but ungraded exercises, and, in the full editions, an individual self-designed project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Taking into account differences in assignments and course material, su21 and su22 were roughly two-thirds of sp21 or wi22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Lectures in sp21 and su21 were delivered remotely over Zoom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' The first three weeks of wi22 were also taught remotely, with the remaining weeks con- ducted in a hybrid format (during which students more often joined via Zoom than in person).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' The su22 edition was taught entirely in person and—with more in-class time (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 3)—included more required group work on practice exercises than other editions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' While the course was designed for those with limited experience, we observed high levels of self-reported prior experience in each edition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' These varying levels color some of our observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Based on our experience teaching them, the high school students in su21 may have been over-confident in their description of their prior experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' However, those in su22 did seem to have non-trivial Students Self-reported Experience Sessions 23 Session Period 10weeks Session Length 50 min including 1-2 short breaks not including lunch break sp21 Edition 13 3 150 su21 26 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='6% total 96 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='8% 23 10 50 wi22 27 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='1% 13 3 270 su22 12 100% 31 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='5% Course Details Student Details including an auditor who finished the course live coding lectures Figure 3: Course details by edition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Experience was found by a pre-course survey that asked “How much programming ex- perience do you have?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' We coded answers into no experience, some (having taken less than a college-level course), or high otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' We merge the latter two levels here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' A Study of Editor Features in a Creative Coding Classroom CHI ’23, April 23–28, 2023, Hamburg, Germany Submissions from wi22 students who opted-in for public release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Figure 4: One assignment in each course involved designing a tree, which exhibits horizontal axial symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' prior experience, perhaps due to a selection bias caused by the course being offered in-person at our university.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Despite higher self-reported prior experience than we expected, in our experience teaching we found that this experience did not necessarily lead to overwhelming mastery of the basic introductory material covered in the course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Therefore, we believe it is fair to view our students, as a group, to be novices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' The progression of assignments was designed to employ funda- mental programming concepts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' variables, function abstraction and decomposition, loops, arrays, and objects) for various media computation tasks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' vector graphics drawing, animation, image manipulation, and basic web development).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Aiming to serve the twin goals of teaching programming and fostering its use for cre- ative expression, most assignments were open-ended (as opposed to being prescribed with easily-testable specifications).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' For exam- ple, one early assignment asked students to make judicious use of variables and arithmetic expressions to implement a symmetric tree drawing of their own design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 4 shows a sample of student submissions from wi22 for this tree assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Additional assign- ments are described in the appendix, and the full course materials are available online at cs111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 4 METHODS We now describe the studies that ran alongside each offering of our creative coding course and provide summary statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' See appen- dix for survey instruments, ethics statement, and other materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='1 Year 1: Editions sp21 and su21 with p5/y1 In the first year of our two-year formative study, we deployed the p5 editor mostly as is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' We added a couple features to support course logistics, but we did not add any new programming affordances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='1 Custom Features in p5/y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Our initial fork added two fea- tures in support of teaching the class online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' The first enabled students to submit assignments from within the editor to GitHub repositories as pull requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Course staff then provided feedback and grading on these pull requests, merging them once complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' The second mechanism allowed students to click any code exam- ple in the online course materials to open the code directly in the editor (without intervening copy-pastes or file-saves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' We removed features which were either not relevant to the class or would have negatively affected the course design (such as project sharing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='2 (Per-User) Log Study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' We ran a study in the first two course offerings (sp21 and su21) to collect information about the coding behavior of students who opted-in to participate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' For these students, we captured the state of each sketch on every execution, save, submission, and structure edit (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' find and replace) throughout the course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Logs were sent to a cloud-based server which only recorded events generated by study participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' This ensured that all students experienced the same level of network traffic regardless of study involvement and thus did not penalize participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Student consent (and parental consent for students under 18) was sought prior to the course as part of a pre-course on-boarding process, which was also used to gather GitHub identification for submitting assignments and gauge prior experience levels (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Students were not compensated for their involvement in the log study as participation did not modify the course experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Stu- dents were able to retract consent at any time during the course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Logs were not analyzed during the course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Although relatively coarse-grained, the logged events capture overall trends and pat- terns in the use of basic editor features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' In contrast, a key-level log study (as in Vihavainen et al.’s [101] study of novices’ first weeks with an IDE) might have enabled more detailed observations at in- creased cost, both in terms of data collection and analysis, without clearly supporting our research questions about feature usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' During this study we collected ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='5 million logged actions spread across ∼5500 sessions, which we define as periods of interactivity with ≤15 minutes between any two actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' On average sessions lasted 𝜇=23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='3 minutes with a standard deviation of 𝜎=38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='9 minutes (listed as 𝜇±𝜎 hereafter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Our analysis of error frequency did not shed light on our research questions, but we provide summary statistics about observed run-time errors in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='3 (Long-Format) Feature Survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' After both sp21 and su21, stu- dents were invited to take part in an online survey soliciting their experiences using p5/y1 and opinions about various features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' They were asked about a series of features (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 6), each presented as a static image with a paragraph of descriptive text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' The feature progression was bookended by free-text questions on more general topics, such as debugging and code organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' A total of 25 sp21 and su21 students participated in the survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Participation in the log study was not a prerequisite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Our survey tool did not report the working time, but based on piloting, we believe that the survey took 20-40 minutes to complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Participants were paid $30 for completing the survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Demographic data was not collected beyond a self-reported experience level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' For each feature, the survey asked both free-text questions and Likert-item style rating questions (how “Useful” is the feature, and how “Often” would they use it).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' The number of surveyed features (17) was rather high, and we did not randomize their order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' So, to help calibrate ratings, at the end of the progression a table summa- rizing the features asked for additional Likert-item style numerical ratings (how “Interested” they were in each feature).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' We found CHI ’23, April 23–28, 2023, Hamburg, Germany McNutt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Figure 5: The Shape Toolbox is used to create simple compo- sitions of shape-drawing code through simple GUI actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (1)-(4): Interaction workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (5): Frequency of shape usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' slightly negative correlations (Spearman’s r) with presentation or- der and rating: Useful: 𝑟=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='117, Interested: 𝑟=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='118, and Often: 𝑟=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='133 with 𝑝≤0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' These metrics exhibited good agreement: 𝑟=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='817 (Useful/Often), 𝑟=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='635 (Useful/Interested), 𝑟=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='655 (Of- ten/Interested) with 𝑝<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Thus calibrated, we focus only on perceived Usefulness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' This selection is further informed by the technology acceptance model [27], which suggests that users’ per- ceived usefulness is indicative of subsequent usage, and is thus a more helpful quality than estimated Interest or frequency of use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' We found that only in-context docs was statistically significantly (𝑟=- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='308, 𝑝<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='01) correlated with self-reported experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' This slightly negative correlation is also reflected in the qualitative comments about that feature (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 9 and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' We included a mixture of features for general computation (such as “Interactive Value Inspector” and “Linked Copy-and-Paste”) as well as creative coding (such as “Coding by Drawing Tools” and “Canvas Ruler”) that would be understandable based on their ex- perience in the course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' The complete list of surveyed features is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 6 and described in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Other features, such as notebook-style programming or multi-canvas editors (such as Stamper [20]) were considered, but not included—we believed that textual descriptions or static renderings would be unlikely to give effective motivation for their utility, and students may incorrectly forecast their experience of such unfamiliar features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' A key limi- tation of our survey is the simple static presentation of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Participant perceptions might have differed if they had watched video demonstrations or been able to experiment with the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='2 Year 2: Editions wi22 and su22 with p5/y2 After gleaning student predilections in our formative Year 1 studies, we modified our editor to investigate these stated preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' We used p5/y2 in wi22 and su22, during which we ran two more studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Whereas we customized p5/y1 to improve course logistics, our changes in p5/y2 were motivated by the first year results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='1 Custom Features in p5/y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' We implemented the top-3 unim- plemented features from the survey (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 6 or Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 13 in the appendix): Color Pickers, Autocomplete, and Shape Toolbox (which was a synthesis of Coding by Drawing Tools and Directly Manip- ulate Shapes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Given limited resources, we forwent the Canvas Ruler (the next most-preferred feature) because there is a sim- ple workaround for identifying positions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' console logging the mouseX and mouseY on mouse movement), although we intend to address it in future editions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Several highly-rated features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Time Travel Slider, p5 State Displays, and Interactive Value Inspector) were more speculative and thus deemed beyond the scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' We made two further modifications based on observations, namely, adjusting Auto-refresh and adding Number Sliders (which are common in interactive documents [99]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Two of these new features are accessed by calling special func- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Editor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='slider(min, max, value) renders a Number Slider ( ) for value in the range from min to max (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 1), with an optional fourth step argument to override the default continuous- dragging behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' One alternative design is to store the metadata (range and step) in special comments, for example, as in Juxta- pose [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Such an approach warrants comparison to our chosen design in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' However, we elected to use the function call API to mimic a common p5 function for creating dynamic sliders, which students already learned (createSlider [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' The most novel feature introduced in p5/y2, the Shape Tool- box, is also accessed through a function call-based workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' The user calls Editor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='shapeToolbox() and clicks a button to open the shape-drawing GUI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' The text area is then disabled and a simple drawing toolbox is overlaid atop the output window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Using the Toolbox, users create compositions in the output pane by adding, translating, rotating, and scaling shapes (including primitive shapes and Bezier segments) with direct manipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Once satisfied, users click “save” to update the code—shape commands are called in the body of a function passed to Editor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='shapeToolbox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 5 depicts this workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Translation back to the code is achieved through a simple template matching method allowed by a one-to-one map- ping between drawn elements and lines of code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' We decided against always displaying the shape-drawing GUI for two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' First, not all calls to shape drawing functions can have GUIs—this is the subject of research on bidirectional editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' At one of the end the spectrum is a very simple approach that maintains a top-level “scratchpad” function (where all new shape-drawing calls would be added), and at the other end are heavyweight and expressive techniques in prior work [44, 47, 49]—the former would be more restrictive than our chosen approach, and the latter beyond the scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Second, shape-drawing is only one aspect of our creative coding tasks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' our approach displays the GUI only when the user explicitly opts to use it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='2 (Aggregate-Use) Log Study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' We collected anonymized us- age of p5/y2 features during wi22 and su22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Logs were collected through a customized version of Umami [22], a self-hosted privacy- minded tracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' We elected not to capture full-sketch snapshots in this study because our planned analyses for the second year did not require them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' We believed lighter weight instrumentation would allow us to capture more fine-grained usage patterns, such as the length of sessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Finally, this reconfiguration to full anonymity gave us leeway to collect data on all course participants, rather than just those who opted-in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' function draw() ( background("pink");' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' SUBMIT 3 Compose changes 5 Editl through direct Canvas Editor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='shapeToolbox manipulation Editor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='slider Type the shapeToolbox command unctior background("pink");' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Editor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='shapeToolbox(() => ( 8 rect(83, 27, 64, 64);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' ellipse(177, 172, 103, 103);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 9 10 rect(213, 38, 70, 64);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 11 bezier(63, 240, 49, 335, 187， 370, 338, 245);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' })open ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 12 13 7 function draw() ( background"pink");' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' reset Editor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='shapeToolbox() open save Click the open s 7 widgetA Study of Editor Features in a Creative Coding Classroom CHI ’23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' April 23–28,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2023,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Hamburg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Germany ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Canvas Ruler ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Time Travel Slider ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='In-context Docs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='p5 State Displays ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Interactive Value Inspector ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Linked Copy-and-Paste ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Code Snippet Templates ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Drag-and-Drop Refactoring ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Very ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='useful ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Useful ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Neither useful ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='nor unuseful ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Hypothetical Features ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Year 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Year 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Survey ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Responses ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Implemented Features ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='p5/y1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='p5/y2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Very ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='useful ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Useful ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Neither useful ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='nor unuseful ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Coding by ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Drawing Tools ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Directly ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Manipulate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Shapes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='These Y1 features ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='were merged in Y2 as ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Shape Toolbox ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='mean ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='standard ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Linters ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Color Picker ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Tidy Code ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Autocomplete ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Shape Toolbox ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Auto-refresh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Code Folding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Number Sliders ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Number Picker* ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='† was present in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' but not asked about in the Y2 survey not asked about in the Y1 surveys p5/y2 Figure 6: The features surveyed across both years and how “Useful” they were deemed to be on a 5-point Likert scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Through this process we collected ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='2 million events across ∼6730 sessions (defined as before).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Due to a configuration error (present only in wi22), events were not collected with unique ses- sion identifiers, although we were able to reconstruct 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='4% of the sessions—the remainder are excluded from analyses requiring spe- cific session information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' While the incomplete data is unfortunate, it still provides a more detailed picture of activity than in Year 1, which saw 68% log study participation across sp21 and su21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Within this reduced sample, sessions lasted 𝜇=24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='1±38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='0 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='3 (Short-Format) Feature Survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Near the end of wi22, students were invited to take an abbreviated version of the Year 1 survey (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='3), containing only features that were added or improved upon in p5/y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Following the structure of the previous survey, we asked about frequency of use and Usefulness for features one at a time, followed by a summary table asking about Interest and a suite of reflection questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Participants were compensated with extra credit roughly equivalent to 1% of the final course grade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' A total of 23 students participated in the survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Our survey provider did not measure time taken to respond, but based on pi- loting we believe that the survey took 10-15 minutes to complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Presentation order was not correlated with any of our metrics (𝑝=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='779-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='939).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Again, the ratings exhibited reasonable agreement: 𝑟=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='727 (Useful/Often), 𝑟=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='607 (Useful/Interested), 𝑟=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='693 (Of- ten/Interested) with 𝑝<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' As with Year 1, we focus only on Usefulness in the body of the text (see the appendix for the others).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' We found prior experience to be statistically significantly (𝑝<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='01) correlated with only a single feature, auto-refresh, for which there was a somewhat negative correlation (𝑟=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='487).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 5 ANALYSIS We now reflect on the features, connecting them to the themes summarized in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 1, denoted T1 through T4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' We consider features implemented in both p5/y1 and p5/y2 (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='1), followed by those added in p5/y2 (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='2), and then introduce concerns that cut across multiple features (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='3 and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Our analysis draws on data from the survey studies (summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 6) and the log studies as appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Participants from the sp21, su21, wi22, and su22 surveys are referred to as A1-16, B1-9, C1-19, and D1-4, respectively, and are colored by year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='1 Features in Both p5/y1 and p5/y2 We begin by considering features present in both editor versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='1 Linting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' This static analysis tool eagerly executes after small code edits, checking simple syntactic assertions akin to spell check for code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' It was well received in both years and was mostly seen as helpful, although sometimes impolite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Students found linting to be “very helpful” (A5,7,17,20, C10,16, D1, 2) and “very useful” (A19, C2,4,6,12,18, D4), because it “saves time and energy” (A4) and shows “where I needed to go to fix simple bugs” (A16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' C10 believed that debugging “would be way more annoying without it” because “it’s not always obvious what you did wrong” (D4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Whereas 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='0% of executions in Year 1 passed lint, in Year 2 (where we had visibility into all lint runs) code passed 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='5% of lint runs (which happened after most small text edits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' This may indicate that students address lint errors before running code as a simple integrity check, or that the analyses are executed too early;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' however, student comments seem to indicate the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Unlike other features, students were incentivized to attend to it, as the absence of lint errors was a small part of homework grades (98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='7% and 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='2% of submissions in Years 1 and 2, respectively, passed lint).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Beyond code style, linting can provide opportunities to expose novices to other best practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' For example, CSSLint [26] (used in p5/y2) explained that the * selector is considered bad practice because it is inefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Indeed, C3 felt that linting “trained me to think and type in a certain way”, and A5 observed that it could be “a nice way to point out when I am making stylistic errors (instead of [Tidy Code] just magically fixing all of them for me).” Utilizing this well-received channel for introducing programming features and practices is an opportunity for future IDE design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' T1 Participants also offered ideas to improve the feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Because the editor eagerly ran the linter, “the yellow line warning[s] often exist all the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' It annoys me” (B4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Instead, some students would have preferred not to see lint errors “until I finish typing” (A13) or “before finishing a line of code” (B5)—the mechanics of exactly when and how to display errors for incomplete code will require careful design (as considered, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' in Hazel [79, 80]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Others expressed a desire for more nuance—“acknowledging the difference between ‘This Must Be Changed To Have Nice Code™’ and ‘hey, maybe consider changing this thing!”’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (A5)—and control—being able to “ignore/exit out of a warning” (A3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Poorly-received default choices and persistent errors can repel users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' As an extreme example, one wi22 student decided to use Replit [89], rather than p5/y2, for their final project because too many (CSSLint) errors seemed irrelevant or unclear how to fix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Linters integrated with editors in this way do not offer mechanisms to override general advice or to indicate that the user knows what they are doing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' This is impolite computing [104]: it CHI ’23, April 23–28, 2023, Hamburg, Germany McNutt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' forgoes user agency and generally is perceived as a pest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Avoiding these pitfalls is important to leverage the instructive opportunities offered by the well-received, static analysis-informed tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' T1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='2 Tidy Code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Auto-formatters provide on-demand code restyling without semantic modification, and are common in professional cod- ing workflows [86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' We often encouraged the use of this tool—called Tidy Code in the p5 editor—in lectures, but we did not incentivize its usage in grading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' It was invoked manually (from the top menu bar or keyboard shortcut) rather than being executed on every save.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Like linting, this feature was generally well received.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Students found auto-formatting to be “super useful” (A15) and “very satis- fying” (A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' The formatting choices were not always appreciated, however.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Whereas C15 “only rarely preferred my own organization”, A12 felt the results “appeared less organized, such as having irregu- lar line breaks” and A10 “worried it would mess up my organization.” We observed that students in Year 1 often (needlessly) invoked auto-formatting twice in a row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' In particular there was a probability of 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='15% and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='65% (in sp21 and su21) of auto-formatted code being auto-formatted again right away—with similar behavior observed for saves (see appendix for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' This suggests that providing clear code-state signals (analogous to linting’s visual indicators) may reduce needless anxiety-motivated saves and tidyings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' The presence of this behavior in Year 1 suggests it was likely repeated in Year 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' however, the aforementioned configuration error prevented us from collecting auto-formatting usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' While simple indicators may seem to be trivial UI modifications, we suggest that it will impact the perception and understanding of such features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Several students would have liked the feature to be customizable, rather than enforcing a fixed set of “preferences that should not be forced by tidy code” (C16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Indeed, some students would have liked auto-formatting better “if it was a little configurable” (A16);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' for example, “if there [were] multiple common/standard rulesets there could be a way to choose which you want to follow” (A5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Further- more, it “would be helpful to be able to specif[y] which block of code to tidy” (A13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Thus, extending well-chosen defaults with ways to selectively customize style preferences—a notion which has been referred to as “code style sheets” [69]—could further increase the politeness of this feature and thus its utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' T1 Like the teachable moments offered by linting, B8 felt they “Learnt a lot about code organization using this feature!”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' As imple- mented, however, the results of auto-formatting are updated in the code box without explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Better would be for the editor to “show you what you are doing ‘incorrectly”’ (C19), for example, using visual highlights and annotations to explain the differences—which could also serve as scaffolding to introduce version control tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='3 Auto-refresh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' This feature re-executes code upon text edits— a workflow demonstrating “level-3 liveness” [96, 97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Auto-refresh was present in both p5/y1 (inherited from the original editor) and in p5/y2 (where it was modified).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' In principle, live feedback would seem particularly helpful in a creative coding context as programs are often updated with small graphical adjustments, and thus well matched with a short edit-run cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' It was also well matched with ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='our setting: the Normalized Programming State model [23] sug- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='gests that spending longer periods of time in syntactically unknown ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='states (such as when the code has not been executed in a while) is ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Year 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Year 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Fraction of sessions using auto-refresh by students ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='0% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='10% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='20% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='30% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='40% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='50% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='60% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='70% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='28 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='80% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='90% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='2 students used auto-refresh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='in 40-50% of their sessions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Figure 7: Histograms of the fraction of sessions where a stu- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='dent used auto-refresh any amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='7% and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='1% of stu- dents never used auto-refresh in Years 1 and 2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' negatively correlated with program success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' This needs to be bal- anced with the cognitive load [54] caused by repeated executions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Auto-refresh in p5/y1 did not achieve a fruitful balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' As indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 7, only a handful of participants regularly used it and most students used it rarely, if at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' The survey responses color this imbalance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Whereas A9 “used this all the time and loved it”, finding it “way easier than clicking the ‘play’ button all the time”, others felt that the keyboard hotkey was sufficient (A16, B1,8, D1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' More important than convenience were differing views on the fundamental interaction model itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' B7 appreciated the ability “to see what I was creating as I coded”, finding it useful even though “error messages that kept popping up got in the way a little”, while others found the errors “very distracting” (A5,6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Participants felt the feature was “running incomplete code unintentionally” (B6) and “when you don’t want it to” (B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Instead, some students felt robbed of their agency over their code, desiring “to be the boss of when my code reran” (A5) and in “control my own pace” (B2), only running the code when “I know I have something that I want to see” (A13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' This suggests that, while spending too much time in syntactically invalid states may be detrimental [23], spending too little time is also problematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Developing a careful understanding of the tradeoffs is an important avenue for future live programming work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' T2 For the purposes of this work, we made only simple changes to auto-refresh in p5/y2 based on our observations from the first year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' We increased the refresh delay from 400ms to 1s, and, more importantly, in the event that executed code had lint errors—a proxy for run-time errors—the editor did not refresh the canvas, instead indicating that it was “stale.” Thus, in the (many) cases when edits are incomplete or erroneous, the canvas remains visually stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' The modified auto-refresh was modestly better received, with its Usefulness increasing from 𝜇=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='1 to 𝜇=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' In addition, per Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 7, it was used more often—although we note that auto-refresh was demonstrated more at the beginning of wi22 and su22 than in prior editions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' A one-sided t-test indicates that students in Year 2 used auto-refresh significantly (𝑝<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='001) more often.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Yet, the overall bal- ance remained far from perfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Some participants were “stressed” at “all the errors that pop up as I implement new things” (C15) and “before I got to fix them” (C4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' These negative views seemed more likely to come from those with prior experience (𝑟=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='487, 𝑝<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='001), which may suggest that expectations are set by experience with A Study of Editor Features in a Creative Coding Classroom CHI ’23, April 23–28, 2023, Hamburg, Germany tools exhibiting a different execution cadence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Others, however, found it “very useful for certain exercises that needed lots of small ad- justments” (C3) and “very helpful when using trial and error” (C16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Overall, we observed no significant changes in user behavior after modifying auto-refresh, despite the improved perception of the feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' This again underscores that designing UIs to be polite (or at least not irritating) is critical to their usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='2 Features Only in p5/y2 Next, we consider the features that were added in p5/y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' While Year 2 survey responses are based on hands-on experience with the features in p5/y2, Year 1 responses are based on descriptions in the survey and experience with other tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Feature use in wi22 is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='1 Shape Toolbox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' The most significant addition to p5/y2 was the Shape Toolbox feature that allowed GUI-based specification of primitive shapes using direct manipulation which generated matching code (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' The constituent parts of this feature were highly perceived in Year 1: 𝜇=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='44 for Coding by Drawing Tools, and 𝜇=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='2±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='0 for Directly Manipulate Shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Some students believed it would be “very beginner friendly” (A3) and would make work “a lot easier and faster” (B7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Others believed it would also reduce errors (A9) and help with debugging (B6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Help programming curved shapes—such as the trees in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 4— was particularly enticing: “for bezier curves, changing the input values rarely produced an expected result” (A12), highlighting a gulf of execution [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' The process usually involved “lots of trial and error” (A3), sometimes resulting in student disengagement: “Coding the bezier curves manually turned me off of them, and I did not attempt them in my work” (A14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' However, that same student noted “If I had had a tool like this, I certainly would have used them.” Several students in Year 2 embraced the feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' For example, C18 found it “EXTREMELY helpful, especially when it came to draw- ing Bezier curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Every time I had to draw a curve, I used the shape toolbox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' I probably would have cried without it.” C10 mentioned that it “Was very nice to use it to get approximate coordinates then fine tune them after.” Although the feature was “very useful for beginner projects” (C2), several students, including C6, “used them less as time progressed.” Shape Toolbox was used often for the tree homework (see HW3 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 8), and use per execution by week was minimal after that assignment, being used in only 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='12% of all (available) sessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Perhaps because the feature did not have a stable visual presence (as with the auto-refresh button), some students “completely forgot this existed, but I think it would have been really really useful if I had remembered” (C4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' In addition, although we expected the feature to be used extensively for HW 2, in wi22 Editor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='shapeToolbox was announced but not demonstrated in class until after the assignment was released.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Bezier curves accounted for the majority of invoca- tions (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Toolbox sessions (from open to save) lasted 𝜇=22±30 seconds, indicating that it may have been used relatively often to make small graphical adjustments, as opposed to building larger compositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Students may have continued to use them later in the course “if it allowed for some of the shapes that are more complicated” (C16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Fur- ther limiting the utility of the feature, within an invocation shape Jan 09, 2022 Jan 17 Jan 25 Feb 01 Feb 09 Feb 17 Feb 25 Mar 05 Mar 13 0 2k 4k 6k 8k 10k 12k 14k Executions Per Day HW 1: Color Wheel HW 2: Freeze Frame HW 3: Trees HW 4: Book of Patterns HW 5: Deck of Cards HW 6: Snake HW 7: Wordle Project: Proposal HW 8: Blackout Poetry Project: Progress Report Project: Final 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='2k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='4k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='6k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='8k 1k 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='2k Feature Use Per Day HW 1: Color Wheel HW 2: Freeze Frame HW 3: Trees HW 4: Book of Patterns HW 5: Deck of Cards HW 6: Snake HW 7: Wordle Project: Proposal HW 8: Blackout Poetry Project: Progress Report Project: Final Auto Autocomplete Color Picker Manual Number Widgets Shape Toolbox Slider Figure 8: Feature use in wi22 was guided by course content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' For instance, autocomplete was demonstrated prior to HW3 and sliders were included in the starter code for HW5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' drawing functions allowed only literals—once a student wanted to use variables and arithmetic expressions, the Toolbox would no longer open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' “[C]reating an object without this feature would be bet- ter because of the precision” (B5) afforded by variables, expressions, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Thus, the feature ultimately fell short of what students imagined: 𝜇=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='8±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Bidirectional updates are being explored in a growing number of systems (as in Sketch-n-Sketch [49]), but there remain significant technical and UI design challenges to explore, before even consider- ing their value to novices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' As predicted by a couple students, more feature-rich bidirectional synchronization would need to reconcile ambiguous graphical interactions (“There are many parameters and it would be hard to make it so it manipulates them one at a time” (B5)) and their effect on other parts of the program (“My only but major concern would be that it doesn’t confuse the other lines of code, and that it may not run the way the programmer wants to use it” (B6)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Nevertheless, the experience suggests that even a simple imple- mentation of this very desirable feature was promising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' However, as we discuss in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='3, many students were skeptical about the effect of this feature on learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' T4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='2 Autocomplete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' We enabled a simple autocomplete menu [45] and populated it with p5-specific identifiers (variables and function names), syntax templates (common patterns, like for-loops with holes), and commands for invoking the Shape Toolbox and Number Sliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Passively supporting learning in this way would seem to be a natural fit for our setting, but some students were leery of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Year 1 survey respondents anticipated autocomplete positively (𝜇=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='6±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='5), believing it would help in several ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' For example, to “increase speed and productivity when coding” (B9) and “make it faster to get debugging done” (B6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' In addition, A13 believed auto- complete would encourage better code style: “not having dynamic autocomplete incentivizes me to write non-descriptive function names and variables for the sake of efficiency.” Participants also believed autocomplete would help “discover new features” (B5), “expose us to new things we didn’t know existed” (B7), and provide “an idea of what to write or what could be written” (B4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' These beliefs are in line with how professional programmers use autocomplete to debug and explore APIs [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' CHI ’23, April 23–28, 2023, Hamburg, Germany McNutt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' However, the experienced reality of p5/y2 fell short (𝜇=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='7±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='0) of anticipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Autocomplete was used in only 12% of sessions (with 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='6% selections being templates), although it was used progres- sively less as wi22 and su22 went on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' While this relative infrequency of use may be related to the simple implementation (which did not include embedded documentation or other common guidance fea- tures) or the emphasis later in the course on web programming (the DOM was not thoroughly reflected in the autocomplete sugges- tions), this trend appears to agree with how Vihavainen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' [101] observed novice usage of autocomplete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' They note that 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='3% of novices initially used autocomplete to create a particular command (Java’s system print), which decreased to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='64% after a week of use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' This appears to suggest that autocomplete can serve as a vehicle for teaching: it is “a useful guide until I was able to type certain things in by memory” (C13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Some perceived the ability to “stop memorizing certain code” (A9) as a benefit, while others thought “it’s a give and take” (C7) and might hinder “programmers’ knowledge about commands and their forms in the long run” (B8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' We return to this concern about the effect on learning in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' T4 Beyond these hesitancies, it is unclear why more students did not engage with the feature, although some noted that it can be “annoying when you already know what you want” (C15)—which suggests that the clutter T3 or cognitive noise T2 may be a factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Given this diversity of opinion, we suggest that configurability is important to designing such features politely, as some students (such as D4) wanted to be able to turn off autocomplete (to limit its disturbances).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='3 Color Pickers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Integrating a color picker into an editor for creative coding was perceived as very useful in the Year 1 surveys (𝜇=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Whereas A4, probably like many students, did not pick colors as much in the second half of the course, A6 said “I had a color picker tab open for every single assignment.” Based on this enthusiasm, we implemented a modal color picker dialog box in p5/y2 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 1), following a sentiment from A13 that such a design “would probably be more helpful than in the code to prevent clutter.” T3 Note that a similar color picker was recently added to the p5 editor, highlighting the value of this feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' However, ours supports more color formats, namely, web color names—driven by a participant suggestion (B9)—and RGB values as numeric lists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' After experiencing color pickers, students viewed them posi- tively (𝜇=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='65), with the wi22 students using them frequently in the first half of the course, as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' They helped make the process “more efficient” (A9), “[taking] out the hassle” (C6) of “open[ing] up another program” (A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Color pickers may foster creativity, as they could “let me pick some irregular colors” (B2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Several participants also voiced support for the idea, suggested in the survey prompt, for an eyedropper tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Others suggested additional features inspired by drawing programs like Illustrator, such as grid (D1), zoom (C15), “better proportions” (A3), or a way to “group lines and shapes and move them all at once” (B7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Such lightweight and familiar tools from creativity domains are natural enhancements—as long as they are not impolitely imposed T1—that we intend to investigate in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='4 Number Pickers and Sliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' “Scrubbers” [100], which allow direct manipulation of numeric values by dragging, are often touted in live programming systems and interactive documents [99] as being representative of the value of those environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Despite the overlap between live programming’s close connection to the visual domain and the interests of creative coding, the clutter T3 and lack of control T2 brought on by these features impeded adoption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Some Year 1 students were positive about hover-based Number Sliders, believing they would allow them to “experiment with the code more quickly” (B6) and “more efficiently” (B9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' However, some worried that “it could make the editor look more crowded” (A3), while A5 noted “I would rather just do it myself.” Nevertheless, we added Number Sliders to p5/y2, which appear (per Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='1) inline via Editor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='slider(min, max, value), as well as Number Pickers (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 1), which are buttons surrounding each number literal that allow it to be incremented and decremented ( ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (Such small modifications explain the large absolute number of Number Picker events in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=') Students found these additions could be a “quick, helpful way to make sure my assignments didn’t break at a larger scale” (C6), as was the case for HW5 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Al- though scrubbers were perhaps most useful toward the latter stages of a task, “when I’m playing around with my final result” (C17), C13 felt they “allowed me to tap into my creativity.” Yet, per Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 6, the feature was not so highly rated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' A recurring theme is that scrubbers—in various configurations—felt “messy” (B4, C9), “disrupted the look of the code” (C4) T3 or were just generally unnecessary (B10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' A14 felt that the transitory changes would be confusing, and hard to maintain a model of different parameter configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' T2 Others wanted more refinement in the numeric type, such as limiting it to numbers “divisible by five” (A5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' While these features are typically well-used in graphical applications like Figma, it seems that this type of feature is “trying to solve or better a process that needs no help” (B10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' While there is evident overlap be- tween our domain with other artistic settings, not every translated feature will match the interests of learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='3 On Skepticism Next, we grapple with the perception that some tools take away learning opportunities that may be needed to “become a good pro- grammer.” T4 Several features were perceived as making things too easy for novices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 9 summarizes how “skeptics” worried about different features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' These perceptions are valuable: as the technol- ogy acceptance model [27] and related theories highlight, perceived usefulness is a central part of whether a system is ultimately used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Several students worried how syntax templates (see appendix) and autocomplete balanced the tradeoff between augmenting their abilities and enfeebling their development of skills.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Whereas A5 was “not sure if it actually matters” to practice memorizing names and function signatures, C17 weighed the tradeoff according to the goals of the student: “I wouldn’t consider it a horrible thing for those who don’t want to go into coding professionally/too much”— implying that a more serious programmer might indeed miss out on practicing an important skill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' A16 thought such features “might reduce some of the learning by doing that you get when coding, so I’m not sure if it’s great for a class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' I learn through my coding mistakes and this would reduce the number of mistakes, so a mixed bag.” There were similar concerns about in-editor documentation (also discussed in the appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' For example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' A15 said that “new coders need to learn the process of going into the manual.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' A4 reconciled the aforementioned tradeoff as follows: “However,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' depending on 600 +A Study of Editor Features in a Creative Coding Classroom CHI ’23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' April 23–28,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2023,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Hamburg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Germany ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='No ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Some ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Prior Experience ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Yes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='C1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='C14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Linting ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='A5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Canvas ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Ruler ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='B8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Number ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Sliders ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='A4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='p5 State ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Displays ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Tidy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Code ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='C1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='C14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='C17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='B8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Auto- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='complete ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='C1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='C7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='C17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='A2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Shape ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Toolbox ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='A5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='A6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='A15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='A13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='A16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='B8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Code ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Snippets ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='A7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='A15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='A4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='A16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='In-Context ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Docs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='A4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='B5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Year 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Year 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Survey Responses: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Total skeptics in all surveys: 14/48 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Skepticism by Feature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Participant ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='wi22 worried that ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='linting would prevent ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='them from learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='C14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Figure 9: Some skeptical survey respondents worried that ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='some features would deleteriously affect learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' the specific goal of this course, if it is to focus more on the creative coding aspect and not necessarily ‘become a good programmer’ then in-context docs would be awesssommeee.” Some of these concerns might be ameliorated by introducing a notion of documentation or autocomplete levels (in a similar style as DrRacket’s language levels [73]), which gradually adjust what information is available as new concepts are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' However, without sufficient signaling, students might construct mental mod- els of the information present in the feature and then dismiss all subsequent configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Conflicting views over the Shape Toolbox in Year 1 were most striking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' “This feature would be so useful and allow for more creative opportunities especially for beginner coders” (A10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' It would also be a “useful learning tool” (A3) by allowing students to “see how the code changes in order to learn how certain parts of the code are working” (B9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' But many students were skeptical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' This “feels like cheating!”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (A6) and “saves way too much work for the new learn- ers” (A15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' “It’s way too useful but can hinder with the learning process of basics of coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' As a student, I won’t want this but as a programmer who knows the basics, it’s a nice feature” (B8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' “I think some of these features while helpful would have discouraged learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Some of the most rewarding parts was sweating through inconve- nient parts” (A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 9, some of this hesitation was self-censorship by students with little or no prior experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' However, among Year 2 participants (all of whom had access to the feature), there were no skeptics of the feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Perhaps the idea of others having improved tools is jarring, while students who are given improved tools simply worry about the plenty of challenging learning left to do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' We view this situation as akin to giving students calculators in a math class: they help with specific classes of tasks that, once simplified, enable learning about richer topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Students seem to construct a naive model of what makes a good programmer, suggested above as being someone who has memo- rized the entire language and does not depend on digital assistants or developer-experience tools, thereby dismissing behaviors besides this as being inauthentic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' We suggest that reorganizing and reform- ing this model is part of the value that classroom-based computing education offers, as it can help to offer a thicker model of what is authentic [91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Enhancements to novice-oriented IDEs such may also help to dispel these notions if they are perceived as realistic tools rather than as something akin to training wheels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='4 On Creativity Finally, we consider the role of creativity in our editor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' While there exists little agreement on what creativity means in HCI re- search [35], we found that students espoused two clear views on how tools might help them creatively: automating tasks that impede of creativity and helping explore unknown functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' For students, creativity often appeared to be something which typical coding tasks stood in the way of;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' obstacles that some techni- cal interventions could ameliorate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Shape Toolbox was emblematic of this style of reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' For instance, A10 believed that such a tool would “allow for more creative opportunities especially for beginner coders,” and B9 believed that it “could help with planning ideas for art projects and increase creativity.” As noted in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='1 students in p5/y2 embraced this feature and appeared to use it to reduce the tedium required to precisely locate shapes, thereby making greater room for artistic expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Others highlighted the value that tools that reduced tedious tasks, such as picking individual coordinates through a ruler (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' A3 and A9) or identifying which lines corre- sponded to which components of the image (B9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Summarizing this view, C16 observed that “I think what this editor did well as an art tool was streamlining certain common processes.” Beyond reducing tedium were opportunities for exploration, which were manifested both as moments of play (A8) or fun (A4, C6), as well as discovering new functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' For instance A12 believed that “comprehensive documentation would have allowed me to be both more creative,”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' a view which was confirmed by C17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' who believed that such features help do “things I don’t yet know how to do by myself” and thereby “help me be more creative.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' A9 believed that surfacing program state might encourage reflection and discovery,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' such as by seeing that “circle has a round stroke cap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' so it might make me wonder what other shapes the stroke cap could have.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' A14 believed that an assistant that made artistic suggestions might be well received,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' “[f]or instance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' if I’m editing text,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' and I was given suggestions for font,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' color,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' etc.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' A4 noted that it would be useful receive suggestions to help inspire their designs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' for example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' through “videos on youtube,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' images,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' articles.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Some features,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' such as number sliders,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' were highlighted as being only valuable “when I’m playing around with my final result” (C17),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' but that they “encourage a lot of experimentation and creativity” (A4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' These observations align with prior work, which highlighted the value of providing assistance in exploring the space of possible designs in creative coding contexts [77] and in creativity support tools generally [93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Whether a student’s primary purpose was closer to coding or to making art was an additional source of skepticism to those in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Again, regarding Shape Toolbox: “This would be great, but would reduce the amount of time figuring out the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' This would make it more an explicit art tool, and less a ‘make art with code’ tool” (A16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' A5 similarly noted that “feels a little too much like draw-ing for my taste” and took the class “with the primary goal of getting better at programming so I’d want to do things the code-y way”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Similarly, C17 felt that it “hinder[ed] the process of creative discovery– including trial and error”, however this was mostly not an issue as they sought to be “to be more accurate than creative” in this course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' While it is natural to want tools to be familiar, we believe that new authoring paradigms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' bidirectional programming) should be viewed as complementary rather than antagonistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' CHI ’23, April 23–28, 2023, Hamburg, Germany McNutt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 6 DISCUSSION This paper explored the observed behaviors and surveyed percep- tions of novice programmers in a creative coding course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' To wrap up, we recap the main themes, reflect on the connection between our work and other domains, describe limitations of our studies, and offer avenues for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='1 Recap: Themes In our analysis, we chose four recurring themes to highlight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' T1 Static Analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' We observed that simple static analyses were seen as supportive of a variety of types of work—notable given that error messages sometimes are obstacles in introductory set- tings [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Polite lightweight assistants that respect user agency, like those expressed through linting or auto-formatting, can be a helpful platform on which to learn and test new skills with confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' On the other hand, A5 noted that they “would also probably prefer to do things by hand” rather than use advanced features because there lacked visual indicators of a particular action’s effect—highlighting the importance of clear effect-forecasting for feature trust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' T2 Liveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' We saw that overeager evaluation can overwhelm and stress users through distracting updates that are unsynchro- nized with their expected edit-run cadence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Live programming of- fers enticing benefits for novice and creative contexts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' feedback immediacy or a closeness of mapping between code and graphics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Yet, these interaction challenges for non-expert settings are not yet thoroughly understood, leaving open questions about how to blend user control with system eagerness in a profitable way that maintains an experience level-attuned sense of agency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' T3 Clutter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' We noted that amateurs are mindful of how the edit- ing space can become overwhelming if too much visual noise or unfamiliar forms of interaction are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' For instance, stu- dents are aware that individual features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Number Scrubbers, lint errors, and autocomplete menus) can break their flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' We highlight the difficulty and importance of developing design guidelines that can aid the development of novel features within these constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' T4 Skepticism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Finally, we discussed how user perceptions of a feature can inspire skepticism about its propriety in learning envi- ronments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Year 1 students believed that the Shape Toolbox would impede learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' however, those who used it in Year 2 did not share that concern, instead viewing it as a convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Year 2 students also saw knowledge assistants such as autocomplete as detrimental to their development as programmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' We believe it is valuable future work to better understand what types of features and knowledge assistants are likely to be viewed as detrimental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='2 Connections to Other Domains Next we reflect on how our findings may apply more broadly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='1 Programming Pedagogy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Our work is merely situated within a classroom;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' we do not seek to make claims about the learning effects of the features we studied—this is an important, separable direction for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Yet, some of our themes may carry over to pedagogically-minded editors in more general learning contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' We suggest that skepticism T4 about features perceived as being too useful, such as autocomplete, may continue to be prevalent in learning contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Such concerns might be circumvented by emphasizing tools that help correct, rather than help complete, such as how linting T1 can identify an error while also providing justification and explanation for that error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' We also note the value of having a programming environment that is perceived as being approachable (A3,5, B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Furthermore, tools having not “got in the way” (C16) or otherwise cluttering T3 the display in unhelpful ways seem intuitively valuable, particularly in learning contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Similarly, live execution T2 may be beneficial in non-visual contexts as it promotes immediate feedback, such as by rerunning a test suite dynamically, as in Jest’s watch mode [32] or Huang et al.’s use of projection boxes [52] in a classroom to expose live values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Finally, like others before us [36, 70, 105], we found that a cur- riculum centered around media-art topics—as opposed to more abstract content often found in intro CS courses—invited a broad range of students who might not otherwise study CS in a formal setting (the appendix lists majors represented in the courses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='2 Other Domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Editors specialized to a given domain can make adaptations that aid that context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' In this work we focused on creative coding and designed affordances specific to this domain, however our findings might be applied in related contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' We highlight the value of bidirectional editing, linting, and designing editors with their effects on creativity in mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Bidirectional synchronization of code and effect (such as in our Shape Toolbox) seems to be an especially valuable approach in domains that have a prominent visual component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' This has been explored by Asai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' [10] as a mechanism to clean and synthesize data for statistical modeling, as well by DeLine [28] and Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' [107] for data science tasks such as modeling and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' We suggest such synchronization might be usefully applied to other visualization contexts (like preparing charts for presentation), as well as other creative coding contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Such interfaces may poten- tially reduce tedium in certain tasks and, more fundamentally, may provide opportunities for learning about the domain, for example, demonstrating how to achieve a particular effect using code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Next, we highlight that linters (or other static analysis tools T1) can provide a straightforward channel for introducing newcom- ers to basic principles and best practices of a particular domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' While they have already been explored in some contexts—such as for spreadsheets [13] and visualizations [51, 74]—additional fields such as data science [75] and music editors might integrate these concepts as well in order to surface best practices, such as high- lighting statistical fallacies, helping guide usage with unusual tools (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Orca [55]), or surfacing accidental discordance or inaudible components in music editors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' As discussed, such ambient assis- tants should be designed in a polite manner (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=', through granular dismissal of advice) to avoid being irritating and then dismissed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' While most technical tasks require some amount of creativity, we argue that features in editors in creativity centered-domains should be constructed in order to align specifically with goals of either reducing tedium or aiding in exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Barke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' [12] observe a similar pattern of exploration vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' acceleration in use of the AI-powered code assistant Copilot for traditional, non-creative soft- ware development tasks, suggesting overlap in editor features that A Study of Editor Features in a Creative Coding Classroom CHI ’23, April 23–28, 2023, Hamburg, Germany support creative coding and coding more generally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Compton [24] argues for IDEs with features that are valuable unto themselves— for example, for being playful or thought-provoking—rather than their use as a means to end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Non-productivity focused techniques may be useful in creative coding contexts more generally, perhaps as a design advisor as A4 suggested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' These additions may drive unexpected patterns of usage, leading to new types of discovery through play—which might even valuable in technical domains like data science or visualization [102].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' At the same time, such inter- ventions may inspire skepticism T4 about their authenticity if they are perceived as too whimsical or unrealistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='3 Limitations and Future Work As described throughout, our study had a number of limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' These included data collection errors (such as the configuration error in wi22) and the relative simplicity of the survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' For in- stance, our use of static images—as opposed to videos or interactive prototypes—limited our ability to accurately explore reactions to proposed features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' However, the use of non-interactive stimuli (fol- lowing Kery et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' [60]) allowed respondents to project their own beliefs about the features and ignore potentially distracting low- level bugs or stylistic issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Further, we only implemented a subset all designs we identified, so we cannot make inferences about what features would be most valuable in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Instead, we focus only on the observed themes and interactions with implemented fea- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' This approach was a coarse and inexpensive way to identify and explore some potentially fruitful features, however not all such features were necessarily identified nor considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Future work could implement more of the identified features—and also augment our observations with lab studies—to better understand the effects of particular features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Furthermore, whereas our work investigated how novices perceive the utility of various editor features, subse- quent work should also investigate their pedagogical effects on learning outcomes—one notable point for comparison is that Oviatt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' [82] found novel interfaces can hinder learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' The biases of our particular student populations may not be re- flective of a more general student population, however the views of the college-aged (sp21, wi22) students seem aligned with those of the high-school students (su22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' In addition, they are in agreement with those of su21 students who, because of pandemic era-distancing, at- tended from around the world and thus drawn from a substantially different population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Our own biases were likely projected onto the students in teaching this material, and different instructors may have inspired different responses in students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' To this end, student perceptions are likely reflective of the context and content of the work they were asked to do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' For instance, the open-ended nature of many assignments likely shaped student opinions of the features we asked about, which may have been different under more structured programming tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' In future work we would like to reexamine our findings by teaching the course to and soliciting feedback from students from other institutions, age groups, and backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Students were generally positive about the editor being online and the way in which our feedback and submission systems were in- tegrated (A3,5), with B1 noting that they were especially beginner- friendly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Nevertheless, the choice to use a web tool had limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Students with inconsistent internet connections struggled with the online environment (prompting B6 to suggest an offline mode), while others had computers that were unable to handle the com- putational weight of a larger web application (which made some students hesitant to explore some editor features).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' For instance, A2 noted that they hesitated to use auto-refresh because “my computer was already very slow and I didn’t want my code to crash while it was running.” These concerns were particularly prominent during the fully online sp21 and su21 editions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' While in-person teaching has resumed (as in wi22 and su22), that consideration of how to build novice-oriented tools that support those with limited internet connectivity or less powerful computers should not cease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' While our target population in this work was students, in future work we wish to understand what features instructors see as valu- able or concerning in such a setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Similarly, it would be useful to consider whether these user interface patterns are applicable to professional artists working in creative coding spaces—questions which are closely connected to Li et al.’s [67] study of the tools that artists make for themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Of particular relevance are artist- designed custom coding environments used for teaching and artistic practice (such as Field [30]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' In sum, creative coding has been, and continues to be, fertile soil for HCI research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' We believe that studying the problems users in these creative domains face is valuable unto itself, and is ever more relevant as creative coding becomes an increasingly common way to introduce computing and to make art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' ACKNOWLEDGMENTS We are grateful to those who made our courses possible, includ- ing the course staff (Brian Hempel, Angela Liu, and Bhakti Shah), Kevin Workman for allowing us to incorporate his Happy Coding tutorials, and the p5 community and developers for building such useful tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' We thank Lilian Huang, Shriram Krishnamurthi, Elsie Lee-Robbins, Justin Lubin, and the anonymous reviewers for their helpful commentary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Finally, we thank our students, without whom this work could not have taken place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' This work was supported in part by the University of Chicago College Innovation Fund.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2011, part of SPLASH ’11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 3–18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='1145/2048237.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='2048241 [74] Andrew McNutt, Gordon Kindlmann, and Michael Correll.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Computational Art: Introduc- ing High School Students to Computing via Art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' In ACM Technical Symposium on Computing Science Education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 261–266.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='1145/2839509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='2844614 [106] Kevin Workman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Happy Coding Tutorials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' https://happycoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='io/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Ac- cessed 4/3/2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' [107] Yifan Wu, Joseph M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Hellerstein, and Arvind Satyanarayan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' B2: Bridg- ing Code and Interactive Visualization in Computational Notebooks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' In ACM Symposium on User Interface Software and Technology (UIST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' ACM, 152–165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='1145/3379337.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='3415851 [108] YoungSeok Yoon and Brad A Myers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Supporting Selective Undo in a Code Editor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' In International Conference on Software Engineering (ICSE), Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' IEEE, 223–233.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='1109/ICSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='43 CHI ’23, April 23–28, 2023, Hamburg, Germany McNutt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' APPENDIX In this appendix we provide supplementary material that fell outside the scope of the main content of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' A we make several notes about the course design and other ancillary details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' B we provide additional details about several of our studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' A THE CREATIVE CODING COURSE Here we provide additional context for our course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 11, we show the schedule for the 3-week su21 edition (link to course site).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' This edition featured only a sequence of homeworks and exercises, and did not include the self-guided project found in the “full” editions of course (sp21, wi22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' A similar version of the course was also taught as su22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' There were six individual homework assignments in su21, the first five of which appeared in all course editions: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Color Wheel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Recreate a given red-yellow-blue color wheel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (function calls, color and shape-drawing APIs, trigonometric expressions) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Freeze Frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Pick a static frame from the “StoryBots: Shapes” video and recreate it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (function calls, color and shape-drawing APIs) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Use variables and arithmetic expressions to implement a symmetric tree drawing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (variables, arithmetic, curves) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Book of Patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Implement several 2-dimensional grid patterns—stripes, polka dots, checks, plaid, chevron, harlequin, argyle, and honeycomb—inspired by the designs in My First Book of Patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (nested loops) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Snake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Make a simple version of the classic snake game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Starter code was provided with function stubs for different aspects of a simple model-view-controller architecture (mutable variables, arrays, objects, animation, mouse and keyboard events) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Subway Font.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Rewrite a webpage using a “font” that resembles the signage of the New York City subway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' As shown on the cover of Subway, some letters are rendered white-on-black and others are set atop colored circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (HTML, CSS, DOM API, dictionaries) As highlighted in the main text, we designed our course primarily for college students with little-to-no programming experience who were not planning to major in computer science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' In sp21, 4 out of the 31 students were undeclared, and among the remaining 27 students 14 different programs of study were represented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' In wi22, 10 out of the 27 students were undeclared, and among the remaining 17 students 12 different programs of study were represented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' All told, students from 23 different departments participated in the course (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Based on both our study and pre-course on-boarding surveys, students self-reported high levels of prior experience (as highlighted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 3) in each edition of the course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' In sp21, students who had previously completed computer science courses at the university—in a couple cases many such courses—were mistakenly allowed to enroll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' This enrollment issue was fixed for wi22, but still nearly half of the students (who completed the course) self-reported prior experience through self-study, courses in high school, and from other university departments or institutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' In su21 and su22, enrollment was unrestricted (the high-school students were not already associated with the university), and a large majority of these students reported prior experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' In any case, the different levels among our student populations helps color some of the observations in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' B ADDITIONAL STUDY DETAILS In this section we present aspects of our studies that did not fit in the main text: the ethics statement for our studies, additional results, followed by descriptions of the hypothetical features from our Year 1 survey that were not implemented in Year 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='1 Ethics Statement All studies were reviewed and determined to be exempt by our university’s institutional review board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' We did not collect demographic data, because it was not a core aspect of our investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Although we designed and taught these courses with an eye towards the associated studies, we believe the course materials we developed and delivered (through lectures, office hours, and online discussions) were minimally affected by the presence of these studies and our use of custom versions of the p5 editor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='2 Additional Results Here we list a series of one-off results that were observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Then in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='1 include an analysis of the code folding feature (original part of the analysis in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Finally we provide an additional analysis of the the auto-refresh feature in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 13, we show an alternative depiction of the results from both years of our survey which includes metrics other than the one used in the main text (namely Usefulness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 14 we provide a simple summary of the volume of code executions across all three editions along with assignment due dates, which highlights that execution volume tended to be higher for earlier graphic-only assignments (compared to later assignments which involved interactivity or HTML/CSS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 15 we provide a related graphic showing execution history for sp21 and su21, along with the relative error rates by day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' In Year 1, our logging scheme did not include a mechanism for explicitly collecting run-time errors, so they were reconstructed post-course by running each logged sketch for 10 seconds and collecting all errors generated during that period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' This approach may exclude errors students saw, such as those generated through interaction with the sketch or through randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' On average each session had 𝜇=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='27±32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='8 errors, with outliers excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Within our reduced sample from Year 2, sessions exhibited 𝜇=30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='7±95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='8 errors, again with outliers excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' A one-sided t-test indicated that there were significantly more mean errors per session in Year A Study of Editor Features in a Creative Coding Classroom CHI ’23, April 23–28, 2023, Hamburg, Germany 2 (𝑝<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' This increase is likely due to the new collection method, which captured errors witnessed by the user rather than just reconstructed errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 16 we provide a figure showing the bi-gram action sequence probability of actions in Year 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' We then provide tables in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 6 for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 17 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='1 Code Folding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' This simple feature, common to most modern editors [33], allows functions and other blocks of code to be collapsed and later expanded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' This feature was generally well liked as it made code “feel more organized” (A9), while helping users avoid “being overwhelmed” (A2) and making things “look neater and less intimidating” (A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' This has the organizational benefit that it is “easier to find specific chunks of code” (A1), which, as noted by A13 and A16, reduces the amount of scrolling—these are well-understood benefits of this feature [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Being able to organize and navigate code are important concerns for novice creative coders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' T3 However, the feature was not universally appreciated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Whereas B8 found that code folding “Helped a lot while debugging and re- organization!”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=', A9 asked “when debugging, what if the problem is in one of the lines of code that are hidden?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' A number of participants noted that they simply did not use it or did not find it helpful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Some students only invoked it accidentally (A10), while others found it confusing (A5,14) because it did not clearly communicate what code was to be folded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Code folding, or other interface-based code organization tools, seem especially valuable in this context as most sketches typically involved only a single file (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' sp21 and wi22 final projects had a median of 1 JavaScript file).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' As the file structure abstraction for code organization is underused, there is opportunity for interface-based abstraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Debugging Programming Year 1/2 Outliers were dropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Year 2 has missing data due to a data collection error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Cycle Frequency (Count) Average number of executions that it takes to switch from one episode type to another 0 5 10 15 Cycle Length (Minutes) Average length of time between code executions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='0 Episode Length (Minutes) Alaboudi & LaToza Auto-refresh Manual Year 1 Year 2 Year 1 Year 2 Period of time that a programmer is either in a debugging or programming state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 0 2 4 6 8 10 12 Q3 mean Q1 Figure 10: Comparing how students shifted between debugging and programming states (using different execution styles) against a baseline of professional programmers [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='2 Live Coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' In addition to the analysis of the auto-refresh feature considered in the main text (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='3), we also sought to understand how student edit-run behavior compared to that of professional programmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 10 shows how auto-refresh usage affected the length, size, and frequency of edit-run cycles with regard to debugging versus programming states adopting the metrics used by Alaboudi and LaToza [7], who studied how professional programmers shift between debugging and programming states during edit-run cycles in their own work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' A salient difference from the baseline was that the number of executions to transition from a programming to a debugging state (and vice versa) was shorter for our students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' This is likely informed by the domain: the professionals were working on projects such as Firefox and Curl, which likely have a different execution cadence than the graphic-oriented work conducted in creative coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' The programming episode length was similar for the professionals and those using manual execution—although debugging episodes for the latter group were much shorter, suggesting that the errors were much less complex for our students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' However, given the differences in expertise and domain between these groups, it is difficult to identify a primary cause of the changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' The key observation is that the usage patterns exhibited by our students were not the same as those of professionals, but were not fundamentally dissimilar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' This suggests the potential transferability of our observations about novices to more experienced users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Using auto-refresh does not appear to have an effect on cycle frequency, although it seems to be associated with shorter episodes and cycle lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' This coheres with our expectation of auto-refresh, as it triggers executions more quickly than one might with manual execution, but suggests a certain consistency related to task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' CHI ’23, April 23–28, 2023, Hamburg, Germany McNutt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Figure 11: Schedule for su21 edition of the course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Readings and their corresponding images were adapted from Workman’s HappyCoding tutorials [106].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' sp21 wi22 Anthropology Chemistry Undeclared Economics English Global Studies Humanities Mathematics Neuroscience Physics Political Science Psychology Visual Arts Exchange Undeclared Economics English Env.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' and Urban General Studies Law Humanities Media Arts Music Physics Public Policy Russia, East Europe 1 1 1 2 2 2 1 2 4 10 Total 27 Total: 31 10 5 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 Comparative Human Dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Comp Science Figure 12: Home departments of students who completed the sp21 (top) and wi22 (bottom) courses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' WEEK 2 WEEK 3 Tu Jul 13 W Jul 14 Th Jul 15 F Jul 16 M Jul 19 Tu Jul 20 W Jul 21 Th Jul 22 F Jul 23 M Jul 26 Tu Jul 27 W Jul 28 Th Jul 29 Class Class Class Random Topics Loops Class Ftomagesit Class Class Class Recursion classes Class stiders (Unidentified Flying) aries L-systems Functions Objects Recursione Exercise 5 Abstract Art oops (classified) Class 1，' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='2，' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='3 Exercise 11 Exercise 4 Objects Class (nothing) Exercise 1 SPRAY Exercise_ 9 Class Class Coding Mondrian Counting Tags Fractals Exercise 2 If statements Arrays PAINT Abstracting Mondrian Exercise _7 taths//Br Reading count[tag]++;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Input (nothing) Exercise 8 Images Exercise 12 Pong Exercise 6 Pixelate Class Reading Reading Reading Exercise 10 Living Line L-systems Exercise 3 Random Reading For Loops Interactive HTML Reading Walk NYC Bouncy Bali Using Objects Project From p5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='js to Web Dev :: Walk ANYC Description Reading html Reading circle Conjecture Reading Creating Functions If statements Array Functions Bonus Reading Creating Classes Reading Homework 3 HTML, Tags, CSs Bonus Reading Homework 5 Counting Pancakes Welcome to Coding Just okay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Trees Reading Local Storage Debugging Snake p5Js Animation Arrays p5Js Homework 4 Calling Functions Homework 2 Book of Patterns SubwayFont Input Freeze Frame cloud Storage Homework 1 SHATES Post-Processing Using Variables Color wheel 1 Creating Variables Post-Processing 8A Study of Editor Features in a Creative Coding Classroom CHI ’23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' April 23–28,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2023,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Hamburg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Germany ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Year 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Year 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Implemented ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Implemented ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Implemented ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Implemented ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Standard ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Standard ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='All features implemented ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Coding by Drawing Tools ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Color Picker ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Linters ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Autocomplete ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Canvas Ruler ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Tidy Code ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Directly Manipulate Shapes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Time Travel Slider ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Code Folding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='In-context Docs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='p5 State Displays ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Interactive Value Inspector ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Linked Copy-and-Paste ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Code Snippet Templates ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Number Sliders ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Auto-refresh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Drag-and-Drop Refactoring ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Interested ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Often ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Useful ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Linters ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Color Picker ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Tidy Code ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Number Sliders ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Coding by Drawing Tools ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Auto-refresh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Autocomplete ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Number Picker ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Figure 13: Participants in both the long (Year 1) and short (Year 2) survey were asked about a variety of features and rated each ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='of them on how Interested they were in it,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' how Often they would use it,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' and how Useful they thought it was.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Most features considered are non-standard, however several were implemented in our editor or standard (but not implemented).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' It is notable that although some of the most commonly instrumented features are not necessarily predictive of perceived utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' CHI ’23, April 23–28, 2023, Hamburg, Germany McNutt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Auto-Refresh Manual Execution Average Executions per Student Mar 29,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2021 Apr 05,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2021 Apr 13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2021 Apr 21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2021 Apr 29,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2021 May 07,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2021 May 15,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2021 May 23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2021 May 31,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2021 0 100 200 300 400 500 600 hw-1-mondrian-nyc hw-2-color-wheel hw-3-freeze-frame hw-4-trees hw-5-sleepy-face hw-6-patterns hw-7-snake hw-8-pancakes hw-9-subway-font hw-10-typoglycemia proposal progress-report final Jul 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2021 Jul 14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2021 Jul 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2021 Jul 18,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2021 Jul 20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2021 Jul 22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2021 Jul 24,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2021 Jul 26,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2021 Jul 28,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2021 0 100 200 300 400 500 600 hw-04-book-of-patterns hw-01-color-wheel hw-03-trees hw-06-subway-font hw-05-snake hw-02-freeze-frame Jan 09,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2022 Jan 17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2022 Jan 25,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2022 Feb 01,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2022 Feb 09,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2022 Feb 17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2022 Feb 25,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2022 Mar 05,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2022 Mar 13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2022 0 100 200 300 400 500 600 hw-1-color-wheel hw-2-freeze-frame hw-3-trees hw-4-book-of-patterns hw-5-deck-of-cards hw-6-snake hw-7-wordle proposal hw-8-blackout-poetry progress-report final Jun 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2022 Jun 14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2022 Jun 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2022 Jun 18,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2022 Jun 20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2022 Jun 22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2022 Jun 24,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2022 Jun 26,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2022 Jun 28,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2022 0 100 200 300 400 500 600 hw-1-color-wheel hw-2-freeze-frame hw-3-trees hw-4-book-of-patterns hw-5-snake hw-6-wordle hw-7-typoglycemia sp21 su21 wi22 su22 Figure 14: A summary of executions for each of the course editions show auto-refresh versus manual execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' A Study of Editor Features in a Creative Coding Classroom CHI ’23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' April 23–28,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2023,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Hamburg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Germany No Error DOMException Error FirebaseError RangeError ReferenceError SyntaxError TypeError Live Programming Executions Manual Executions Normalized Error Type sp21 su21 0 50k 100k 0 2k 4k 6k Week 1 Week 2 Week 3 Week 4 Week 5 Week 6 Week 7 Week 8 Week 9 Week 10 0% 50% 100% 0 1k 2k 0 1k 2k 3k 4k Week 1 Week 2 Week 3 0% 50% 100% Figure 15: Errors over time in the first year of courses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Outliers have not been dropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Our research questions are generally not motivated 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='4% 0% 100% sp21 su21 Figure 16: The bi-gram action sequence probability in Year 1 shows the rate at which a given action is followed by another particular action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' We do not show wi22 because we mistakenly did not collect Tidy Code executions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Feature Name rating mean 𝜎 Q1 Q3 Auto-refresh 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='26 3 4.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='08 4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='00 Figure 18: The computed values for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 13, the survey results relating to Usefulness from Year 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' CHI ’23, April 23–28, 2023, Hamburg, Germany McNutt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='3 Hypothetical Features Here we return to the hypothetical features asked about in Year 1 surveys but not implemented in p5/y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Because responses were based only on a brief description and static image, we limit our discussion of each feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Canvas Ruler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' As mentioned earlier, the Canvas Ruler was widely viewed as a useful tool to add for creative coding—however, B8 felt “it would take away the fun of mouseX and mouseY!”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Several additional suggestions were made, such as being able to “measure angles, so a ruler and compass.”(A16) In future editions of the course we intend to return to this feature, as it seems like a natural next step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' The primary concern in implementing such an addition would be that it does not clutter the interface T3, and perhaps, per commentary on Syntax Templates, be controllable with a keyboard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' In-context Docs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Many full-featured editors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' VS Code) include relevant documentation about language features and user-defined variables as a tooltip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' As with autocomplete, B4 believed In-context Docs would be helpful because “gives me an idea of what to [write].” Several participants echoed this sentiment, believing that it “would have drastically widened my skill set” (A14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' On the other hand, A4 was “actually a little torn by [it] because I think googling and traveling to the reference is really important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' It may start off as inconvenient but just becomes more natural with practice”—which is in line with our observations about student skepticism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' T4 Among the quantitative ratings from the surveys in the first year, this feature was the only one that had a statistically significant relationship (𝑝<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='01) with self reported experience was in-context docs, in particular exhibiting a negative correlation (𝑟=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='308).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' It is possible that an alternative presentation of this feature (perhaps in the search-based style of Blueprint [19]) might elicit more positive responses, however, based on these results we believe that users might be similarly skeptical, although exploration of such responses could be usefully explored in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='js Saved: 2 minutes ago Preview const size=5o;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2 const gap = 10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 37 functionsetup()( 4 createCanvas(400,400);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 5 9 7vfunctiondraw()( 8 background(220);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 9 10V for（leti=0:i<10i++)( 11V for（letj=o;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='j<10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='j++) 12 fill((i+j)%2？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content="'black' 'white') 13 square(i *(size + gap),j *(size + gap), size);" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 14 15 16 17 Left ruleredge (0,170)Right ruler edge (400,170)background(colorstring, [a]) background(gray,[a]) background(v1,v2,v3,[a]) background(values) background(image,[a]) 1function se The background() function sets the color used for the background of the p5js canvas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' The default background is "t transparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' This function is typically used within draw() to 5V function dr clear the display window at the beginning of each frame can be usedlinside setup( to set the backaround o 6 background(250);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 7 rotateY(frameCount*0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='01);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 8 for letj=;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='j<5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='j++)( 10 push();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 11V for(leti=0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='i<80;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='i++)( 12 translate( 13 sin(frameCount*0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='001+j)*100 14 sin(frameCount*0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='001+j)*100 15 i*0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='1 16A Study of Editor Features in a Creative Coding Classroom CHI ’23, April 23–28, 2023, Hamburg, Germany Syntax Templates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' The syntax templates we implement in our autocomplete were similar to our proposed Code Snippet Templates, however the latter feature was docked (in the manner of Google Colab’s Code Snippet library), and thus required mouse clicks, which may have dampened enthusiasm for the feature: “I think if there were keyboard shortcuts for these then I would use them extensively” (A13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Some thought these features would be an “easy way to get students started with no experience” (A24), but as discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 6 others were skeptical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' We still believe this feature would be valuable to implement in the future, possibly integrated into the autocomplete, in order to keep the interface tidy and unencumbered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' T3 Time Travel Slider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' This proposed feature would allow the state of the code execution to be paused and rewound in order to support debugging tasks—which a majority of respondents either understood as a GUI-based shortcut for p5’s frameRate setting (which specifies how many times per second the draw loop is called) or as a mechanism for version control, both of which, while interesting, are not the feature we intended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' While several students expressed enthusiasm for this latter idea (indicating the potential utility of a Variolite-style [59] or other selective undos, such as that of Yoon and Myers [108] or Mikami et al.’s [76] Micro-Versioning), this did not yield coherent feedback, beyond confusion about unfamiliar features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Auto-refresh Sketch name Wood cushion SUBMIT Sketch Files V sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='js Preview index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='html sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='js 1functionsetup() D style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='css 2 createCanvas(400,400);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 3 t 4 5V function draw()( 9 background(220);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 7 bezier(x1, y1, x2, y2, x3, y3, x4, y4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 8 7 9 Code Snippets 10 Filter code snippets 11 Add color Add colored rectangle Add beziercurve 4 Insertedlineofcode Add mouseDrags /ent Add mouseClicked event > Load image Access web cam Adid custom snippet + Consolesketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='js Saved: 2 minutes ago Preview constsize=50;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' const gap =10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 3V function setup()( createCanvas(400,400);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' function draw() 8 background(220);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 9 10V for(leti=0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='i<10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='i++)( 11V for(letj=o;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='j<1o;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='j++) 12 fill((i+j)%2?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='black\' "white\';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 13 square(i * (size + gap),j * (size + gap), size);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 14 15 16 17 Frame 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='2kCHI ’23, April 23–28, 2023, Hamburg, Germany McNutt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' p5 State Displays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' A display of current values for common library variables, such as strokeWidth and fill color, at particular lines of code—similar in spirit to object value displays in creativity tools like Illustrator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Some students were enthusiastic about this feature, noting that it “would be extremely useful to be able to see all this information in one place” (B7), while others felt it might enrich creativity by showing what options are available (A9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Others were less enthusiastic, noting that it would be “a little redundant” (A1) with running the code, or that it would be tedious (A7) compared to simply writing code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Interactive Value Inspector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' A growing thread of research allows users to inspect the current value of variables at various lines of code on demand, such as in Lerner’s Project Boxes [65] or Kang and Guo’s [58] “DISPLAY ALL THE VALUES!”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' approach to novice coding in Omnicode, as well through live probes [85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' In this feature we proposed a Projection Box style feature that included a customizable inspector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Students were generally enthusiastic about this feature, noting that it would be helpful for beginners (B3,8) as it would make “loop definitions” (A16) and debugging (A1,10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Yet some worried that the implementation might be overwhelming (A5) or distracting (A6), or would not substantially improve over console.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='log-based debugging (A14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' T2 In addition one skeptical student believed that it might “make the coder (especially early learner) to be lazy” (A15), and prevent them from learning good debugging skills T4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Auto-refresh Sketch name Woodcushion SUBMIT 8 Sketch Files < sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='js` Preview index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='html sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='js 1vfunctionsetup()( D style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='css 2 createCanvas(360,280);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 3 nostroke();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 4 noLoop();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 5 6 77 functiondraw()( 8 drawcircle(width/2,280/2,6);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 9 10 P5 State Values 11 function drawcircle(x,radius,level)( 12 consttt=（126*leve1)/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Stroke None 13 fill(tt);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Stroke Weiqht 1px 14 ellipse( height/2,radius*2,radius*2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Stroke Cap Round 15V if (leve) 1) ( Fill Multiple 16 level =Ievel - 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Translate 0, 0 17 drawcircle(x-radius /2,radius /2,level) 18 Rotation drawcircle(x + radius/2,radius/2,level) 19 20 21 22 Watch additional values + Console< sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='js Saved: 9 days ago Preview 175 //Acounterto help safeguardagainstinfiniteloops 176 //during development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Youmaytry adjusting this 177 //valueif there is toolittleflooding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 178 let fuel = 100000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 179 180vwhile(worklist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='length>0&&fuel>0)( 181 //Gettheposition atthefront oftheworklist 182 const pos=worklist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='shift();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 183 const x = pos[o];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 184 const y= pos[1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 185 186 const withinBounds=!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (x<0Ilx>img.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='width Ilyimg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='height);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 187 constnotAlreadyFilled=!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content="alreadyFilled['$(x}-$(y}'J;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 188 189V if(withinBounds&¬AlreadyFilled)( 190 const c = img.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='get(x,y);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 191 const [r,g,b,a］ = c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' C: [255,0,0,0] 192 constiswhitish=r>240&&g>240&&b>240;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' r: 255 193 constisTransparent=a<5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=" g: 0 194V if(iswhitis isTransparent) 195 ['$(x}-$ty=true;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' b: 0 alreadyFil 196 img.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='set(x, ，color(fiilcolor));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' a: 0 197 iswhitish: true 198 //Addneighboringpixelstoendofworklist isTransparent: false 199 worklist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='push([x +1,yJ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='A Study of Editor Features in a Creative Coding Classroom CHI ’23, April 23–28, 2023, Hamburg, Germany Linked Copy-and-Paste.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Vihavainenet al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' [101] note that novices tend to make heavy use of copy-paste, so a natural point of enhancement then would be to embed variable-style abstraction into copy-paste itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' This idea has been discussed in research works previously [31, 98], however is not typically seen in this style of editor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Some students thought this would be helpful, by “facilitat[ing] better organizational practices” (A12) or in niche situations (A7,13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' However, most others were apprehensive about the feature’s value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Some noted that it seemed to be a more oblique version of creating a variable (A1,16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Some thought that what was already in the editor sufficiently addressed any tasks linked copy-and-paste might accomplish, through regular copy-paste (A9) and Find & Replace (B8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' A13 argued that a Sublime-style multi-cursor selection would be more flexible and preferable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' We note that multi-cursor support was enabled in our editor (as part of CodeMirror), although students were not explicitly made aware of this functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Others still simply thought it would not be useful, and would “creat[e] mess for me” (B4) or otherwise be confusing (A10, B5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' T3 Drag-and-Drop Refactoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Clicking and dragging values to create arguments, variables, and other functions in a technique that has been previously explored to useful effect [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' In this feature we proposed a simple version of this feature, however our presentation lacked the nuance of the presentations used by Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' [64], which may have led feature being rated lowest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Although a few respondents were intrigued, some said they would prefer copy and paste (A12, B1,5,6,8), most were disinterested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' For example: “I personally don’t like dragging and dropping things because there is room for dragging and dropping into the wrong section especially if your computer is slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' I don’t think copy paste was too time consuming and encourages greater accuracy” (A2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' several others shared these views about efficiency and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' In addition, there were concerns about the usability of the feature: “Clicking and dragging is not an ergonomic motion on a laptop touchpad” (A6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' We highlight this as an especially valuable concern, as dragging may not be an accessible motion for some users, although something like Kobayashi and Igarashi’s suspendable drag-and-drop interactions [62] may usefully address these concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Other Suggested Features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Beyond the hypothetical features we presented, some respondents suggested ideas like scratchpads or selective execution contexts similar to some of the ideas expressed in Code Bubbles [18] or Jupyter notebooks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Others suggested course-specific affordances, such as hints relevant to the assignment or integration of the assignment directly into the editor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' This has a similar flavor as DrRacket’s language levels, and Marceau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' [72] briefly sketched out learner-attuned error messaging levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' This is similar to Interactive Tutor Systems [53] which integrate curriculum and course work into a single environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' While this level of integration can be helpful, it may undermine the utility of an in-class instruction model because such interfaces are naturally self- rather than group-paced, although that should be investigated in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' sketch.' metadata={'source': 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+page_content=' 9 10 11V function drawcircle(x,radius,level)( 12 consttt=（126*level)/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 13 fill(tt);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 14 ellipse(x,height/2,radius*2,radius *2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 15V if(level>1) 16 level = level - 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 17 drawCircle(x + radius / 2, radius / 2,level) 18 drawcircle(x -radius /2,radius / 2,level);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Drag a line of code to move it 19 drawCircle(x radius radius Level 20 21 22 23 ConsoleCHI ’23, April 23–28, 2023, Hamburg, Germany McNutt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' C SURVEY INSTRUMENT FOR INITIAL SURVEY — YEAR 1 (sp21, su21) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='1 Page 1: Consent to Participate in a Research Study Research Project Title: Post-Course Survey of Students in Creative Coding (2021) Principal Investigator: Ravi Chugh Graduate Student: Andrew McNutt IRB Protocol: IRB21-1062 This form is designed for students younger than 18 years of age who took the Creative Coding Pre-College Immersion class in Summer 2021 and their parents, respectively referred to as “you” and “your child” below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' You (or your child) is being asked to take part in a research study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' This form has important information about the reason for doing this study, what we will ask you (or your child) to do, and the way we would like to use information about you (or your child) if you choose to allow yourself (or your child) to be in the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Purpose of Research Study: You are (or your child is) being asked to participate in a research study regarding the usability of editors for creative coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' In our recently completed course we used an in-browser editor that was slightly modified from the publicly available p5 editor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' We are interested in understanding what editor features might be useful to someone learning to code (particularly in the context of a creative coding course) or otherwise making digital art works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Ultimately, this research may be published and presented at scientific conferences to improve the community’s knowledge about editors for creative coding, and may be used to improve the editor used in future iterations of our course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Participation Procedures and Activities: The full extent of the procedure will involve completing this survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' We anticipate that completion of this survey will take up to 60 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Due to the difficulty of determining credit for partial completion, no compensation will be provided for partial completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' At the end of the form you (or your child) will provide a student id and preferred email address, and you (or your child) will receive a $30 Amazon gift card for participating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Consent and Assent Process: If you are (or your child is) 18 years or older, you (or your child) can provide the consent required to opt-in to the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' If you are (or your child is) under 18 years of age, you can give your assent (or you can give your parental consent) to join the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' For students under 18 years of age, participation in this study requires both consent from a parent as well as assent from the student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Risks/Discomforts of Being in this Study: The risks to your participation in the survey are those associated with basic computer tasks, including boredom, fatigue, or mild stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Benefits of Being in this Study The only benefit to you (or your child) is the learning experience from participating in a research study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' The benefit to society is the contribution to scientific knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Confidentiality of Data and Limits to Confidentiality: Any reports and presentations about the findings from this study will not include your (or your child’s) name or any other identifying information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Use of Your Research Data: We will never share the data beyond the University of Chicago research team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' However, an analysis of the data may be analyzed and published in scientific conference proceedings or journal articles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' The free-text responses provided to any portion of this survey may be quoted in part or in whole in this publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' We will remove any information from the analysis that could identify you (or your child) before providing the analysis for publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Voluntary Participation and Right to Refuse or Withdraw: Participation in this study is voluntary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' The decision to participate in this study is entirely up to you and your child.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' You (or your child) may refuse to take part in the study at any time without prejudice or penalties and will not result in any loss of benefits to which you (or your child) are otherwise entitled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Mandatory Reporting of Child Abuse or Neglect: The research study staff are mandated reporters and are required to report suspected child abuse or neglect to the Illinois Department of Child and Family Services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' For more information, please see the University policy: https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='com/mr26uazn Contact Information for Research Questions and Participation: If you have questions or concerns about the study, you can contact the researchers at: Principal Investigator Ravi Chugh, Associate Professor John Crerar Library University of Chicago 5730 S Ellis Ave Chicago, IL 60637 Email: rchugh@uchicago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='edu Graduate Student Andrew McNutt, PhD student John Crerar Library A Study of Editor Features in a Creative Coding Classroom CHI ’23, April 23–28, 2023, Hamburg, Germany University of Chicago 5730 S Ellis Ave Chicago, IL 60637 Email: mcnutt@uchicago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='edu If you have any questions about your rights as a participant in this research, feel you have been harmed, or wish to discuss other study-related concerns with someone who is not part of the research team, you can contact the University of Chicago Social and Behavioral Sciences Institutional Review Board (IRB) Office by phone at (773) 702-2915, or by email at sbs-irb@uchicago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Parental Consent (1) Parent full name (Last, First) (2) Parent Email address (3) I have read and understood this consent form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Yes ⃝ no ⃝ (4) I am a parent and give consent for my child, under 18 years of age, to participate in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Yes ⃝ no ⃝ Student Assent (1) Student full name (Last, First) (2) Student Email address (3) Student GitHub username (same as used for homework submission in this class) (4) Student CNetID (the username before your @uchicago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='edu email address) (5) I have read and understood this consent form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Yes ⃝ no ⃝ (6) I am a student, under 18 years of age, and give assent to participate in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Yes ⃝ no ⃝ CHI ’23, April 23–28, 2023, Hamburg, Germany McNutt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='2 Page 2: Introduction and Reflection In this section, we’ll ask you some questions about your programming background, and to reflect on your experience during the course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (1) Pre-Course Experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' How much programming experience did you have prior to taking the course?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (2) Post-Course Confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' How confident do you feel in your programming skills after taking this course?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Have they improved?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (3) Challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' What aspect of coding or learning to program gave you the most trouble?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' As a way to help organize your thinking, consider the assignment that you had the most difficulty with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Could the editor have done anything to help you with that?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (4) Debugging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Think about the experience of debugging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' How did you go about doing it?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' If you used console.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='log to help debug, did you find it helpful?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Did you use any other strategies?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Is there anything about it or the debugging process that you wish could have been different?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (5) Error Messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Think about the error messages you encountered (inline in the code box, in the console area under the code box, in the browser console, or elsewhere).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Were they useful?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' How did you deal with them?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Do you wish they were presented differently?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (6) Code Organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' How did you go about organizing your code?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' For instance, how did you decide where to place variables, create functions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Was there ever a point when your organizational scheme ran into problems, if so how did you handle it?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Is there anything the editor could have done to help you during these organizational tasks?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (7) Freeze Frame Homework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Think about the freeze frame assignment (or any other time during the course when you needed to repeatedly edit and re-run the code in order to get particular positions or other values to achieve a desired effect).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Is there anything the editor could have done to help you get your image to be just right?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (8) External Tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' It’s natural to use other tools as part of the programming process, such as color eye droppers or p5’s online documentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Do you think it would be useful to integrate these tools as part of the editor?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' What other tools can you imagine wanting to be part of your in-editor coding workflow?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (9) Desired Features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' What sorts of editor features might have allowed you to be more effective in your coding?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' What sorts of editor features might have allowed you to be more creative?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' A Study of Editor Features in a Creative Coding Classroom CHI ’23, April 23–28, 2023, Hamburg, Germany C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='3 Pages 3-18: Editor Features In this section, we’ll ask for your thoughts and opinions about some features that appeared in the editor as well as some hypothetical features that we may implement for future iterations of the course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (1) Autocomplete Imagine an editor feature which provides autocomplete suggestions as you type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' This would be akin to the predictive text feature found in many messaging applications, but would be sensitive to variables you’ve created and functions available from imported libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' While this feature appears in some other editors it did not appear in our editor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (a) Do you think this would be useful?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (1) Not Very Useful ⃝, (2) ⃝, (3) Neither useful nor unuseful ⃝, (4) ⃝, Very Useful ⃝ (b) How often do you think you would use this feature?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (1) Never ⃝, (2) ⃝, (3)Occasionally ⃝, (4) ⃝, All the time ⃝ (c) Why or why not?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Is there any way you would like to modify this feature to make it more useful?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (2) Linters Our editor featured a tool called a “linter” that surfaced stylistic or coding errors through on-screen alerts, as in the image below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' This tool is analogous to spell- and grammar-checkers in standard word processors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' The particular linter used in our editor, called JSHint, tends not to give many warnings for stylistic errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Other available linters give many more warnings for stylistic errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (a) Do you think this is useful?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (1) Not Very Useful ⃝, (2) ⃝, (3) Neither useful nor unuseful ⃝, (4) ⃝, Very Useful ⃝ (b) How often do you think you used this feature?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (1) Never ⃝, (2) ⃝, (3)Occasionally ⃝, (4) ⃝, All the time ⃝ (c) Why or why not?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Is there any way you would like to modify this feature to make it more useful?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (3) Tidy Code Our editor featured a button called “Tidy Code” which automatically reorganized your code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' This feature is sometimes seen in other editors and is more commonly known as an “auto-formatter”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' These can typically be configured to enforce a particular coding style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (a) Do you think this is useful?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (1) Not Very Useful ⃝, (2) ⃝, (3) Neither useful nor unuseful ⃝, (4) ⃝, Very Useful ⃝ (b) How often do you think you used this feature?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (1) Never ⃝, (2) ⃝, (3)Occasionally ⃝, (4) ⃝, All the time ⃝ (c) Why or why not?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Is there any way you would like to modify this feature to make it more useful?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=" 1 function hello() 2 alert('Hello world!" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content="') 3 let counter = 0;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 4 for(let idx = ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' idx < 10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' idx++) ( 5 counter += idx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 6 7 console.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='l 8 log method)Console.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='log(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='data:any):.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='. timeLog1vfunctionsetup()( 2 createCanvas(710,40o,WEBGL);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 3 4 5vfunctiondraw()( 6 background(250) × Missing semicolon rotateY(frameCount * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='01)s Expected an assignment or function call and instead saw an expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 8 9V for(letj=o;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='j<5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='j++)( 10 push();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 11 for(leti=0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='i<80;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='i++) 12 translate( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='2 100 Ap5 * cs11 File Edit Help 0 Tidy Code ++Tab duct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Find 3+FCHI ’23, April 23–28, 2023, Hamburg, Germany McNutt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (4) Auto-refresh There is a feature in our editor called “Auto-refresh.” When selected, it re-runs your code every time you finish typing (or sometimes before).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' This enables small update cycles as you code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (a) Do you think this is useful?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (1) Not Very Useful ⃝, (2) ⃝, (3) Neither useful nor unuseful ⃝, (4) ⃝, Very Useful ⃝ (b) How often do you think you used this feature?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (1) Never ⃝, (2) ⃝, (3)Occasionally ⃝, (4) ⃝, All the time ⃝ (c) Why or why not?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Is there any way you would like to modify this feature to make it more useful?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (5) Code Folding There is a feature in our editor, and many other editors, called “code folding” as in the screenshot below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' This allows you to collapse certain sections of code, such as functions and loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' The “folded” code is still there and can be referenced from other places, but it’s temporarily hidden and replaced with “.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='..”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (a) Do you think this is useful?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (1) Not Very Useful ⃝, (2) ⃝, (3) Neither useful nor unuseful ⃝, (4) ⃝, Very Useful ⃝ (b) How often do you think you used this feature?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (1) Never ⃝, (2) ⃝, (3)Occasionally ⃝, (4) ⃝, All the time ⃝ (c) Why or why not?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Is there any way you would like to modify this feature to make it more useful?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (6) Canvas Ruler Imagine a feature which allows you to place a draggable ruler into the drawing side of the editor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' You can use it as a way to visually identify screen coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' This feature might involve a way to display the current direction and placement of the coordinate origin, especially with regard to translation and rotation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (a) Do you think this would be useful?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (1) Not Very Useful ⃝, (2) ⃝, (3) Neither useful nor unuseful ⃝, (4) ⃝, Very Useful ⃝ (b) How often do you think you would use this feature?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (1) Never ⃝, (2) ⃝, (3)Occasionally ⃝, (4) ⃝, All the time ⃝ (c) Why or why not?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Is there any way you would like to modify this feature to make it more useful?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='js Saved: 2 minutes ago Preview const size=5o;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2 const gap = 10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 37 functionsetup()( 4 createCanvas(400,400);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 5 9 7vfunctiondraw()( 8 background(220);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 9 10V for（leti=0:i<10i++)( 11V for（letj=o;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='j<10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='j++) 12 fill((i+j)%2？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content="'black' 'white') 13 square(i *(size + gap),j *(size + gap), size);" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 14 15 16 17 Left ruleredge (0,170)Right ruler edge (400,170)p5 * cs111 File v Edit Help 0 Auto-refresh Sketch name Small mapusaurus functionsetupO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=') 4 5V function draw()( 6 background(25o);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' rotateY(frameCount * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='01);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 8 9V for (let j = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' j< 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' j++)( 10 push();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 11V for(leti=0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='i<80;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='i++)( 12 translate( 13 sin(frameCount *0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='001 +j)* 100 14 sin(frameCount * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='001 +j)* 100 15 i*0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='1 16A Study of Editor Features in a Creative Coding Classroom CHI ’23, April 23–28, 2023, Hamburg, Germany (7) Number Sliders Imagine a feature which allows you to modify the numeric values in the code without typing or re-running the program (such as in the image below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' With this feature, you click a value of interest and then drag a slider that appears above it to change it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' The canvas is continuously re-rendered as you drag the slider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' This would be similar to using p5’s slider function, but, rather than just changing the value in the running code, it would also modify the text of the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (a) Do you think this would be useful?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (1) Not Very Useful ⃝, (2) ⃝, (3) Neither useful nor unuseful ⃝, (4) ⃝, Very Useful ⃝ (b) How often do you think you would use this feature?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (1) Never ⃝, (2) ⃝, (3)Occasionally ⃝, (4) ⃝, All the time ⃝ (c) Why or why not?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Is there any way you would like to modify this feature to make it more useful?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (8) Color Picker Imagine having a color picker integrated into the editor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' When selecting color values in the code, the color picker could appear on hover (as in the image below) to modify the value, or the tool could be docked into the bottom of the editor (allowing it to be always on).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' This could include pre-configured or document-based palettes, as in Illustrator or Photoshop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (a) Do you think this would be useful?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (1) Not Very Useful ⃝, (2) ⃝, (3) Neither useful nor unuseful ⃝, (4) ⃝, Very Useful ⃝ (b) How often do you think you would use this feature?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (1) Never ⃝, (2) ⃝, (3)Occasionally ⃝, (4) ⃝, All the time ⃝ (c) Why or why not?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Is there any way you would like to modify this feature to make it more useful?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' > sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='js Saved: 2 minutes ago Preview 1 1 const size = 50;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 2 const gap = 10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 3V+ functionsetup 4 createCanvas(400,400);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 5 口 6 7vfunctiondraw() 8 background(220);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 10V for（leti=0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='i<10:i++)( 11V for(letj=o;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='j<10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='j++)( 12 fill((i+j)%2？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content="'black' 'white');" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 13 square(i * (size + gap),j * (size + gap), size);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 14 t 15 16 17sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='js* esago Preview const size = 5o;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' const gap = 10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 3V functionsetup(） 4 createCanvas(400,400);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 5 6 7vfunction draw()( 8 background(220);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' #000000 100 10V for(leti=0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='i<10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' HEX H S Alpha 11V for(letj=0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='j<10 12 fill((i+j)%2？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='\'black\' "white\');' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 13 square(i * (size + gap),j * (size + gap), size) 14 15 7 16 17CHI ’23, April 23–28, 2023, Hamburg, Germany McNutt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (9) p5 State Displays In p5 it is common to set values for variables such as strokeWidth (which describes the width of subsequently drawn lines), or fill (which describes the interior color of subsequently drawn shape).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' These are examples of ""state variables"".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' There are a variety of such variables in p5, however (in contrast with digital drawing tools like Photoshop), these variables are not displayed anywhere in the editor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='

Imagine a feature where all of the relevant state values are shown, such that when you move the text cursor to a line in your code, the display shows the state values at that point in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' This would allow you to evaluate if your drawing tools are configured as you want them to be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (a) Do you think this would be useful?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (1) Not Very Useful ⃝, (2) ⃝, (3) Neither useful nor unuseful ⃝, (4) ⃝, Very Useful ⃝ (b) How often do you think you would use this feature?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (1) Never ⃝, (2) ⃝, (3)Occasionally ⃝, (4) ⃝, All the time ⃝ (c) Why or why not?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Is there any way you would like to modify this feature to make it more useful?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (10) Interactive Value Inspector Imagine a feature which allows you to see the value of the current program execution by hovering over chunks of the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' It would provide similar information as when inserting console.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='log statements into your code, but instead you would extract that same information through hovering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' In contrast with ""p5 State Displays"" (which only shows p5 state variables like fill and strokeWidth) this feature would allow you to see both state variables as well as the value of all variables, including ones you’ve defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' This information could be presented through a tooltip (as in the below image) or through a docked panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (a) Do you think this would be useful?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (1) Not Very Useful ⃝, (2) ⃝, (3) Neither useful nor unuseful ⃝, (4) ⃝, Very Useful ⃝ (b) How often do you think you would use this feature?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (1) Never ⃝, (2) ⃝, (3)Occasionally ⃝, (4) ⃝, All the time ⃝ (c) Why or why not?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Is there any way you would like to modify this feature to make it more useful?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Auto-refresh Sketch name Woodcushion SUBMIT 8 Sketch Files < sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='js` Preview index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='html sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='js 1vfunctionsetup()( D style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='css 2 createCanvas(360,280);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 3 nostroke();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 4 noLoop();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 5 6 77 functiondraw()( 8 drawcircle(width/2,280/2,6);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 9 10 P5 State Values 11 function drawcircle(x,radius,level)( 12 consttt=（126*leve1)/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Stroke None 13 fill(tt);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Stroke Weiqht 1px 14 ellipse( height/2,radius*2,radius*2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Stroke Cap Round 15V if (leve) 1) ( Fill Multiple 16 level =Ievel - 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Translate 0, 0 17 drawcircle(x-radius /2,radius /2,level) 18 Rotation drawcircle(x + radius/2,radius/2,level) 19 20 21 22 Watch additional values + Console< sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='js Saved: 9 days ago Preview 175 //Acounterto help safeguardagainstinfiniteloops 176 //during development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Youmaytry adjusting this 177 //valueif there is toolittleflooding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 178 let fuel = 100000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 179 180vwhile(worklist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='length>0&&fuel>0)( 181 //Gettheposition atthefront oftheworklist 182 const pos=worklist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='shift();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 183 const x = pos[o];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 184 const y= pos[1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 185 186 const withinBounds=!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (x<0Ilx>img.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='width Ilyimg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='height);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 187 constnotAlreadyFilled=!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content="alreadyFilled['$(x}-$(y}'J;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 188 189V if(withinBounds&¬AlreadyFilled)( 190 const c = img.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='get(x,y);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 191 const [r,g,b,a］ = c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' C: [255,0,0,0] 192 constiswhitish=r>240&&g>240&&b>240;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' r: 255 193 constisTransparent=a<5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=" g: 0 194V if(iswhitis isTransparent) 195 ['$(x}-$ty=true;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' b: 0 alreadyFil 196 img.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='set(x, ，color(fiilcolor));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' a: 0 197 iswhitish: true 198 //Addneighboringpixelstoendofworklist isTransparent: false 199 worklist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='push([x +1,yJ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='A Study of Editor Features in a Creative Coding Classroom CHI ’23, April 23–28, 2023, Hamburg, Germany (11) In-context Docs Imagine an editor feature which gives you access to the documentation while you are writing code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' This might involve a tooltip that appears on hover (as in the image below) which describes the usage of a particular function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' It could also involve showing the description in a dedicated pane on the side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' While such features appear in some other editors it did not appear in our editor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (a) Do you think this would be useful?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (1) Not Very Useful ⃝, (2) ⃝, (3) Neither useful nor unuseful ⃝, (4) ⃝, Very Useful ⃝ (b) How often do you think you would use this feature?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (1) Never ⃝, (2) ⃝, (3)Occasionally ⃝, (4) ⃝, All the time ⃝ (c) Why or why not?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Is there any way you would like to modify this feature to make it more useful?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (12) Code Snippet Templates Imagine a feature which allows you to paste in common code snippets from a list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' After clicking one of the desired options (such as in the image below) a piece of code achieving that functionality will be added to your code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' These snippets could include small structures, such as for-loops, or larger structures, such as particular API uses or classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' This feature sometimes appears in other coding systems, but was not implemented in our system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (a) Do you think this would be useful?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (1) Not Very Useful ⃝, (2) ⃝, (3) Neither useful nor unuseful ⃝, (4) ⃝, Very Useful ⃝ (b) How often do you think you would use this feature?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (1) Never ⃝, (2) ⃝, (3)Occasionally ⃝, (4) ⃝, All the time ⃝ (c) Why or why not?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Is there any way you would like to modify this feature to make it more useful?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (13) Coding by Drawing Tools Imagine an editor feature which allows you to fill out the arguments to particular functions graphically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' For instance, you might indicate to the editor that you are interested in drawing a bezier curve, and then draw each of the vertices in the curve directly on the editor, just as you would in a GUI-based tool like Illustrator, which in turn inserts a corresponding line of bezier command in your code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Unlike in the previous feature, which just inserted code templates, this feature allows you to specify the values of the inserted code with your mouse on the output canvas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (a) Do you think this would be useful?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (1) Not Very Useful ⃝, (2) ⃝, (3) Neither useful nor unuseful ⃝, (4) ⃝, Very Useful ⃝ (b) How often do you think you would use this feature?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (1) Never ⃝, (2) ⃝, (3)Occasionally ⃝, (4) ⃝, All the time ⃝ (c) Why or why not?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Is there any way you would like to modify this feature to make it more useful?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' background(colorstring, [a]) background(gray,[a]) background(v1,v2,v3,[a]) background(values) background(image,[a]) 1function se The background() function sets the color used for the background of the p5js canvas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' The default background is "t transparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' This function is typically used within draw() to 5V function dr clear the display window at the beginning of each frame can be usedlinside setup( to set the backaround o 6 background(250);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 7 rotateY(frameCount*0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='01);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 8 for letj=;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='j<5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='j++)( 10 push();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 11V for(leti=0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='i<80;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='i++)( 12 translate( 13 sin(frameCount*0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='001+j)*100 14 sin(frameCount*0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='001+j)*100 15 i*0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='1 16 Auto-refresh Sketch name Wood cushion SUBMIT Sketch Files V sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='js Preview index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='html sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='js 1functionsetup() D style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='css 2 createCanvas(400,400);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 3 t 4 5V function draw()( 9 background(220);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 7 bezier(x1, y1, x2, y2, x3, y3, x4, y4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 8 7 9 Code Snippets 10 Filter code snippets 11 Add color Add colored rectangle Add beziercurve 4 Insertedlineofcode Add mouseDrags /ent Add mouseClicked event > Load image Access web cam Adid custom snippet + Console> sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='js Saved: 2 minutes ago Preview Drawing tools NTHE+ Additional drawing tools 1function setup()( click Add Slider + 2 createCanvas(400,400);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Add Button + 7 click 4 Add Radio + 5function draw()( Adid Vector Shane 6 background(220);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 7 noFiil();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 8 stroke(255,102,0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 9 10 stroke(0,0,);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 11 bezier(285,20,10,10,90,90,15,80);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 12 13 click click Insertedlineofcode ConsoleCHI ’23, April 23–28, 2023, Hamburg, Germany McNutt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (14) Linked Copy-and-Paste A common abstraction mechanism that we used in class is to create variables or functions rather than copy-pasting chunks of code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' While variables and functions are a useful form of computational thinking, there are other ways to approach this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='

Imagine a feature which keeps track of your copy-and-pastes: whenever you edit a value you’ve copied and pasted, all pieces of code which were copied are also changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' This special linked copy-paste can be selectively turned on and off so that you can make edits without changing all copies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (a) Do you think this would be useful?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (1) Not Very Useful ⃝, (2) ⃝, (3) Neither useful nor unuseful ⃝, (4) ⃝, Very Useful ⃝ (b) How often do you think you would use this feature?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (1) Never ⃝, (2) ⃝, (3)Occasionally ⃝, (4) ⃝, All the time ⃝ (c) Why or why not?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Is there any way you would like to modify this feature to make it more useful?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (15) Drag-and-Drop Refactoring Refactoring is the process of changing the way a piece of code is organized such that the functionality remains the same, but the code is easier to work with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' You probably did this during the course by making a variable to capture repeated code or by creating a function to represent some repeated functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='

Imagine a feature which allows you to click and drag values to create arguments, variables, and functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' This might allow you to reorder lines of code by clicking and dragging them, or to highlight a series of repeated values and drag them to automatically create a new variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (a) Do you think this would be useful?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (1) Not Very Useful ⃝, (2) ⃝, (3) Neither useful nor unuseful ⃝, (4) ⃝, Very Useful ⃝ (b) How often do you think you would use this feature?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (1) Never ⃝, (2) ⃝, (3)Occasionally ⃝, (4) ⃝, All the time ⃝ (c) Why or why not?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Is there any way you would like to modify this feature to make it more useful?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (16) Time Travel Slider Imagine an editor feature which allows you to go back to earlier points in time of your code’s execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' With such a feature you’d press Play, as normal, and watch your code execute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' If there was an intermediate state you were curious about you can pause the execution and go back (by dragging a slider) to an earlier state of the canvas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Once you are finished inspecting you can resume execution without rerunning the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' This would allow you to inspect how your code was adding shapes to the canvas over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (a) Do you think this would be useful?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (1) Not Very Useful ⃝, (2) ⃝, (3) Neither useful nor unuseful ⃝, (4) ⃝, Very Useful ⃝ (b) How often do you think you would use this feature?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (1) Never ⃝, (2) ⃝, (3)Occasionally ⃝, (4) ⃝, All the time ⃝ (c) Why or why not?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Is there any way you would like to modify this feature to make it more useful?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='js Saved: 2 minutes ago Preview constsize=50;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' const gap =10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 3V function setup()( createCanvas(400,400);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' function draw() 8 background(220);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 9 10V for(leti=0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='i<10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='i++)( 11V for(letj=o;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='j<1o;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='j++) 12 fill((i+j)%2?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='black\' "white\';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 13 square(i * (size + gap),j * (size + gap), size);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 14 15 16 17 Frame 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='2ksketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='js Saved: 13 minutes ago Preview 1functionsetup()( 2 createCanvas(360,280);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' nostroke();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 4 noLoop();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 5 6 7vfunction draw()( 8 drawcircle(width /2,280/2,6);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 6 10 11V functiondrawcircle(x,radius,level)( 12 consttt=(126*level)/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 13 fill(tt);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 14 ellipse(x, height/2,radius*2,radius*2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 15V if(level>1)( 16 level = level -1 17 drawcircle(x - radius / 2， radius / 2, level);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 18 drawcircle(x+radius/2, radius/ level);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 19 20 21 23 Thepurplevalueswerecopy+pastedfrom thegreenvalue,linkingthosevalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='Achange Console to any linked value will change all linked values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='sketchjs Saved: 24 minutes ago Preview 17 function setup()( 2 createCanvas(360,280);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 3 nostroke();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 4 noLoop();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 5 7 6 77 functiondraw()( 8 drawcircle(width/2,280/2,6);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 9 10 11V function drawcircle(x,radius,level)( 12 consttt=（126*level)/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 13 fill(tt);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 14 ellipse(x,height/2,radius*2,radius *2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 15V if(level>1) 16 level = level - 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 17 drawCircle(x + radius / 2, radius / 2,level) 18 drawcircle(x -radius /2,radius / 2,level);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Drag a line of code to move it 19 drawCircle(x radius radius Level 20 21 22 23 ConsoleA Study of Editor Features in a Creative Coding Classroom CHI ’23, April 23–28, 2023, Hamburg, Germany (17) Directly Manipulate Shape Attributes on Canvas Imagine being able to edit the output canvas and have that change the corresponding JavaScript code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' This would involve making changes to specific values graphically, such as changing the size of circle or end points of lines by dragging them to a desired position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' This differs from the functionality of the previously described ""Code by Drawing Tools"" feature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' that one allowed clicking and dragging to add new shapes to the code, whereas this one allows clicking and dragging to dynamically update and modify existing shapes in the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (a) Do you think this would be useful?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (1) Not Very Useful ⃝, (2) ⃝, (3) Neither useful nor unuseful ⃝, (4) ⃝, Very Useful ⃝ (b) How often do you think you would use this feature?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' (1) Never ⃝, (2) ⃝, (3)Occasionally ⃝, (4) ⃝, All the time ⃝ (c) Why or why not?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' Is there any way you would like to modify this feature to make it more useful?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' > sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='js Saved: 15 seconds ago Preview 5functiondraw()( 6 background(102);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 7 8 push();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 9 translate(width*0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='2,height*0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='5): 10 rotate(30);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 11 star(0,0,5 Wethinkyoumeanforthis 12 pop();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' shape to be rotated, but 13 perhapsyou meanforittobe 14 push();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' translated?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' If so, click here 15 translate(wi 16 star(0,0,80,100,40);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 17 pop();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 18 19 push();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 20 transiate(width * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='8, height * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 21 star(0,0,30,70,5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 22 pop();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 23 24 25vfunction star(x,y,radius1,radius2,npoints)( 26 letangle=Two_PI/npoints;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 27 let halfAngle = angle / 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 28 beginShape（);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content=' 29V for(leta=o;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FQT4oBgHgl3EQfVza8/content/2301.13302v1.pdf'} +page_content='a n2|τ|. That +is, vT lies in the interior of hτ. +Lemma 4.1 follows from a standard induction argument. +Proof of Lemma 4.2. If T is not a collection of tubes that do proper tubing, then at least +one of the following two cases holds: +(1) There is a pair of non-nested and non-disjoint tubes τ1, τ2 in T. +(2) There is a sequence of disjoint tubes τ1, ..., τk such that τ1 ≺ · · · ≺ τk ≺ τ1. +The idea of the proof is as follows: For S ⊆ P, define the convex hull of S as +conv(σ) := {b ∈ P | ∃a, c ∈ S : a ≤ b ≤ c}. +Observe that if p ∈ L (P), then diamS(p) ≤ diamconv(S)(p). Take σ = conv(τ1 ∪ · · · ∪ τk). +One can show that σ is a tube, so Lemma 3.3 tells us that for each τi, diamτi(p) is very +small compared to n2|σ|. As the tubes either intersect or are cyclic, one can show this forces +diamσ(p) to also be small, so ασ(p) < n2|σ|. +More concretely, suppose that +p ∈ +� +Hτi ∩ L (P). +Note that for all i, |σ| > |τi| + 1 and diamτi(p) ≤ n2(|σ|−1). In case (1), let a, b ∈ σ. There +exists some x ∈ τ1 ∩ τ2, so +|pa − pb| ≤ |pa − px| + |px − pb| +≤ diamτ1(p) + diamτ2(p) +≤ 2n2(|σ|−1) +< n2(|σ|). +Hence diamσ(p) < n2|σ|, so by Lemma 3.3, p /∈ hσ. +Now we move to case (2). Suppose there is a sequence of disjoint tubes τ1, ..., τk such that +for each i there exists xi, yi ∈ τi where xi ≺ yi+1 where we take the indices mod k. Then: +pyi − diamτi(p) ≤ pxi +pxi ≤ pyi+1 +pyi+1 ≤ pxi+1 + diamτi+1 +Furthermore, since τi and τi+1 are disjoint, |τi| ≤ |σ|−2 and diamτi ≤ n2(|σ|−2). Combining +these we get +pyi ≤ pyi+1 + 2n2(|σ|−2). +Then we have: +py1 ≤ pyi + 2in2(|σ|−2) +and +pyi + 2in2(|σ|−2) ≤ py1 + 2(k + 1)n2(|σ|−2). +7 + +4 +5 +2 +1 +3 +6 +7 +8 +9 +τ +4 +5 +2 +1 +3 +6 +7 +8 +9 +τ +σ +A +B +Maximal Tubing T and tube τ +σ, A, and B labelled +Figure 5. An example illustrating the proof of Lemma 4.3. +A +B +σ +τ +Figure 6. +If diamA(p) and diamB(p) are small and diamσ(p) is large, then +diamτ(p) is large. +These yield +py1 − pyi ≤ 2in2(|σ|−2) +and +pyi − py1 ≤ 2(k − i + 1)n2(|σ|−2). +As i, k − i + 1 ≤ k ≤ n +2, we have |py1 − pyi| ≤ n2(|σ|−1). Finally, if zi ∈ τi, zj ∈ τj, then +|pzi − pzj| ≤ |pzi − pyi| + |pyi − py1| + |py1 − pyj| + |pyj − pzj| +≤ 4n2(|σ|−1) +< n2|σ|. +Hence diamσ(p) < n2|σ|, and by Lemma 3.3, p /∈ hσ. +□ +Proof of Lemma 4.3. Let T be a maximal tubing of P and let τ /∈ T be a tube. Define the +convex hull of τ relative to T by +convT(τ) := min{σ ∈ T | τ ⊂ σ}. +Let σ = convT(τ). T partitions σ into a lower set A and an upper set B where A and B +are either tubes or singletons. Furthermore, A and B both intersect τ. See Figure 5 for an +example illustrating this. +The idea of the proof is as follows: Let p = vT. By Lemma 3.3, diamA(p) and diamB(p) +are both very small compared to diamσ(p). Then for any a ∈ A, b ∈ B, |pa − pb| must be +large. As τ intersects both A and B, diamτ(p) must be large and hence p ∈ hτ. See Figure 6 +for an illustration of this. More precisely, we show that for any i ∈ A, j ∈ B, pj −pi > (n2)|τ|, +which implies that p lies in the interior of hτ. +8 + +Observe that:� +x≺·y +py − px = +� +x≺·y +x,y∈A +(py − px) +� +�� +� +≤(n2)|σ|−1 +< 1 +8 (n2)|σ| ++ +� +x≺·y +x,y∈B +(py − px) +� +�� +� +≤(n2)|σ|−1 +< 1 +8 (n2)|σ| ++ +� +x≺·y +x∈A,y∈B +(py − px). +Fix i ∈ A and j ∈ B. By Lemma 3.3, for any x ∈ A, y ∈ B, +py − px ≤ pj − pi + diamA(p) + diamB(p) +≤ pj + pi + 2n2(|σ|−1). +Again, noting that the number of covering relations in σ is at most n2 +4 we obtain: +� +x≺·σy +x∈A,y∈B +(py − px) ≤ +� +x≺·σy +x∈A,y∈B +(pj − pi + 2(n2)|σ|−1) +≤ n2 +4 +� +pj − pi + 2(n2)|σ|−1� += n2 +4 (pj − pi) + 1 +2(n2)|σ|. +Combining all of this we get: +� +x≺·σy +py − px = (n2)|σ| +< 1 +8(n2)|σ| + 1 +8(n2)|σ| + 1 +2(n2)|σ| + n2 +4 (pj − pi) +≤ 3 +4(n2)|σ| + n2 +4 (pj − pi) +Then (n2)|σ|−1 < (pj − pi) and as |τ| ≤ |σ| − 1, p is in the interior of hτ. +□ +Remark 4.4. A similar approach for realizing graph associahedra is taken by Devadoss [4]. +One difference is that Devadoss realizes graph associahedra by cutting off slices of a simplex +whereas we cut off slices of an order polytope. +5. Realizing affine poset cyclohedra +The proofs in the affine case are nearly identical to the finite case with some additional +technical components. The similarity comes from the fact that Lemma 3.3 still applies. We +highlight where the proofs are different. Let ˜P be an affine poset of order n. +Define +C ( ˜P) := +� +σ⊂P +hσ +and +L ( ˜P) := {p ∈ R +˜P/C | pi ≤ pj for all i ⪯ j}. +where the intersection is over all tubes of ˜P. Note that C ( ˜P) ⊆ L ( ˜P) as if i ≺· j is a +covering relation, then for p ∈ h{i,j}, pi ≤ pj. Theorem 2.8 follows as a result of 3 lemmas: +9 + +Lemma 5.1. If T is a maximal tubing, then +vT := +� +τ∈T +Hτ +is a point. +Lemma 5.2. If T is a collection of tubes that do not form a proper tubing, then +� +τ∈T +Hτ ∩ C ( ˜P) = ∅. +Lemma 5.3. If T is a maximal tubing and τ /∈ T is a proper tube, then ατ(vT) > n2|τ|. That +is, vT lies in the interior of hτ. +Proof of Lemma 5.1. Let T be a maximal tubing and take any σ ∈ T such that |τ| = n. +Then restricting to ˜P|σ, Lemma 4.1 implies that +� +τ∈T +τ⊆σ +Hτ +is a point. However, as T is n-periodic, +� +τ∈T +τ⊆σ +Hτ = +� +τ∈T +Hτ. +□ +Proof of Lemma 5.2. By Remark 2.7, we can assume T is n-periodic. The proof is almost +identical to the proof of Lemma 4.2. Define +L ( ˜P) := {p ∈ R +˜P/C | pi ≤ pj for all i ⪯ j}. +and note that +L ( ˜P) ⊆ R +˜P/C +� +i,j∈ ˜P +i≺·j +h{i,j}. +Let +p ∈ +� +Hτi ∩ L ( ˜P). +We again break into two cases: +(1) There is a pair of non-nested and non-disjoint tubes τ1, τ2 in T. +(2) All tubes in T are pairwise nested or disjoint and there is a sequence of disjoint tubes +τ1, ..., τk such that τ1 ≺ · · · ≺ τk ≺ τ1. +The only difference in the proof occurs in case (1). Here, it is possible that there exists +x ∈ τ1 ∪τ2 such that x+n ∈ τ1 ∪τ2 as well. In this case, the proof of Lemma 4.2 still implies +that diamτ1∪τ2(p) ≤ diamτ1(p) + diamτ2(p) ≤ 2n2n. However, |px+n − px| = n2(n+1). +□ +Proof of Lemma 5.3. Let T be a maximal tubing and τ /∈ T be a proper tube. Let p = vT. +We claim that ατ(p) > n2|τ|. +The only difference from the proof of Lemma 4.3 is that τ may not be contained by any +tube in τ so convT(τ) may not be well-defined. In this case, there exists A ∈ T such that +10 + +ˆ0 +a +b +c +ˆ1 +P +ε = 1 +3 +ε = 1 +9 +ε = +1 +27 +Figure 7. +O(P) as a limit of A (P) +|A| = n, A ∩ τ ̸= ∅, and (A + n) ∩ τ ̸= ∅. Here, (A + n) acts the same as B in the finite +case, except the argument is much simpler. +Let i ∈ A ∩ τ, j ∈ (A + n) ∩ τ. Observe that diamA(p), diam(A+n)(p) ≤ n2n and that +i + n ∈ (A + n). Then +|pj − pi| ≥ (pj − n2n) − pi +≥ pi+n − pi += n2(n+1). +Hence diamτ(p) > n2|τ| and by Lemma 3.3, ατ(p) > n2|τ|. +□ +6. Remarks and Questions +Remark 6.1. Let (P, ⪯) be a bounded poset. In Remark 3.1, we discuss how O(P) can be +realized as the set of all p ∈ RP such that pˆ0 = 0, pˆ1 = 1, and pi ≤ pj whenever i ⪯ j. We +can similarly realize A (P) as follows: Fix 0 < ε < +1 +n2. +For a proper tube τ ⊂ P, let +h′ +τ = {p ∈ RP | ατ(p) < εn−|τ|}. +Then A (P) is realized as the intersection over all h′ +τ with the hyperplanes +{pˆ0 = 0} and {pˆ1 = 1}. +Letting ε → 0, we obtain O(P) as a limit of A (P) as shown in Figure 7. +Remark 6.2. The key piece to the realizations in Theorems 2.3 and 2.8 is the linear form +ατ, where ατ acts as an approximation of diamτ. In particular, let τ be a tube and let +p ∈ L (P). Then: +• ατ(p) ≥ 0. +• ατ(p) = 0 ⇔ p|τ is constant. +• If σ ⊆ τ is a tube, then ασ(p) ≤ ατ(p). +However, there are many other options for choice of ατ that could fill this role. Some other +options include: +11 + +(1) Sum over all pairs i ≺ j in τ. +ατ(p) = +� +i≺j +i,j∈τ +pj − pi. +(2) Let A and B be the set of minima and maxima of the restriction P|τ respectively. +ατ(p) = +� +i≺j +i∈A,j∈B +pj − pi. +(3) Fix a spanning tree T in the Hasse diagram of τ. Let E = {(i, j) | i ≺·T j} be the set +of edges in T. +ατ(p) = +� +(i,j)∈E +pj − pi. +An advantage of this option is that we would have +diamτ(p) ≤ ατ(p) ≤ (n − 1) diamτ(p). +A similar realization can be obtained for each choice of of ατ. +Question 6.3. Recall that for a simple d-dimensional polytope P, the f-vector and h-vector +of P are given by (f0, . . . , fd) and (h0, . . . , hd) where fi is the number of i-dimensional faces +and +d +� +i=0 +fiti = +d +� +i=0 +hi(t + 1)i. +Postnikov, Reiner, and Williams [12] found a statistic on maximal tubings of graph associa- +hedra of chordal graphs where +� +T +tstat(T) = +� +hiti. +In particular, they define a map T �→ wT from maximal tubings of a graph on n vertices +to the set of permutations Sn such that stat(T) = des(wT), the number of descents of wT. +It would be interesting to find a similar statistic on maximal tubings of poset associahedra. +For a simple polytope P, one can orient the edges of P according to a generic linear form +and take stat(v) = outdegree(v) [17, §8.2]. It may be possible to use our realization to find +the desired statistic. +Acknowledgements +The author is grateful to Pavel Galashin for his many helpful comments and suggestions +and to Stefan Forcey for fruitful conversations. +References +[1] +Scott Axelrod and Isadore M Singer. “Chern-Simons perturbation theory. II”. In: Jour- +nal of Differential Geometry 39.1 (1994), pp. 173–213. +[2] +Raoul Bott and Clifford Taubes. “On the self-linking of knots”. In: Journal of Mathe- +matical Physics 35.10 (1994), pp. 5247–5287. +[3] +Michael Carr and Satyan L Devadoss. “Coxeter complexes and graph-associahedra”. +In: Topology and its Applications 153.12 (2006), pp. 2155–2168. +12 + +[4] +Satyan L Devadoss. “A realization of graph associahedra”. In: Discrete Mathematics +309.1 (2009), pp. 271–276. +[5] +Sergey Fomin and Nathan Reading. “Root systems and generalized associahedra”. In: +arXiv preprint math/0505518 (2005). +[6] +Pavel Galashin. “Poset associahedra”. In: arXiv preprint arXiv:2110.07257 (2021). +[7] +Mark Haiman. “Constructing the associahedron”. In: Unpublished manuscript, MIT +(1984). +[8] +Pascal Lambrechts, Victor Turchin, and Ismar Voli´c. “Associahedron, cyclohedron and +permutohedron as compactifications of configuration spaces”. In: Bulletin of the Belgian +Mathematical Society-Simon Stevin 17.2 (2010), pp. 303–332. +[9] +Guillaume Laplante-Anfossi. “The diagonal of the operahedra”. In: Advances in Math- +ematics 405 (2022), p. 108494. +[10] +Chiara Mantovani, Arnau Padrol, and Vincent Pilaud. “Acyclonestohedra: when ori- +ented matroids meet nestohedra”. in prep. +[11] +Kyle Petersen. Eulerian Numbers. Oct. 2015. isbn: 978-1-4939-3090-6. doi: 10.1007/ +978-1-4939-3091-3. +[12] +Alex Postnikov, Victor Reiner, and Lauren Williams. “Faces of Generalized Permuto- +hedra”. In: Documenta Mathematica 13 (2008), pp. 207–273. +[13] +Dev P Sinha. “Manifold-theoretic compactifications of configuration spaces”. In: Selecta +Mathematica 10.3 (2004), pp. 391–428. +[14] +Richard P Stanley. “Two poset polytopes”. In: Discrete & Computational Geometry +1.1 (1986), pp. 9–23. +[15] +Jim Stasheff. “From Operads to ‘Physically’ Inspired Theories”. In: (Sept. 1996). +[16] +Dov Tamari. “Mono¨ıdes pr´eordonn´es et chaˆınes de Malcev”. In: Bulletin de la Soci´et´e +math´ematique de France 82 (1954), pp. 53–96. +[17] +G¨unter M Ziegler. Lectures on polytopes. Vol. 152. Springer Science & Business Media, +2012. +Department of Mathematics, University of California, Los Angeles, CA 90095, USA +Email address: andrewsack@math.ucla.edu +13 + diff --git a/NdFJT4oBgHgl3EQfGyyu/content/tmp_files/load_file.txt b/NdFJT4oBgHgl3EQfGyyu/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..33cd4220111147f347dcc72ad2bd8f5eac3065e7 --- /dev/null +++ b/NdFJT4oBgHgl3EQfGyyu/content/tmp_files/load_file.txt @@ -0,0 +1,438 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf,len=437 +page_content='A REALIZATION OF POSET ASSOCIAHEDRA ANDREW SACK Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Given any connected poset P, we give a simple realization of Galashin’s poset associahedron A (P) as a convex polytope in RP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' The realization is inspired by the de- scription of A (P) as a compactification of the configuration space of order-preserving maps P → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' In addition, we give an analogous realization for Galashin’s affine poset cyclohedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Introduction Given a finite connected poset P, the poset associahedron A (P) is a simple, convex polytope of dimension |P| − 2 introduced by Galashin [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Poset associahedra arise as a natural generalization of Stasheff’s associahedra [7, 11, 15, 16], and were originally discovered by considering compactifications of the configuration space of order-preserving maps P → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' These compactifications are generalizations of the Axelrod–Singer compactification of the configuration space of points on a line [1, 8, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Galashin constructed poset associahedra by performing stellar subdivisions on the polar dual of Stanley’s order polytope [14], but did not provide an explicit realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Various poset associahedra and cyclohedra have already been studied including permutohedra, associahedra, operahedra [9], type B permutohedra [5], and cyclohedra [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Poset associahedra bear resemblance to graph associahedra, where the face lattice of each is described by a tubing criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' However, neither class is a subset of the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' When Carr and Devadoss introduced graph associahedra in [3], they distinguish between bracketings and tubings of a path, where the idea of bracketings does not naturally extend to any simple graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' In the case of poset associahedra, the idea of bracketings does extend to every connected poset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Galashin [6] also introduces affine posets, and analagous affine order polytopes and affine poset cyclohedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' In this paper, we provide a simple realization of poset associahedra and affine poset cyclohedra as an intersection of half spaces, inspired by the compactification description and by a similar realization of graph associahedra due to Devadoss [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' In inde- pendent work [10], Mantovani, Padrol, and Pilaud found a realization of poset associahedra as sections of graph associahedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' The authors of [10] also generalize from posets to oriented building sets (which combine a building set with an oriented matroid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Date: January 30, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Poset, associahedron, cyclohedron, realization, configuration space, compactification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' This material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' DGE-2034835.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Any opinions, findings, and conclusions or recommen- dations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='11449v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='CO] 26 Jan 2023 1 2 3 4 5 1 2 3 4 1 2 3 4 5 1 2 3 4 5 1 2 3 4 1 2 3 4 5 Examples Non-examples Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Examples and non-examples of proper tubings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Background 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Poset Associahedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' We start by defining the poset associahedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Let (P, ⪯) be a finite poset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' We make the following definitions: A subset τ ⊆ P is connected if it is connected as an induced subgraph of the Hasse diagram of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' τ ⊆ P is convex if whenever a, c ∈ τ and b ∈ P such that a ⪯ b ⪯ c, then b ∈ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' A tube of P is a connected, convex subset τ ⊆ P such that 2 ≤ |τ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' A tube τ is proper if |τ| ≤ |P| − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Two tubes σ, τ ⊆ P are nested if σ ⊆ τ or τ ⊆ σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Tubes σ and τ are disjoint if τ ∩ σ = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' For disjoint tubes σ, τ we say σ ≺ τ if there exists a ∈ σ, b ∈ τ such that a ≺ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' A proper tubing T of P is a set of proper tubes of P such that any pair of tubes is nested or disjoint and the relation ≺ extends to a partial order on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' That is, whenever τ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' , τk ∈ T with τ1 ≺ · · · ≺ τk then τk ̸≺ τ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' This is referred to as the acyclic tubing condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' A proper tubing T is maximal if it is maximal by inclusion on the set of all proper tubings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Figure 1 shows examples and non-examples of proper tubings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' For a finite poset P, the poset associahedron A (P) is a simple, convex polytope of dimension |P| − 2 whose face lattice is isomorphic to the set of proper tubings ordered by reverse inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' That is, if FT is the face corresponding to T, then FS ⊂ FT if one can make S from T by adding tubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Vertices of A (P) correspond to maximal tubings of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' We realize poset associahedra as an intersection of half-spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Let P be a finite poset and let n = |P|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' We work in the ambient space RP Σ=0, the space of real-valued functions on P that sum to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' For a subset τ ⊆ P, define a linear function ατ on RP Σ=0 by ατ(p) := � i≺·j i,j∈τ pj − pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Here the sum is taken over all covering relations contained in τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' We define the half-space hτ and the hyperplane Hτ by hτ := {p ∈ RP Σ=0 | ατ(p) ≥ n2|τ|} and Hτ := {p ∈ RP Σ=0 | ατ(p) = n2|τ|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' The following is our main result in the finite case: 2 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 11 12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 11 12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' ˜P A maximal tubing of ˜P Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' An affine poset of order 4 and a maximal tubing Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' If P is a finite, connected poset, the intersection of HP with hτ for all proper tubes τ gives a realization of A (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Affine Poset Cyclohedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Now we describe affine poset cyclohedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' An affine poset of order n ≥ 1 is a poset ˜P = (Z, ⪯) such that: (1) For all i ∈ Z, i ⪯ i + n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' (2) ˜P is n-periodic: For all i, j ∈ Z, i ⪯ j ⇔ i + n ⪯ j + n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' (3) ˜P is strongly connected: for all i, j ∈ Z, there exists k ∈ Z such that i ⪯ j + kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' The order of ˜P is denoted | ˜P| := n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Following Galashin [6], we give analagous versions of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' We give them only where they differ from the finite case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Let ˜P = (Z, ⪯) be an affine poset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' A tube of ˜P is a connected, convex subset τ ⊆ P such that 2 ≤ |τ| and either τ = ˜P or τ has at most one element in each residue class modulo n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' A collection of tubes T is n-periodic is for all τ ∈ T, k ∈ Z, τ + kn ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' A proper tubing T of ˜P is an n-periodic set of proper tubes of ˜P that satisfies the acyclic tubing condition and such that any pair of tubes is nested or disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Figure 2 gives an example of an affine poset of order 4 and a maximal tubing of that poset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' For an affine poset ˜P, the affine poset cyclohedron C ( ˜P) is a simple, convex polytope of dimension | ˜P| − 1 whose face lattice is isomorphic to the set of proper tubings ordered by reverse inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Vertices of C ( ˜P) correspond to maximal tubings of ˜P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' We also realize affine poset cyclohedra as an intersection of half-spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Let ˜P be an affine poset and let n = | ˜P|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Fix some constant c ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' We define the space of affine maps R ˜P as the set of bi-infinite sequences ˜x = (˜xi)i∈Z such that ˜xi = ˜xi+n + c for all i ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Let C ⊂ R ˜P be the subspace consisting of all constant maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' We work in the ambient space R ˜P/C where the constant c in the definition of affine maps is given by c = n2(n+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' 3 a b c d e → ab c de → ab cde → abcde a b c d e f → ab c d e f → abc d e f → abcd e f → abcde f → abcdef Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Multiplication of a word and of a generalized word For a finite subset τ ⊆ P, define a linear function ατ on R ˜P/C by ατ(˜x) := � i≺·j i,j∈τ ˜xj − ˜xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Again, the sum is taken over all covering relations contained in τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' We define the half-space hτ and the hyperplane Hτ by hτ := {p ∈ R ˜P/C | ατ(p) ≥ n2|τ|} and Hτ := {p ∈ R ˜P/C | ατ(p) = n2|τ|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Observe that for any tube τ and k ∈ Z, hτ = hτ+kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' The following is our main result in the affine case: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' If ˜P is an affine poset, the intersection of hτ for all proper tubes τ gives a realization of C ( ˜P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' An interpretation of tubings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' When P is a chain, A (P) recovers the classical as- sociahedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' There is a simple interpretation of proper tubings that explains all of the conditions above in terms of generalized words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' We can understand the classical associahedron as follows: Let P = ({1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=', n}, ≤) be a chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' We can think of the chain as a word we want to multiply together with the rule that two elements can be multiplied if they are connected by an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' A maximal tubing of P is a way of disambiguating the order in which one performs the multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' If a pair of adjacent elements x and y have a pair of brackets around them, they contract along the edge connecting them and replace x and y by their product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Similarly, we can understand the Hasse diagram of an arbitrary poset P as a generalized word we would like to multiply together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Again, we are allowed to multiply two elements if they are connected by an edge, but when multiplying elements, we contract along the edge connecting them and then take the transitive reduction of the resulting directed graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' That is, we identify the two elements and take the resulting quotient poset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' A maximal 4 tubing is again a way of disambiguating the order of the multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' See Figure 3 for an illustration of this multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' This perspective is discussed in relation to operahedra in [9, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='1] when the Hasse diagram of P is a rooted tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Configuration spaces and compactifications We turn our attention to the relationship between poset associahedra and configuration spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' For a poset P, the order cone L (P) := {p ∈ RP Σ=0 | pi ≤ pj for all i ⪯ j} is the set of order preserving maps P → R whose values sum to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Fix a constant c ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' The order polytope, first defined by Stanley [14] and extended by Galashin [6], is the (|P| − 2)-dimensional polytope O(P) := {p ∈ L (P) | αP(p) = c}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' When P is bounded, that is, has a unique maximum ˆ1 and minimum ˆ0, this construction is projectively equivalent to Stanley’s order polytope where we replace the conditions of the coordinates summing to 0 and αP(p) = c with the conditions pˆ0 = 0 and pˆ1 = 1, see [6, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Galashin [6] obtains the poset associahedra by an alternative compactification of O◦(P), the interior of O(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' We describe this compactification informally, as it serves as motivation for the realization in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' A point is on the boundary of O(P) when any of the inequalities in the order cone achieve equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' The faces of of O(P) are in bijection with proper tubings of P such that all tubes are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Let T be such a tubing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' If p is in the face corresponding to T and τ ∈ T then pi = pj for i, j ∈ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' We can think of the point p in the face corresponding to T as being “what happens in O(P)” when for each τ ∈ T, the coordinates are infinitesimally close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' However, by taking all coordinates in τ to be equal, we lose information about their relative ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' In A (P), we still think of the coordinates in τ as being infinitesimally close, but we are still inter- ested in their configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Upon zooming in, this is parameterized by the order polytope of the subposet (τ, ⪯).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' We iterate this process, allowing points in τ to be infinitesimally closer, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' We illustrate this in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' This idea is a common explanation of the Axelrod–Singer compactification of O◦(P) when P is a chain, see [1, 8, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' The idea of the realization in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='3 is to replace the notions of infinitesimally close and infinitesimally closer with being exponentially close and exponentially closer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' For p ∈ L (P), ατ acts a measure of how close the coordinates of p|τ are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' We can make this precise with the following definition and lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' For S ⊆ P and p ∈ RP, define the diameter of p relative to S by diamS(p) = max i,j∈S |pi − pj|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' That is, diamS(p) is the diameter of {pi : i ∈ S} as a subset of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Let τ ⊆ P be a tube and let p ∈ L (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Then diamτ(p) ≤ ατ(p) ≤ n2 4 diamτ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' 5 1 2 3 4 5 6 123 456 1 2 3 4 5 6 1 2 3 5 4 6 Tubing in O(P) Point in O(P) Tubing in A (P) Point in A (P) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' A vertex in O(P) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' A (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' By the triangle inequality and as τ is connected, diamτ(p) ≤ ατ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' For the other inequality, ατ(p) = � i≺·j i,j∈τ pj − pi ≤ � i≺·j i,j∈τ diamτ(p) ≤ 1 4n2 diamτ(p) The inequality in the last line comes from the fact that there are at most n2 4 covering relations in P, which follows from Mantel’s Theorem and the fact that Hasse diagrams are triangle-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' □ In particular, for p ∈ L (P), if p ∈ Hτ, then {pi | i ∈ τ} is clustered tightly together compared to any tube containing τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' If p ∈ hτ, then {pi | i ∈ τ} is spread far apart compared to any tube contained in τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Realizing poset associahedra We are now prepared to prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Define A (P) := � σ⊂P hσ ∩ HP where the intersection is over all tubes of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Note that A (P) ⊆ L (P) as if i ≺· j is a covering relation, then for p ∈ h{i,j}, pi ≤ pj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='3 follows as a result of three lemmas: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' If T is a maximal tubing, then vT := � τ∈T∪{P} Hτ is a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' 6 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' If T is a collection of tubes that do not form a proper tubing, then � τ∈T Hτ ∩ A (P) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' If T is a maximal tubing and τ /∈ T is a proper tube, then ατ(vT) > n2|τ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' That is, vT lies in the interior of hτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='1 follows from a standard induction argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' If T is not a collection of tubes that do proper tubing, then at least one of the following two cases holds: (1) There is a pair of non-nested and non-disjoint tubes τ1, τ2 in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' (2) There is a sequence of disjoint tubes τ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=', τk such that τ1 ≺ · · · ≺ τk ≺ τ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' The idea of the proof is as follows: For S ⊆ P, define the convex hull of S as conv(σ) := {b ∈ P | ∃a, c ∈ S : a ≤ b ≤ c}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Observe that if p ∈ L (P), then diamS(p) ≤ diamconv(S)(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Take σ = conv(τ1 ∪ · · · ∪ τk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' One can show that σ is a tube, so Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='3 tells us that for each τi, diamτi(p) is very small compared to n2|σ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' As the tubes either intersect or are cyclic, one can show this forces diamσ(p) to also be small, so ασ(p) < n2|σ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' More concretely, suppose that p ∈ � Hτi ∩ L (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Note that for all i, |σ| > |τi| + 1 and diamτi(p) ≤ n2(|σ|−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' In case (1), let a, b ∈ σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' There exists some x ∈ τ1 ∩ τ2, so |pa − pb| ≤ |pa − px| + |px − pb| ≤ diamτ1(p) + diamτ2(p) ≤ 2n2(|σ|−1) < n2(|σ|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Hence diamσ(p) < n2|σ|, so by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='3, p /∈ hσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Now we move to case (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Suppose there is a sequence of disjoint tubes τ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=', τk such that for each i there exists xi, yi ∈ τi where xi ≺ yi+1 where we take the indices mod k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Then: pyi − diamτi(p) ≤ pxi pxi ≤ pyi+1 pyi+1 ≤ pxi+1 + diamτi+1 Furthermore, since τi and τi+1 are disjoint, |τi| ≤ |σ|−2 and diamτi ≤ n2(|σ|−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Combining these we get pyi ≤ pyi+1 + 2n2(|σ|−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Then we have: py1 ≤ pyi + 2in2(|σ|−2) and pyi + 2in2(|σ|−2) ≤ py1 + 2(k + 1)n2(|σ|−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' 7 4 5 2 1 3 6 7 8 9 τ 4 5 2 1 3 6 7 8 9 τ σ A B Maximal Tubing T and tube τ σ, A, and B labelled Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' An example illustrating the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' A B σ τ Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' If diamA(p) and diamB(p) are small and diamσ(p) is large, then diamτ(p) is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' These yield py1 − pyi ≤ 2in2(|σ|−2) and pyi − py1 ≤ 2(k − i + 1)n2(|σ|−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' As i, k − i + 1 ≤ k ≤ n 2, we have |py1 − pyi| ≤ n2(|σ|−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Finally, if zi ∈ τi, zj ∈ τj, then |pzi − pzj| ≤ |pzi − pyi| + |pyi − py1| + |py1 − pyj| + |pyj − pzj| ≤ 4n2(|σ|−1) < n2|σ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Hence diamσ(p) < n2|σ|, and by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='3, p /∈ hσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' □ Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Let T be a maximal tubing of P and let τ /∈ T be a tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Define the convex hull of τ relative to T by convT(τ) := min{σ ∈ T | τ ⊂ σ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Let σ = convT(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' T partitions σ into a lower set A and an upper set B where A and B are either tubes or singletons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Furthermore, A and B both intersect τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' See Figure 5 for an example illustrating this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' The idea of the proof is as follows: Let p = vT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='3, diamA(p) and diamB(p) are both very small compared to diamσ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Then for any a ∈ A, b ∈ B, |pa − pb| must be large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' As τ intersects both A and B, diamτ(p) must be large and hence p ∈ hτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' See Figure 6 for an illustration of this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' More precisely, we show that for any i ∈ A, j ∈ B, pj −pi > (n2)|τ|, which implies that p lies in the interior of hτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' 8 Observe that:� x≺·y py − px = � x≺·y x,y∈A (py − px) � �� � ≤(n2)|σ|−1 < 1 8 (n2)|σ| + � x≺·y x,y∈B (py − px) � �� � ≤(n2)|σ|−1 < 1 8 (n2)|σ| + � x≺·y x∈A,y∈B (py − px).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Fix i ∈ A and j ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='3, for any x ∈ A, y ∈ B, py − px ≤ pj − pi + diamA(p) + diamB(p) ≤ pj + pi + 2n2(|σ|−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Again, noting that the number of covering relations in σ is at most n2 4 we obtain: � x≺·σy x∈A,y∈B (py − px) ≤ � x≺·σy x∈A,y∈B (pj − pi + 2(n2)|σ|−1) ≤ n2 4 � pj − pi + 2(n2)|σ|−1� = n2 4 (pj − pi) + 1 2(n2)|σ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Combining all of this we get: � x≺·σy py − px = (n2)|σ| < 1 8(n2)|σ| + 1 8(n2)|σ| + 1 2(n2)|σ| + n2 4 (pj − pi) ≤ 3 4(n2)|σ| + n2 4 (pj − pi) Then (n2)|σ|−1 < (pj − pi) and as |τ| ≤ |σ| − 1, p is in the interior of hτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' A similar approach for realizing graph associahedra is taken by Devadoss [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' One difference is that Devadoss realizes graph associahedra by cutting off slices of a simplex whereas we cut off slices of an order polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Realizing affine poset cyclohedra The proofs in the affine case are nearly identical to the finite case with some additional technical components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' The similarity comes from the fact that Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='3 still applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' We highlight where the proofs are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Let ˜P be an affine poset of order n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Define C ( ˜P) := � σ⊂P hσ and L ( ˜P) := {p ∈ R ˜P/C | pi ≤ pj for all i ⪯ j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' where the intersection is over all tubes of ˜P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Note that C ( ˜P) ⊆ L ( ˜P) as if i ≺· j is a covering relation, then for p ∈ h{i,j}, pi ≤ pj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='8 follows as a result of 3 lemmas: 9 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' If T is a maximal tubing, then vT := � τ∈T Hτ is a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' If T is a collection of tubes that do not form a proper tubing, then � τ∈T Hτ ∩ C ( ˜P) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' If T is a maximal tubing and τ /∈ T is a proper tube, then ατ(vT) > n2|τ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' That is, vT lies in the interior of hτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Let T be a maximal tubing and take any σ ∈ T such that |τ| = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Then restricting to ˜P|σ, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='1 implies that � τ∈T τ⊆σ Hτ is a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' However, as T is n-periodic, � τ∈T τ⊆σ Hτ = � τ∈T Hτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' □ Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' By Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='7, we can assume T is n-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' The proof is almost identical to the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Define L ( ˜P) := {p ∈ R ˜P/C | pi ≤ pj for all i ⪯ j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' and note that L ( ˜P) ⊆ R ˜P/C � i,j∈ ˜P i≺·j h{i,j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Let p ∈ � Hτi ∩ L ( ˜P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' We again break into two cases: (1) There is a pair of non-nested and non-disjoint tubes τ1, τ2 in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' (2) All tubes in T are pairwise nested or disjoint and there is a sequence of disjoint tubes τ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=', τk such that τ1 ≺ · · · ≺ τk ≺ τ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' The only difference in the proof occurs in case (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Here, it is possible that there exists x ∈ τ1 ∪τ2 such that x+n ∈ τ1 ∪τ2 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' In this case, the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='2 still implies that diamτ1∪τ2(p) ≤ diamτ1(p) + diamτ2(p) ≤ 2n2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' However, |px+n − px| = n2(n+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' □ Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Let T be a maximal tubing and τ /∈ T be a proper tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Let p = vT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' We claim that ατ(p) > n2|τ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' The only difference from the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='3 is that τ may not be contained by any tube in τ so convT(τ) may not be well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' In this case, there exists A ∈ T such that 10 ˆ0 a b c ˆ1 P ε = 1 3 ε = 1 9 ε = 1 27 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' O(P) as a limit of A (P) |A| = n, A ∩ τ ̸= ∅, and (A + n) ∩ τ ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Here, (A + n) acts the same as B in the finite case, except the argument is much simpler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Let i ∈ A ∩ τ, j ∈ (A + n) ∩ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Observe that diamA(p), diam(A+n)(p) ≤ n2n and that i + n ∈ (A + n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Then |pj − pi| ≥ (pj − n2n) − pi ≥ pi+n − pi = n2(n+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Hence diamτ(p) > n2|τ| and by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='3, ατ(p) > n2|τ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Remarks and Questions Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Let (P, ⪯) be a bounded poset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' In Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='1, we discuss how O(P) can be realized as the set of all p ∈ RP such that pˆ0 = 0, pˆ1 = 1, and pi ≤ pj whenever i ⪯ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' We can similarly realize A (P) as follows: Fix 0 < ε < 1 n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' For a proper tube τ ⊂ P, let h′ τ = {p ∈ RP | ατ(p) < εn−|τ|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Then A (P) is realized as the intersection over all h′ τ with the hyperplanes {pˆ0 = 0} and {pˆ1 = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Letting ε → 0, we obtain O(P) as a limit of A (P) as shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' The key piece to the realizations in Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='3 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='8 is the linear form ατ, where ατ acts as an approximation of diamτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' In particular, let τ be a tube and let p ∈ L (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Then: ατ(p) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' ατ(p) = 0 ⇔ p|τ is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' If σ ⊆ τ is a tube, then ασ(p) ≤ ατ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' However, there are many other options for choice of ατ that could fill this role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Some other options include: 11 (1) Sum over all pairs i ≺ j in τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' ατ(p) = � i≺j i,j∈τ pj − pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' (2) Let A and B be the set of minima and maxima of the restriction P|τ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' ατ(p) = � i≺j i∈A,j∈B pj − pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' (3) Fix a spanning tree T in the Hasse diagram of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Let E = {(i, j) | i ≺·T j} be the set of edges in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' ατ(p) = � (i,j)∈E pj − pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' An advantage of this option is that we would have diamτ(p) ≤ ατ(p) ≤ (n − 1) diamτ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' A similar realization can be obtained for each choice of of ατ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Question 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Recall that for a simple d-dimensional polytope P, the f-vector and h-vector of P are given by (f0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' , fd) and (h0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' , hd) where fi is the number of i-dimensional faces and d � i=0 fiti = d � i=0 hi(t + 1)i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Postnikov, Reiner, and Williams [12] found a statistic on maximal tubings of graph associa- hedra of chordal graphs where � T tstat(T) = � hiti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' In particular, they define a map T �→ wT from maximal tubings of a graph on n vertices to the set of permutations Sn such that stat(T) = des(wT), the number of descents of wT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' It would be interesting to find a similar statistic on maximal tubings of poset associahedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' For a simple polytope P, one can orient the edges of P according to a generic linear form and take stat(v) = outdegree(v) [17, §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' It may be possible to use our realization to find the desired statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Acknowledgements The author is grateful to Pavel Galashin for his many helpful comments and suggestions and to Stefan Forcey for fruitful conversations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' References [1] Scott Axelrod and Isadore M Singer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' “Chern-Simons perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' II”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' In: Jour- nal of Differential Geometry 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='1 (1994), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' 173–213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' [2] Raoul Bott and Clifford Taubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' “On the self-linking of knots”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' In: Journal of Mathe- matical Physics 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='10 (1994), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' 5247–5287.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' [3] Michael Carr and Satyan L Devadoss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' “Coxeter complexes and graph-associahedra”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' In: Topology and its Applications 153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='12 (2006), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' 2155–2168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' 12 [4] Satyan L Devadoss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' “A realization of graph associahedra”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' In: Discrete Mathematics 309.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='1 (2009), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' 271–276.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' [5] Sergey Fomin and Nathan Reading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' “Root systems and generalized associahedra”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' In: arXiv preprint math/0505518 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' [6] Pavel Galashin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' “Poset associahedra”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' In: arXiv preprint arXiv:2110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='07257 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' [7] Mark Haiman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' “Constructing the associahedron”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' In: Unpublished manuscript, MIT (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' [8] Pascal Lambrechts, Victor Turchin, and Ismar Voli´c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' “Associahedron, cyclohedron and permutohedron as compactifications of configuration spaces”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' In: Bulletin of the Belgian Mathematical Society-Simon Stevin 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='2 (2010), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' 303–332.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' [9] Guillaume Laplante-Anfossi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' “The diagonal of the operahedra”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' In: Advances in Math- ematics 405 (2022), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' 108494.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' [10] Chiara Mantovani, Arnau Padrol, and Vincent Pilaud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' “Acyclonestohedra: when ori- ented matroids meet nestohedra”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' [11] Kyle Petersen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Eulerian Numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Oct.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' In: Bulletin de la Soci´et´e math´ematique de France 82 (1954), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' 53–96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' [17] G¨unter M Ziegler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Lectures on polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' 152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Springer Science & Business Media, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content=' Department of Mathematics, University of California, Los Angeles, CA 90095, USA Email address: andrewsack@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='ucla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} +page_content='edu 13' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'} diff --git a/PtE2T4oBgHgl3EQfVgea/content/tmp_files/2301.03824v1.pdf.txt b/PtE2T4oBgHgl3EQfVgea/content/tmp_files/2301.03824v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3f67a4a7d7f328ae1904902b7ea8a8353fb74975 --- /dev/null +++ b/PtE2T4oBgHgl3EQfVgea/content/tmp_files/2301.03824v1.pdf.txt @@ -0,0 +1,2109 @@ +arXiv:2301.03824v1 [astro-ph.EP] 10 Jan 2023 +Draft version January 11, 2023 +Typeset using LATEX preprint style in AASTeX63 +Photosynthetic Fluorescence from Earth-Like Planets around Sun-Like and Cool Stars +Yu Komatsu,1, 2 Yasunori Hori,1, 2 Masayuki Kuzuhara,1, 2 Makiko Kosugi,1, 2, 3 +Kenji Takizawa,1, 3 Norio Narita,4, 1, 5 Masashi Omiya,1, 2 Eunchul Kim,3 +Nobuhiko Kusakabe,1, 2 Victoria Meadows,6, 7 and Motohide Tamura1, 2, 8 +1Astrobiology Center, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan. +2National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan. +3National Institute for Basic Biology, 38 Nishigonaka, Myodaiji, Okazaki, Aichi 444-8585, Japan. +4Komaba Institute for Science, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8902, Japan. +5Instituto de Astrof´ısica de Canarias (IAC), 38205 La Laguna, Tenerife, Spain +6Department of Astronomy and Astrobiology Program, University of Washington, Box 351580, Seattle, Washington +98195, USA. +7NASA Nexus for Exoplanet System Science, Virtual Planetary Laboratory Team, Box 351580, University of +Washington, Seattle, Washington 98195, USA. +8Department of Astronomy, Graduate School of Science, The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo +113-0033, Japan. +(Received August 15; Revised November 12; Accepted November 15, 2022) +Submitted to ApJ +ABSTRACT +Remote sensing of the Earth has demonstrated that photosynthesis is traceable as the +vegetation red edge (VRE), which is the steep rise in the reflection spectrum of vegeta- +tion, and as solar-induced fluorescence. This study examined the detectability of bio- +logical fluorescence from two types of photosynthetic pigments, chlorophylls (Chls) and +bacteriochlorophylls (BChls), on Earth-like planets with oxygen-rich/poor and anoxic +atmospheres around the Sun and M dwarfs. Atmospheric absorption, such as H2O, CH4, +O2, and O3, and the VRE obscure the fluorescence emissions from Chls and BChls. We +found that BChl-based fluorescence for wavelengths of 1000–1100 nm, assuming the +spectrum of BChl b-bearing purple bacteria, could provide a suitable biosignature but +only in the absence of the water cloud coverage or other strong absorbers near 1000 nm. +The Chl fluorescence is weaker for several reasons, e.g., spectral blending with the +VRE. The apparent reflectance excess is greatly increased in both Chl and BChl cases +around TRAPPIST-1 due to fluorescence and stellar absorption lines. This could be a +promising feature for detecting the fluorescence around ultracool red dwarfs by follow- +up ground-based observations with high spectral resolution; however, it requires a long +time around Sun-like stars, even for a LUVOIR-like space mission. Moreover, the simul- +taneous detection of fluorescence and VRE is key to identifying traces of photosynthesis +Corresponding author: Yu Komatsu +yu.komatsu@nao.ac.jp + +2 +Komatsu et al. +because absorption, reflectance, and fluorescence are physically connected. For further +validation of fluorescence detection, the nonlinear response of biological fluorescence as +a function of light intensity could be considered. +Keywords: astrobiology, planets and satellites: atmospheres, planets and satellites: sur- +faces, planets and satellites: terrestrial planets +1. INTRODUCTION +The ultimate goal of characterizing rocky planets is to identify potential biosignatures, spec- +tral fingerprints of atmospheric gases, and surface features produced by biological activities +(Des Marais et al. 2002; Schwieterman et al. 2018; Meadows et al. 2018). The simultaneous identifi- +cation of oxygen, ozone, and methane on rocky habitable planets shows promise as a way to detect +Earth-like life. Oxygenic photosynthesis produces a unique feature in the reflection spectrum on a +planetary surface, called the vegetation red edge (VRE), as well as biosignature gases (Kiang et al. +2007a). The VRE is the steep difference in the reflection spectrum of the surface vegetation around +700 nm due to chlorophyll (Chl) absorption in the visible region and the large reflectance by cell +structures in the near-infrared (NIR) region (Gates et al. 1965; Jacquemoud & Baret 1990). Remote +sensing of the Earth and Earthshine observations provide spectral indices involved in the VRE, such +as the NDVI, which is a normalized difference in the reflection spectrum of the Earth between the +visible and NIR wavelength regions. The Moderate Resolution Imaging Spectroradiometer (MODIS) +onboard NASA’s Terra satellite at 16-day intervals at 500 m and 1 km resolutions shows that the +NDVI varies from +0.05 to nearly 0.9, whose upper limit is obtained at a dense forest site during +the peak growing season (Huete et al. 2002). Whereas remote sensing observes local areas on Earth, +Earthshine observations provide disk-averaged spectra of the Earth, leading to fruitful insights into +exoplanet applications. The apparent reflectance change in the Earth’s disk-averaged spectrum due +to surface vegetation is less than 2% (Monta˜n´es-Rodr´ıguez et al. 2006). The NDVI calculated from +the Earthshine observations varies up to ∼0.10, depending on different views of the Earth, and is +reduced by cloud coverage (Tinetti et al. 2006). The application of NDVI to disk-averaged spectra +assuming Earth-like exoplanets requires caution because remote sensing observes only local areas on +the Earth to map vegetation. For instance, Livengood et al. (2011) found that additional spectral +bands to NDVI are required to distinguish between the Earth vegetation and the Moon surface. +The VRE signals from exoplanets around stars other than a Sun-like star are challenging to predict +due to the complexity of photosynthetic mechanisms in different light environments. However, the +VRE on exoplanets may still be recognizable as an anomalous time-varying due to seasonal variability +of the vegetation, and step-function-like spectroscopic feature at wavelengths different from those on +the Earth (Seager et al. 2005). Tinetti et al. (2006) proposed that if a three-photon photosynthetic +scheme were working on exoplanets around M dwarfs, where there was little or no visible light, then +the red edge of vegetation could also be shifted into the NIR. However, according to Takizawa et al. +(2017), even around M dwarfs, the evolution of photosynthesis in water may drive a preference for +using visible light rather than NIR, even after organisms colonize land surfaces. Moreover, the light +absorption properties of land vegetation could be optimized after long-term adaptive evolution de- +pending on stellar irradiations as estimated by Lehmer et al. (2021). Anoxygenic photosynthesis as +performed by organisms such as purple bacteria, is thought to precede the emergence of oxygenic + +Photosynthetic fluorescence on Exoplanets +3 +photosynthesis, whose global effect was characterized by the great oxidation event (∼2.3 billion years +ago (Ga)). Sanrom´a et al. (2013) discussed the detectability of light reflected from purple bacteria +with bacteriochlorophyll (BChl) as a photosynthetic pigment. They showed that purple bacteria +exhibit detectable features, and their VRE peak is redder than higher plants, assuming an Earth- +like planet before the rise of oxygen. +In a comprehensive study of different pigment reflectivity, +Schwieterman et al. (2015) showed that both nonphotosynthetic pigments and photosynthetic pig- +ments affect the disk-averaged spectra. Furthermore, as for false positive detection, the reflectance +features of some minerals on the Earth are similar to the VRE ones (Seager et al. 2005; Schwieterman +2018). Thus, extracting the VRE signal from reflected light should require knowledge of the surface +environment on an exoplanet and high-resolution spectroscopic observations. +Fluorescence is another photosynthesis-related phenomenon that could also be a remote-sensing +biosignature. Fluorescence is one of the de-excitation processes of photosynthetic pigments from the +excited states to the ground state, along with intersystem crossing and inner conversion. Photosyn- +thetic organisms on the Earth use Chls or BChls as light-absorbing pigments and electron donors/ac- +ceptors in the primary reactions of photosynthesis. The photon energy captured by Chls/BChls is +mainly transferred to the reaction center (RC), which is the pigment-protein complex at the center of +the photosystem used for photochemical reactions. A part of photon energy is, however, dissipated +as heat or emitted as fluorescence from light-harvesting antenna systems, which are pigment-protein +complexes surrounding RC that capture light energy and deliver the energy to the RC. Excess photon +energy is preferentially removed as heat dissipation, rather than fluorescence. As a result, fluores- +cence yield tends to be a smaller percentage of the excess energy and fluctuates with the degree +of the excitation energy transfer (EET) between Chls, and heat dissipation. The fluorescence yield +of photosynthetic organisms is estimated to be ∼5%, whereas that of free Chls/BChls in organic +solvents is ∼ 30% (Grimm et al. 2006). +Plants and other oxygenic phototrophs use two different photosystems in sequence, that is, pho- +tosystem II (PSII) and photosystem I (PSI). The energy level of the RC of PSII is higher, being +equivalent to 680 nm, than that of PSI. In general, Chl fluorescence is mainly emitted from PSII +because the excess light energy in PSI is immediately dissipated as heat. Therefore, the fluorescence +spectrum of a cell has a peak at 680 nm, and the distribution of fluorescence emission extends to +wavelengths up to 780 nm. Note that fluorescence emissions at 680 nm under highly concentrated +Chls conditions, such as a leaf structure, decrease due to reabsorption by peripheral Chls with a +red-absorption band. Conversely, the six BChls (BChl a, b, c, d, e, and g) used in non-oxygenic +photosynthetic bacteria, such as purple bacteria, green sulfur and nonsulfur bacteria and heliobac- +teria (Kiang et al. 2007b), mainly absorb far-red light in vivo. The BChl b in purple bacteria has +the longest wavelength absorbance (1010 nm) and fluorescence (1050 nm) emissions. However, the +detailed characteristics of fluorescence from BChls, such as fluorescence yield and its variation in +light environments, remain poorly understood. +In contrast to the VRE which tracks the vegetation mass in the remote sensing of the Earth, fluo- +rescence can be used as an indicator of active photosynthesis. The fluorescence signal emitted from +the global ground vegetation, which is called solar-induced fluorescence (SIF), can be detected by +remote sensing from satellites as excess light seen in the absorption of Fraunhofer lines in sunlight +reflected from the Earth, which is the apparent increase in the reflectance spectrum due to fluores- +cence (Maier et al. 2004). The observation of SIF is fundamentally challenging because the small SIF + +4 +Komatsu et al. +signal is overwhelmed by large background signals in the reflected sunlight. Then, high-resolution +spectroscopy utilizes specific wavelengths with large solar absorption, which means the low intensity +of reflected light. The SIF is observed as the in-filling effect at these wavelengths. This methodology +works because a large contrast is ensured between the Sun and the reflected light from the Earth at +specific wavelengths. Thus, SIF has been observed in absorption bands by the Fourier high-dispersion +spectrometers onboard many environmental satellites (e.g., GOSAT (Hamazaki et al. 2005; Lee et al. +2013), GOME-2 (Callies et al. 2000), and GOSAT-2 (Nakajima et al. 2012)), which produce the time- +series SIF map of Earth (Frankenberg et al. 2014; Sun et al. 2018). We can extract information on +the ground vegetation and atmospheric/surface environment, especially the gross primary production +(GPP), from the changes in the fluorescence map by calibrating the remote observations with the +results of local ground observations (Sun et al. 2018). Such as the SIF in Earth observations, the +detection of photosynthetic fluorescence in a planet around stars will investigate the surface envi- +ronment and vegetation conditions on exoplanets. High-resolution spectroscopy would be inevitable +for the exofluorescence detection, and the contrast between a planet and its host star should be high +enough at specific wavelengths. Biofluorescence, similar to that shown by coral reefs on Earth, has +been suggested as a new potential biosignature for exoplanets experiencing strong UV radiation from +F stars (O’Malley-James & Kaltenegger 2018) and M stars (O’Malley-James & Kaltenegger 2019). +It might work if the fluorescence were emitted very efficiently according to gained photons in their +habitats. As mentioned above, photosynthetic pigments are a potential emitter of biofluorescence. +However, the yield and detectability of photosynthetic fluorescence on the surface of exoplanets have +not yet been examined. +Finding surface biosignatures on Earth-like exoplanets, including the potential detectability of +biofluorescence, would be one of the important goals of future astronomy and may become possible +with future space missions such as the Large UV/Optical/IR Surveyor (LUVOIR) or the Habit- +able Exoplanet Observatory (HabEx), and next-generation extremely large ground-based telescopes +(TMT, ELT, and GMT) observing in reflected light. Thus, it is important to quantitatively evaluate +the detectability of any potential surface biosignature using expected specifications of specific future +missions. +This study made the first attempt to investigate the detectability of photosynthetic fluorescence on +Earth-like exoplanets. The remainder of this paper is structured as follows: Section 2 describes the +surface vegetation model for an Earth-like planet in the habitable zone and fluorescence emissions +based on the photoresponse of photosynthetic organisms. Section 3 shows the expected fluorescence +emissions in the reflected light spectra on an Earth-like planet around an M dwarf or the Sun. In Sec- +tion 4, we discuss the physiological conditions of photosynthesis that enhance fluorescence emissions +and its unique features for future detection, including false-positive signals and seasonal changes. +Additionally, we present the detectability of biofluorescence by a future space-based telescope assum- +ing the LUVOIR telescope parameters, and the key spectral feature possibly useful for the detection +by follow-up observations with high-dispersion spectroscopy. In the last section, we summarize our +paper. +2. MATERIALS AND METHODS +We assume that the radiation from a planetary surface is the sum of the reflected light on the +surface and the fluorescence emission from photosynthesis. The outgoing flux on the surface and at + +Photosynthetic fluorescence on Exoplanets +5 +Figure 1. +(a) Incident radiation and (b) photon flux density at the top of atmosphere (TOA) of an +Earth-like planet around the Sun, GJ667C, and TRAPPIST-1. The spectral data of the Sun, GJ667C, and +TRAPPIST-1 were obtained from Meftah et al. (2018), France et al. (2016), and Lincowski et al. (2018), +respectively. +the top of atmosphere (TOA) is given by: +F ↑ +surface(λ)=F ↓ +surface(λ)R(λ) + Ffluor.(λ), +(1) +F ↑ +TOA(λ)=F ↑ +surface(λ)T(λ), +(2) +where λ is the wavelength, T(λ) is the atmospheric transmittance (see Section 2.1), R(λ) is the +surface reflectance of a planet (see Section 2.3), F ↑ +surface(λ) is the upward flux from a planetary +surface, F ↑ +TOA(λ) is the reflected flux at the TOA, and Ffluor.(λ) is the net fluorescence emission from +photosynthesis. F ↓ +surface(λ) = F ↓ +TOA(λ)T(λ) is the downward flux from the planetary atmosphere to +the surface, where F ↓ +TOA is the incident flux from a host star at the TOA (see Figure 1 and Section +2.1 below). +We neglect the effects of thermal emission in all the cases and Rayleigh scattering +in most cases, as both processes contribute little radiation to our spectral region of interest (600- +1000 nm). The transmittance T(λ) in the atmosphere of an Earth-like planet through geological +evolution was obtained from Rugheimer & Kaltenegger (2018), which was calculated by a 1D coupled +radiative/convective-photochemical model for a planetary atmosphere (see also Pavlov & Kasting +2002; Kasting & Ackerman 1986; Segura et al. 2005). +2.1. Stellar Radiation +Two nearby M dwarfs, GJ667 C and TRAPPIST-1, have candidate planets in a habitable zone (HZ). +We considered fluorescence emissions from photosynthesis on an Earth-like planet in an HZ around +GJ667 C, TRAPPIST-1, and the Sun. We extracted the incident stellar flux from high-resolution +spectral data for the Sun (Meftah et al. 2018), GJ667 C (France et al. 2016), and TRAPPIST-1 +(Lincowski et al. 2018). The incident flux F ↓ +TOA received by an Earth-like planet around GJ667 C, +and TRAPPIST-1 is scaled by the current location of GJ667C c, and TRAPPIST-1e. GJ667 C, and +TRAPPIST-1 are modeled as M1V and M8V stars. Figure 1 shows the incident flux received by an +Earth-like planet at the TOA around the Sun, GJ667 C, and TRAPPIST-1. +2.2. Fluorescence from Photosynthesis + +le3 +tons/s/m2/μm) +lel +Sun +(un/zw/m) +2.0 +GJ667C +TRAPPIST-1 +0.8 +1.5 +(mmol-pho +0.6 +Density +1.0 +0.4 +Flux +0.5 +Photon Flux +0.2 +0.0 +0.0 +0 +250 +500 +750 +10001250 +15001750 +2000 +0 +250 +500 +750 +10001250150017502000 +Wavelength(nm) +Wavelength[nm]6 +Komatsu et al. +Fluorescence emissions from a planetary surface F +′ +fluor. are expressed as: +F +′ +fluor.(λ) = scvπF std +fluor. × f(λ), +(3) +where cv is the surface coverage of vegetation (see Section 2.3), s is the scaling factor from the stan- +dard observed fluorescence emission reflecting the photosynthetic activity, and F std +fluor. is the standard +fluorescence intensity from vegetation based on field measurements. The spectral shape of fluores- +cence emissions from a photosynthetic organism at wavelength λ is defined by f(λ). In this study, +F std +fluor. = 1.0 (W m−2 µm−1 sr−1) (Du et al. 2019; Yao et al. 2021) and s = 0, 1, 5, and 10. The net +fluorescence intensity Ffluor. is calculated by considering acquired photons at the habitat using F +′ +fluor. +in Equation (3) as: +Ffluor.(λ) = χ +χ0 +F +′ +fluor.(λ), +(4) +χ ≡ +� +n(λ)σ(λ)dλ, +(5) +χ0 ≡ +� +nsun,ref.(λ)σchls(λ)dλ, +(6) +where χ is the light absorption efficiency, n(λ) is the photon flux density at the planetary surface, and +σ(λ) is the absorption coefficient of a photosynthetic pigment. χ0 represents the standard absorption +efficiency on Earth. The subscript chls on σ(λ) represents chlorophylls (see Chl:abs in Figure 2). +nsun,ref.(λ) is the photon flux density on the surface of the Earth from the reference solar spectral +irradiance at an air mass of 1.5 (National Renewable Energy Laboratory), which corresponds to a +typical irradiance for Earth vegetation. +We considered an incident flux from a star under two sky conditions, a clear sky and 60% cloud +cover, to estimate the reflectance at the TOA and χ on the ground, in accordance with the setup +for the simulation. We assumed the clear sky condition, if not specified, and the cloud condition +appeared only in Section 4.3.1 (Figure 12). In the cloudy condition, 60% of the radiation is reflected +in three kinds of clouds, and 40% of the radiation reaches the ground. For the clouds, we assumed that +40% are low water clouds, 40% are high water clouds, and 20% are high ice clouds (Gao & Kaufman +2003) at 1, 6, and 12 km altitude, respectively. To model Earth-like conditions, the effect of Rayleigh +scattering in a planetary atmosphere was also considered in the cloudy condition using a previously +described empirical approach by Bucholtz (1995) (see Appendix A for more details). +Equation (4) indicates that F +′ +fluor. is linearly scaled to the number of incoming photons that are +absorbed by chlorophylls at the planetary surface. +In other words, chlorophylls can emit strong +fluorescence if the spectral shapes of n(λ) and σ(λ) match well. +Note that n(λ) is exactly the +same as F ↓ +surface(λ), and its unit is shown in Figure 1(b) (F ↓ +TOA in the figure). This treatment in +Equations (3) and (4) can be applied to the relationship between the incoming photons and the +photons emitted as fluorescence on an Earth-like planet around various stars other than the Sun. +Figure 2 shows the normalized spectra of fluorescence f(λ) and absorption coefficient σ(λ) for Chls +and BChls. The peak wavelength of f(λ) is red-shifted from that of σ(λ), which is called the Stokes +shift (Lakowicz 2006). There are two absorption bands in the σ(λ) of chlorophylls: the B band +(known as the Soret band) in the short-wavelength region and the Q band in the long-wavelength +region. The primary fluorescence emission is derived from the Q absorption band. In this study, + +Photosynthetic fluorescence on Exoplanets +7 +Figure 2. Fluorescence (f(λ): solid curves) and photoabsorption (σ(λ): dashed curves) spectra for chloro- +phylls (Chl: black) and bacteriochlorophylls (BChl: red). The absorption coefficient of chlorophylls in units +of cm2 µg−1 was obtained from Feret et al. (2008). The fluorescence spectrum is expressed by the Gaussian +functions given in Frankenberg et al. (2012) and Guanter et al. (2010). For bacteriochlorophylls, f(λ) and +σ(λ) adopt those of the LH1–RC complex of a bacteriochlorophyll b containing purple photosynthetic bac- +teria (Magdaong et al. 2016). The nondimensional absorption spectrum for BChl is normalized at the peak +value in the longest absorption band, the Q band, of the Chl. Two fluorescence spectra are normalized at +their peak values. +we modeled f(λ) for Chls as the superposition of two Gaussian distributions (Frankenberg et al. +2012; Guanter et al. 2010) with means of 680 nm (PSII) and 740 nm (PSI and PSII). σ(λ) for Chl +uses the model vegetation with chlorophylls (σchls(λ)) (Feret et al. 2008). We obtained f(λ) and +σ(λ) for BChls from the spectral data for the LH1–RC complex, the supramolecular complex of +the light-harvesting core antenna (LH1), and the RC in a bacteriochlorophyll b containing purple +photosynthetic bacteria (see Figure 3 in Magdaong et al. 2016)). Note that we used only σ(λ) in +the Q band for calculating χ and χ0 because free Chls and BChl–protein complexes in each solution +affect each spectrum in the B band to different degrees. +2.3. Surface Vegetation +To determine the detectability of vegetation fluorescence, we use two leaf models for our exper- +iments: one which assumes the reflectance spectrum and fluorescence of standard chlorophyll and +another that uses a scaled version of the spectrum of bacteriochlorophyll. The reflectance of a planet +is expressed as R(λ) = � +i ciri(λ), where i denotes the surface type, ci is the fraction of the surface +coverage of type i, and ri the reflectance of type i. We obtained the reflection spectra for various +surface types including vegetation, ocean, and coast from the USGS Digital Spectral Library and +the ASTER Spectral Library (Baldridge et al. 2009). The detailed compositions used in this paper +are summarized in Table 1. The reflectance of the surface vegetation rv is estimated from radiation +transfer calculations for a modeled leaf (Jacquemoud & Baret 1990; Feret et al. 2008), using σ(λ) +over all the wavelengths shown in Figure 2. Figure 3 shows the reflectance of a Chl-based leaf (“stan- + +Absorption coefficient (cm2/μg) +Normalized fluorescence +0.16 +1.0 +0.14 +0.8 +0.12 +0.10 +0.6 +0.08 +0.4 +0.06 +0.04 +0.2 +0.02 +0.00 +0.0 +400 +600 +800 +1000 +1200 +Wavelength (nm) +Chl: abs +Chl: fl +BChl: abs +BChl: fl8 +Komatsu et al. +Figure 3. The reflectance of vegetation estimated from radiation transfer calculations for two leaf models: +Chl (“standard”) and BChl (“hypothetical”) +(Jacquemoud & Baret 1990; Feret et al. 2008). +The light +absorption spectrum for Chls and BChls uses σ(λ) in Figure 2. +dard”) and a BChl-based leaf (“hypothetical”). In the latter case, we assumed the vegetation on a +different planet has a photosynthetic pigment whose optical property is the same as BChl exhibiting +the VRE in the longer wavelength region as shown in Figure 3. As the input to the radiative transfer +calculations, we used the absorption spectra of Chl (Feret et al. 2008) and BChl (Magdaong et al. +2016). The unitless absorption spectrum for BChl is normalized at the peak in that for Chl, unlike +the calculations of χ and χ0 in Section 2.2. As shown in Figure 3, both Chl- and BChl-based leaves +show a large reflectance (i.e., the VRE) in the wavelength ranges around 700–750 and 1000–1100 nm. +The green bump around 500 nm is observed in the reflectance for Chl, and larger and broader bumps +are observed from ∼500 to 950 nm for BChls by the larger difference in the wavelength between the +B and Q bands than observed for Chl. Like many kinds of photosynthetic organisms, the organisms +with BChl could have acquired accessory pigments such as carotenoids (Cars) that absorb photons +with wavelengths between the B and Q bands of Chl (Cogdell 1978). The effective light absorption by +accessory pigments can suppress the increase in reflectance. With or without accessory pigments, the +bump for BChl does not affect fluorescence emissions in the wavelength (see Figures 7, 8, and 10). +The low reflectance from ∼500 to 700 nm (1000 nm), due to the light absorption by Chls (BChls), +affects the reflectance of the planet. The degree of reduction in the overall planetary reflectance +varies depending on the surface coverage by vegetation. +3. RESULTS +We considered three fluorescence cases on an Earth-like planet at different stages of atmospheric +evolution around the Sun, GJ667 C, and TRAPPIST-1 for different surface biosignatures: Earth-like +(Chl) vegetation, hypothetical BChl-based vegetation, and biological fluorescence without any surface +vegetation. Our models for the surface compositions, vegetation, fluorescence types, and atmospheric +compositions, i.e., transmittance, are summarized in Table 1. Mod-earth corresponds to the surface +condition for the Modern Earth, leading to a lesser contribution of fluorescence emissions than in the +other two cases. The veg-only models are considered optimistic conditions for fluorescence emissions +where vegetation covers the whole planetary surface. The veg-land models, with 70 % ocean, 2% +coast, and 28% land covered with the vegetation, lie between the mod-earth and veg-only models. As +mentioned in Section 2, we considered two leaf models for land vegetation: Chl-based vegetation and + +0.5 +Reflectance +0.4 +0.3 +0.2 +Chl +0.1 +BChl +500 +1000 +1500 +2000 +Wavelength (nm)Photosynthetic fluorescence on Exoplanets +9 +BChl-based vegetation. For the atmospheric compositions of an Earth-like planet, we adopted the +Modern Earth model at 0.0 Ga (oxygen-rich atmosphere), the Paleoproterozoic Earth model at 2.0 Ga +(oxygen-poor atmosphere), and the Archean Earth model at 3.9 Ga (anoxic atmosphere) (see Table 1 +in Rugheimer & Kaltenegger 2018). As an extreme case, we assumed the presence of photosynthetic +bacteria with BChl spread over the land and ocean on an Archean-Earth-like planet with no surface +vegetation. We assumed a clear sky for all atmospheric conditions in Section 3. +Model name +Surface compositions +Surface vegetation +Fluorescence type +T(λ) +cv +veg-only 0C +Chl surf. +Chl fluor. +0.0 Ga +veg-only 2C +100% vegetation +2.0 Ga +1.00 +veg-only 0B +BChl surf. +BChl fluor. +0.0 Ga +veg-only 2B +2.0 Ga +veg-land 0C +Chl surf. +Chl fluor. +0.0 Ga +veg-land 2C +70% ocean, 2% coast +2.0 Ga +0.28 +veg-land 0B +and 28% vegetation +BChl surf. +BChl fluor. +0.0 Ga +veg-land 2B +2.0 Ga +mod-earth 0C +Chl surf. +Chl fluor. +0.0 Ga +mod-earth 2C +70% ocean, 2% coast +2.0 Ga +0.168 +mod-earth 0B +and 28 % mixed land +BChl surf. +BChl fluor. +0.0 Ga +mod-earth 2B +(incl. 16.8% vegetation) +2.0 Ga +anoxic B +70% ocean, 2% coast and +- +BChl fluor. +3.9 Ga +0.72 +28% mixed land at 3.9 Ga +Table 1. Surface composition, vegetation, its fluorescence types, and atmospheric transmittance (T(λ)) +for all the cases in this paper. Mixed land is composed of 60% vegetation (16.8% in total), 15% snow, 9% +granite, 9% basalt, and 7% sand (Baldridge et al. 2009); mixed land at 3.9 Ga means the land model of the +Archean Earth at 3.9 Ga, which is composed of 35% basalt, 40% granite, 15% snow, and 10% sand. Chl +surf. and BChl surf. correspond to reflection spectra of Chl and BChl in Figure 3, respectively. The spectral +shapes of fluorescence emissions f(λ) for Chl fluor. and BChl fluor. correspond to the fluorescence spectra +of Chl and BChl in Figure 2, respectively; their intensities Fflour. are scaled in Equations (3) and (4). cv is +given by the relationship between the surface coverage of vegetation and the fluorescence emission. s={0, +1.0, 5.0, 10.0}. We obtained T(λ) at 0.0, 2.0, and 3.9 Ga from Rugheimer & Kaltenegger (2018). +3.1. Case-1: Planets with Earth-Like Vegetation +In case-1, Earth-like vegetation (Chl) emits fluorescence on the surface of an Earth-like planet. The +fluorescence emissions from chlorophyll are visible at the wavelengths from 650 to 800 nm, as shown +in Figure 2. To determine the contribution of fluorescence from planets, the reflectance is defined +as F ↑ +TOA(λ)/F ↓ +TOA(λ) and calculated. Figures 4 and 5 show the reflectance of an Earth-like planet +with the Modern Earth’s atmosphere (0.0 Ga) and an oxygen-poor atmosphere (2.0 Ga), respectively. +The O2, O3, CH4, and H2O absorption features in the atmosphere are imprinted in the reflectivity +in the visible–NIR wavelengths from 600–800 nm. The oxygen-poor-atmosphere models show less +conspicuous patterns in the reflectance profile in the 700 to 750 nm wavelength region. The reflec- +tivity between 600 and 700 nm is nearly constant but increases with decreasing surface coverage of + +10 +Komatsu et al. +vegetation. The VRE is observed as the steep rise in the reflectance from 700 to 750 nm (also see +Figure 3), whereas the reflectance excess due to fluorescence is quite small, even in optimistic condi- +tions (veg-only models). Note that the red curve with 1Ffluor. around the Sun in the mod-earth model +(Figure 4), corresponding to the modern earth fluorescence, is hardly seen. Around TRAPPIST-1, +however, sharp increase in the reflectance around 770 nm is due to the strong absorption of potassium +in the stellar atmosphere. As a result, we observed similar features in the light reflected from an +Earth-like planet with different atmospheric compositions around TRAPPIST-1 (see Section 4.3.2 +for further discussion). +Figure 6 shows the reflectance excess due to fluorescence emissions on an Earth-like planet with +the Modern Earth’s atmosphere. Atmospheric absorptions, such as H2O, O2, and O3, weaken the +Gaussian features in the fluorescence emissions from an Earth-like planet around the Sun. +The +fluorescence from chlorophylls around 740 nm is less pronounced for a planet around M dwarfs than +one around the Sun because of weaker radiation flux in the wavelength region of 700–750 nm (see +Figure 1). In addition, a sudden increase in reflectance due to the VRE obscures the fluorescence +emission around 740 nm (see Figures 4 and 5). As a result, the Chl fluorescence around 680 nm +emitted from PSII on an Earth-like planet would be the most promising feature for detection (see +Figure 2). Note that nonphotochemical quenching processes can decrease the fluorescence intensity +around 680 nm, and the fluorescence emission is further reduced by the reabsorption of photons within +the canopy (Porcar-Castell et al. 2021). +3.2. Case-2: Planets with Bacteriochlorophylls-Based Vegetation +In case-2, BChl-based vegetation, as the major photosynthetic pigment, covers the surface of a +planet. The BChls are assumed to emit the same degree of fluorescence intensity as the Earth’s +vegetation. As shown in Figure 2, fluorescence from BChls occurs in the wavelength range from 1000 +to 1100 nm. In contrast to case-1, fluorescence emissions with 5 and 10Ffluor. show strong features +around 1050 nm in almost all conditions in Figures 7 and 8. Identifying the fluorescence on the Earth’s +vegetation level (≲ Ffluor.) is still challenging even in the optimistic case, that is, (a) veg-only 0B. +The reflectivity between 1000 and 1050 nm becomes slightly higher for mod-earth models with less +surface vegetation coverage. As shown in Figure 9, the BChl organisms efficiently absorb photons and +emit fluorescence with less absorption and scattering in the planetary atmosphere. The fluorescence +emissions from BChls that we assumed are invulnerable to blending with the steep increase in the +reflectance by the VRE. As a result, we found a more significant fluorescence contribution to the +reflected light in case-2. +Atmospheric properties, such as chemical compositions and cloud coverage, change the fluorescence +profile. The water absorption is weak for wavelengths from 1000 to 1100 nm. If the major absorption +bands of a photosynthetic pigment lie in wavelengths longer or shorter than 1000–1100 nm, the pres- +ence of water vapor in the atmosphere complicates the detection of fluorescence emissions. A strong +absorption due to CH4 in an oxygen-poor atmosphere also hides fluorescence near 1000 nm (see the +GJ667C models in Figure 8). The BChl organisms bearing BChl b and their Stokes shift are ideal for +detecting fluorescence in wavelengths longer than the characteristic wavelength of fluorescence from +Chls. Thus, fluorescence in the wavelength range of 1000 -1100 nm could be a suitable biosignature +for photosynthetic organisms, such as bacteriochlorophylls, on planetary surfaces unless they coexist +with strong absorbers near 1000 nm. + +Photosynthetic fluorescence on Exoplanets +11 +Figure 4. Reflectance of an Earth-like planet with the Modern Earth’s atmosphere (0.0Ga) around the +Sun, GJ667C, and TRAPPIST-1. The three colors represent the reflected light from a planet with Ffluor. +(s = 1: red), 5Ffluor. (s = 5: blue), and 10Ffluor. (s = 10: green), where Ffluor. is the fluorescence emission +from chlorophylls observed on the Earth. No-fluorescence emission models are also indicated by gray lines. +We assumed Earth-like vegetation (chlorophylls) covers the planetary surface (see Table 1 for model details). +The reflectance is defined here as F ↑ +TOA(λ)/F ↓ +TOA(λ), where F ↑ +TOA(λ) is the light reflected from the ground +at the top of atmosphere (TOA), and F ↓ +TOA(λ) is the flux at TOA induced by stars. For each case around +TRAPPIST-1, the reflectance with a logarithmic scale is also shown as the inset plot. +The VRE with a sharp rise in the reflectance is observed in the wavelength range from 1050 to +1100 nm in case-2, as shown in Figure 3. Reflectance excess due to BChl fluorescence is 0.01–0.05 +for the Modern Earth atmosphere models (see Figure 9), whereas that due to the VRE is 0.4–0.5 +(0.1–0.15) for veg-only models (veg-land and mod-earth models). Bacteriochrolophylls’ fluorescence +causes a slight increase in reflectance around 1000 -1100 nm compared to the VRE. Such nonprominent +fluorescence emission with a Gaussian shape in the wavelength different from the VRE feature can +be extracted from the reflectance profile using data processing such as principal component analysis +(PCA). Photosynthetic organisms different from those around the Sun are expected to exhibit VRE +and fluorescence features in different wavelengths. Thus, not only spectral features due to atmospheric +molecules but also the simultaneous detection of the VRE and the fluorescence will help identify +traces of photosynthesis on an exoplanet. Probably, when we found a possible signal of VRE, the +fluorescence would be useful for further validation, because the VRE signal is stronger than the +fluorescence one. +3.3. Case-3: Anoxic World (without VRE) + +(a) veg-only 0C +(b) veg-land oC +(c) mod-earth 0C +0.8 +0.25 +0.25 +10 Fflour. +0.20 +0.20 +5 Fflour. +1 Fflour. +0.15 +0.15- +0.4 +O Fflour. +0.10 +0.10 +0.05 +0.05- +0.0 +0.0Q +0.0Q +009 +650 +700 +750 +800 +600 +650 +700 +750 +800 +600 +650 +700 +750 +800 +0.8 +0.25 +0.25 +0.20 +0.20 +J667C +0.15 +0.15 +0.4- +0.10 +0.10 +0.2 +0.05 +0.05 +0.0 +0.0Q +0.00 +600 +650 +700 +750 +800 +600 +650 +700 +750 +800 +009 +650 +700 +750 +800 +0.8 +0.25 +0.25 +0.2010 +0.2010 +TRAPPIST-1 +0.1510- +0.15 +0.4 +600 +700 +800 +600 +700 +800 +600 +700 +800 +0.10 +0.10 +20.2 +0.05 +0.05 +0.0 +0.0Q +0.0Q +600 +650 +700 +750 +800 +600 +650 +700 +750 +800 +600 +650 +700 +750 +800 +wavelength (nm) +wavelength (nm) +wavelength (nm)12 +Komatsu et al. +Figure 5. The same as Figure 4, but for an Earth-like planet with an oxygen-poor atmosphere (2.0 Ga). +Figure 6. Reflectance excess due to chlorophyll fluorescence emissions on an Earth-like planet with the +Modern Earth’s atmosphere. + +(a) veg-only 0C +(b) veg-land 0C +(c) mod-earth 0C +0.10 +10 Fflour. +0.03 +0.03 +5 Fflour. +1 Fflour. +0.02 +0.02 +Sun +0.05 +0.01 +0.01 +0.00 +0.00 +0.00 +650 +700 +750 +800 +650 +700 +750 +600 +800 +650 +600 +600 +700 +750 +800 +0.10 +0.03 +0.03 +GJ667C +0.02 +0.02 +0.05 +0.01 +0.01 +0.0Q +0.0Q +0.00 +600 +650 +700 +750 +800 +600 +650 +700 +750 +800 +600 +650 +700 +750 +800 +0.10- +0.03 +0.03 +10- +Difference in +10- +Reflectance +TRAPPIST-1 +0.02 +10-5 +10-5 +600 +700 +800 +600 +700 +800 +600 +700 +800 +0.05 +0.01 +0.01 +0.0Q +0.0Q +0.0Q +600 +650 +700 +750 +800 +600 +650 +700 +750 +800 +600 +650 +700 +750 +800 +wavelength (nm) +wavelength (nm) +wavelength (nm)(a) veg-only 2C +(b) veg-land 2C +(c) mod-earth 2C +0.8 +0.25 +0.25 +10 Fflour. +9 0.6 +0.20 +0.20 +5 Fflour. +0.15 +0.15- +sun +1 Fflour. +0.4 +O Fflour. +0.10. +0.10- +0.05 +0.05- +0.9 +0.00 +0.00 +009 +650 +700 +750 +800 +600 +650 +700 +750 +800 +600 +650 +700 +750 +800 +0.8 +0.25 +0.25 +0.20 +0.20 +J667C +0.15 +0.15 +0.4 +0.10. +0.10 +0.2 +0.05 +0.05. +0.0 +0.0Q: +0.00 +600 +650 +700 +750 +800 +600 +650 +700 +750 +800 +600 +650 +700 +750 +800 +0.8 +0.25 +0.25 +10 +100 +0.20+ +0.20{0-1 +TRAPPIST-1 +10 +0-1 +0.15↓0 +-2 +0.4 +600 +700 +800 +600 +700 +800 +600 +700 +800 +0.10 +0.10 +0.2 +0.05 +0.05 +0.0 +0.0Q: +0.0Q +600 +650 +700 +750 +800 +600 +650 +700 +750 +800 +600 +650 +700 +750 +800 +wavelength (nm) +wavelength (nm) +wavelength (nm)Photosynthetic fluorescence on Exoplanets +13 +Figure 7. The same as Figure 4 but for the reflectance of a planet covered with bacteriochlorophyll-based +vegetation. +Figure 8. The same as Figure 7, but for a planet with an oxygen-poor atmosphere (2.0 Ga). + +(a) veg-only 2B +(b) veg-land 2B +(c) mod-earth 2B +0.6 +0.20 +0.20 +10 Fflour. +0.5 +5 Fflour. +0.15 +0.15 +0.4 +sun +1 Fflour. +0.3 +0.10 +0.10 +O Fflour. +0.2 +0.05 +0.05 +0.1 +0.0 +0.00 +0.00 +900 +950 +1000 1050 1100 1150 1200 +900 +950 +1000 1050 1100 1150 1200 +900 +950 1000 1050 1100 1150 1200 +0.6 +0.20 +0.20 +0.5 +nce +0.15 +0.15 +J667C +0.4 +ctal +0.3 +0.10 +0.10 +D +G +0.2 +0.05 +0.05 +0.1 +0.0 +0.00 +0.00 +900 +950 +1000 1050 1100 1150 1200 +900 +950 1000 1050 1100 1150 1200 +900 +950 1000 1050 1100 1150 1200 +0.6 +0.20 +0.20 +0.5 +ince +0.15 +0.15 +TRAPPIST-1 +0.4 +Reflectar +0.3 +0.10 +0.10 +0.2 +0.05 +0.05 +0.1 +0.0 +0.00 +0.00 +900 +950 1000 1050 1100 1150 1200 +006 +950 1000 1050 1100 1150 1200 +006 +950 1000 1050 1100 1150 1200 +wavelength (nm) +wavelength (nm) +wavelength (nm)(a) veg-only OB +(b) veg-land OB +(c) mod-earth 0B +0.6 +0.20 +0.20 +10 Fflour. +0.5 +5 Fflour. +0.15 +0.15 +0.4 +sun +1 Fflour. +0.3 +0.10 +0.10 +O Fflour. +0.2 +0.05 +0.05 +0.1 +0.0 +0.00 +0.00 +900 +950 1000 1050 1100 1150 1200 +900 +950 1000 1050 1100 1150 1200 +900 +950 1000 1050 1100 1150 1200 +0.6 +0.20 +0.20 +0.5 +nce +0.15 +0.15 +0.4 +ctal +0.3 +0.10- +0.10- +D +G +0.2 +0.05 +0.05 +0.1 +0.0 +0.0Q +0.00 +900 +950 1000 1050 1100 1150 1200 +900 +950 1000 1050 1100 1150 1200 +900 +950 1000 1050 1100 1150 1200 +0.6 +0.20 +0.20 +0.5 +ince +0.15 +0.15 +TRAPPIST-1 +0.4 +Reflectar +0.3 +0.10 +0.10 +0.2 +0.05 +0.05 +0.1 +0.0 +0.00 +0.00 +900 +950 1000 1050 1100 1150 1200 +006 +950 1000 1050 1100 1150 1200 +006 +950 1000 1050 1100 1150 1200 +wavelength (nm) +wavelength (nm) +wavelength (nm)14 +Komatsu et al. +Figure 9. Reflectance excess due to bacteriochlorophyll fluorescence emissions on an Earth-like planet with +the Modern Earth’s atmosphere. +In case-3, an Earth-like planet has the same reduced atmosphere as the Archean Earth at 3.9 Ga. +Anoxic bacteria with photosynthetic pigments such as bacteriochlorophylls may spread over the +surface of a planet with a CO2-rich atmosphere. Anoxic bacteria are assumed to live in the ocean +and coast (i.e., cv = 0.72) and emit only fluorescence whose intensity is comparable to the standard +emission from land plants, without the distinct reflectance of a vegetation surface. +Fluorescence +emissions from anoxic bacteria adopt those from bacteriochlorophylls on the Earth. Figure 10 shows +the reflectance of an Archean-Earth-like planet with BChl-based bacteria. In the reflection spectra, a +strong water absorption appears around 950 and 1150 nm. The relatively high reflectance across the +wavelength range is mainly from the light reflected by the land. We observe fluorescence emissions +in the wavelength range between 1000 and 1100 nm owing to the lack of light reflected from BChl- +bearing oceanic bacteria, including the VRE feature. Intense absorption in the stellar atmosphere +enhances the apparent reflectance of a planet around TRAPPIST-1 (see also Figures 4 and 5, and +Section 4.3.2). +4. DISCUSSION +This study demonstrated reflectance with photosynthetic fluorescence on an Earth-like planet +around the Sun and two M dwarfs. This section reviews the biological processes of photosynthe- +sis and then considers the future detection of biofluorescence on an exoplanet. In Section 4.1, we +discuss the possible physiological conditions that enhance the fluorescence emissions on a planet +based on our understanding of Chl fluorescence. In Section 4.2, we discuss the possible false positive +or negative detection of fluorescence (Section 4.2.1), and the potential usage of the nonlinear photore- + +(a) veg-only OB +(b) veg-land OB +(c) mod-earth 0B +0.15 +0.04 +0.04 +10 Fflour. +nce +5 Fflour. +0.03 +0.03- +0.10 +1 Fflour. +sun +0.02 +0.02 +0.05 +0.01 +0.01 +0.00: +0.00 +0.00 +900 +1000 +1100 +1200 +900 +1000 +1100 +1200 +900 +1000 +1100 +1200 +0.15 +0.04 +0.04 +0.03 +0.03 +J667C +0.02 +0.02 +G +0.01 +0.01 +0.00 +0.00 +0.00 +900 +1000 +1100 +1200 +900 +1000 +1100 +1200 +900 +1000 +1100 +1200 +0.15 +0.04 +0.04 +Difference in +TRAPPIST-1 +ince +0.03 +0.03- +0.10 +Reflectal +0.02 +0.02- +0.05 +R +0.01 +0.01- +0.00 +0.00 +0.00 +900 +1000 +1100 +1200 +900 +1000 +1100 +1200 +900 +1000 +1100 +1200 +wavelength (nm) +wavelength (nm) +wavelength (nm)Photosynthetic fluorescence on Exoplanets +15 +Figure 10. The reflectance of an Earth-like planet with an anoxic atmosphere and no land vegetation +(anoxic B). + +anoxic B +0.15 +Reflectance +0.10 +Sun +0.05 +0.0Q +900 +1000 +1100 +1200 +0.15 +Reflectance +10 Fflour. +GJ667C +0.10 +5 Fflour. +1 Fflour. +0.05 +O Fflour. +0.00 +006 +1000 +1100 +1200 +0.15 +Reflectance +TRAPPIST-1 +0.10 +0.05 +0.00 +006 +1000 +1100 +1200 +wavelength (nm)16 +Komatsu et al. +sponse in fluorescence yield to excitation light intensity to distinguish between biofluorescence and +the false positive/negative signals of fluorescence (Section 4.2.2). Finally, in Section 4.3, we show the +fluorescence detection with telescopes. We present the detectability of fluorescence from an Earth +twin around a Sun-like star u sing the noise model for a LUVOIR-A-like mission (Section 4.3.1), and +the remarkable enhancement in the reflectance due to the absorption lines of stars, which could be +a promising feature for detection by high-dispersion spectroscopy, especially around ultracool stars +(Section 4.3.2). +4.1. Possible Physiological Conditions for Supporting Fluorescence Detection +This study adopted the typical fluorescence spectrum of Chl-containing plants and LH1–RC purified +from BChl b-bearing purple bacteria. The fluorescence spectrum of the LH1–RC complex suspended +in buffer solution was measured under laboratory conditions with a low concentration of LH1–RC in +the solution to avoid the reabsorption of fluorescence. Cells having LH1-RC in vivo would result in +an ∼ 50 nm shift in the spectral peak wavelength toward longer wavelengths under dense conditions, +because the reabsorption of fluorescence reduces the shorter-wavelength part of fluorescence. A red- +shifted fluorescence spectrum should still be observable because it is located within the atmospheric +window. For the fluorescence intensity of vascular plants on the ground, we referred to the standard +value (Ffluor.) for the fluorescence model on exoplanets in our simulations. The possible detection of +fluorescence emissions on exoplanets would require ≳ 5Ffluor. with BChl (see Figures 7, 8, and 10). +There are four potential factors that increase the fluorescence yield in photosynthetic organisms from +the biophysical viewpoint of photosynthetic studies on existing phototrophs on the Earth: +1. Increasing Chl/BChl concentration per land area +A high concentration of Chls and BChls enhances their fluorescence intensity. In general, the +Chl/BChl concentration in a cell increases for capturing as many photons as possible under +low light conditions. Fluorescence increases linearly with Chl/BChl concentration when cell +density is low. In contrast, the fluorescence intensity reaches a saturation level in highly dense +environments due to the reabsorption of fluorescence by cells (Du et al. 2017). +2. Small spectral overlap between absorption and fluorescence +The large separation between the main absorption band and its fluorescence band increases +the fluorescence intensity of concentrated cells. In photosynthetic organisms, the excitation +energy is transferred between Chls, and the Chl fluorescence tends to be emitted from long- +wavelength Chls (LWC), which has the reddest absorption band in a photosystem because +the excess excitation energy is easily trapped at the lowest energy level. A redshift in the +peak wavelength of fluorescence and a blueshift in absorption, which can be caused by the +modification of the vibronic interactions of pigments between surrounding proteins and solvent, +reduce the spectral overlap between fluorescence and absorption. The fluorescence emission +from LWCs is red-shifted to over 50 nm from that of bulk Chls in some conditions. Although +most plants have a small amount of LWCs in PSII and the Chl fluorescence is absorbed well +under high Chl concentrations, far-red absorbable LWC contributing to PSII has been reported +in some eukaryote algae (Fujita & Ohki 2004; Wilhelm & Jakob 2006; Kotabov´a et al. 2014; +Wolf et al. 2018; Kosugi et al. 2020). These algae show a significant fluorescence emission at +far-red-light wavelengths (700–800 nm) at room temperature, and some of them decrease the +overlap (Fujita & Ohki 2004; Kosugi et al. 2020). + +Photosynthetic fluorescence on Exoplanets +17 +3. Low photosynthetic efficiency +Photon loss in photosynthetic processes reduces the photon yield of fluorescence. Excitation +yield in PSII has increased throughout the evolutionary processes of photosystems. For ex- +ample, the increase in light use efficiency in oxygenic photosynthesis on Earth was achieved +by changing the light-harvesting antenna protein from the membrane superficial phycobili- +some in cyanobacteria to the light-harvesting Chl binding protein in eukaryotic algae. Fur- +thermore, the subsequent modification of LHCs achieved a higher photosynthetic quantum +yield in the evolution process. The maximum excitation yield in PSII of vascular plants is +estimated to be ∼ 0.9, whereas that of green algae and cyanobacteria is ∼ 0.8 and ∼ 0.6, +respectively (Schuurmans et al. 2015). Suppose phototrophs on an exoplanet are in the early +stage of evolution. In that case, the expected fluorescence yield may be high to compensate for +the low efficiency of photon yields in primitive photosynthesis. +4. Suppression of heat dissipation +Photon loss by the heat dissipation in photosynthetic pigments suppresses the photon yield of +fluorescence. Heat dissipation occurs in the vibrational relaxation of excited pigment molecules, +Chls, or accessory pigments such as carotenoids. Additionally, light-dependent protection mech- +anisms to dissipate the excess light energy as heat are inherent in all the cyanobacteria, algae, +and plants. The efficiency of heat dissipation largely depends on the molecular configuration +and the environment of pigments binding to proteins. The energy conversion rate from light to +heat in photosystems is crucial in estimating photosynthetic fluorescence on other planets. +Therefore, the fluorescence yield in photosynthetic pigments should fluctuate over time due to pho- +tosynthetic activity and heat dissipation. +4.2. Further Identification for Confirming Photosynthetic Fluorescence +4.2.1. Potential false positive/negative of biological fluorescence detection from exoplanets +Photosynthetic pigments on an exoplanet may be different from those on Earth, and the wavelength +relevant to fluorescence emission from exovegetation remains to be unknown. A possible fluorescence +signal on other planets can be a false positive or negative detection of biological activities. Poten- +tial main sources causing false positive/negative could be surface reflectance or fluorescence from +minerals on exoplanets. +Both Chl and BChl fluorescence in our study can be contaminated by +mineral fluorescence, but it is not plausible to expect the fluorescent minerals to cover a fraction +of a planetary surface comparable to Earth’s vegetation as far as our knowledge of the Earth’s +environment. +Recently, solar-induced mineral luminescence (SML) has been extracted from SIF +data obtained by remote sensing of the Earth (K¨ohler et al. 2021). They revealed that about 10% +of non-vegetated areas are weakly luminescent and speculated that luminescence came from some +spots covered by carbonate with Mn2+ and was comparable to SIF (or Chl fluorescence). However, +those areas are negligible on the planetary scale. On the other hand, mineral fluorescence could +pollute, to an extent, fluorescence in near-infrared, which includes the BChl fluorescence. For in- +stance, silicate (e.g., pyroxene and olivine) shows a prominent absorption around 1000 nm caused by +Fe2+ (Bishop et al. 2019; Klima et al. 2011; Sunshine & Pieters 1998). Its fluorescence could appear +in a slightly longer wavelength from the absorption, whose energy corresponds to the Stokes shift, like +other near-infrared fluorescent materials (Jackson et al. 2021; Selvaggio et al. 2020). While there are + +18 +Komatsu et al. +a variety of fluorescent minerals (e.g., fluorite, calcite, corundum), we do not deny the possibility that +the unexpectedly strong mineral fluorescence could be observed on exotic planets such as a carbide +exoplanet (Allen-Sutter et al. 2020) whose surface could be covered by diamond with lattice defects, +e.g., due to nitrogen-vacancy center (Schirhagl et al. 2014). To understand potential fluorescence fea- +tures from surface components of an exoplanet, e.g., rocks and minerals, characterizing atmospheric +features is helpful. Besides, as mentioned so far, the simultaneous detection of vegetation reflectance +(VRE) and fluorescence features could help identify photosynthesis. +4.2.2. Nonlinear photoresponse in photosynthesis +Photosynthetic organisms regulate metabolic processes to maximize the use of available photons +under light conditions and emit biological fluorescence by converting light energy via photochemical +reactions. The nonlinear response of the fluorescence yield to the excitation light intensity would +be a clue to finding the presence of photosynthetic organisms. If a planet is in an elliptical orbit, +the incident flux received by the planet from its host star varies with time. Fluorescence emissions +from nonbiological processes increase with incident light intensity. In contrast, a saturation level +of the fluorescence intensity from biological activities, such as photosynthesis, exists because the +quantum yields of Chl fluorescence vary according to the light environment and atmospheric CO2 +concentrations. +The quantum yields of Chl fluorescence are primarily involved in the reduction +states of electron acceptors of photosystems for electron transports and excitation energy quenching +by photoprotection mechanisms (see Genty et al. 1989; Krause & Weis 1991; Baker 2008). A sudden +intense light can induce the reduction in the electron acceptors of PSII, where oxidation of water to +generate O2 occurs as a primary step in photosynthesis. The presence of photoprotection mechanisms +also modulates the quantum yields of Chl fluorescence. When dark- or dim-light-adapted leaves are +suddenly irradiated with intense light, Chl fluorescence quantum yields rapidly increase by up to five +times. Accordingly, the relationship between fluorescence yield and excitation light intensity (i.e., +the number of absorbed photons) provides a hint to explore the origin of fluorescence on a planet. +4.3. Detectability of Biological Fluorescence by Future Telescopes +4.3.1. The Earth-Sun System as an Earth Twin in a LUVOIR-A-Like Mission +We investigated the detectability of fluorescence from an Earth twin around a Sun-like star at 10 pc +from the Earth, assuming a LUVOIR-A-like space telescope. Figure 11 presents the simulated spectra +of a second Earth around a Sun-like star at 10 pc with the biological fluorescence. We applied the +noise model used in Robinson et al. (2016) and Kopparapu et al. (2021), which accounts for planet +photons, stellar photon noise, and background noise, e.g., zodi, exozodi, read-out, and dark current +noises with the throughput assuming the LUVOIR-A telescope. The parameters and the formalism +used in this paper are presented in Appendix B. Figures 11(a–c) show the results of the most optimistic +model for the fluorescence signal (veg-only 0B) from the Earth-Sun system observed from 10 pc with +a 15 m space telescope. +The original data are the same as those of the Sun in Figure 7(a). In +Figure 11(a), Fp/Fs observed at the telescope for each wavelength bin is shown as solid lines, with +the random noise as the 1σ error bars for each bin, in 9000 hours of exposure time, where Fp is +the reflected light from the planet and Fs is the starlight. Figure 11(b) depicts a magnification of +the spectrum in Figure 11(a). Some error bars are outside the solid line, but the spectral feature of +fluorescence emission is recognizable for each case in the figure. Figure 11(c) shows the SNR with +the same observation time as that in Figure 11(a). The difference between 0 and 5 Ffluor. is larger + +Photosynthetic fluorescence on Exoplanets +19 +than 1σ. To detect the fluorescence with 3σ error, ∼ 50000 hours of exposure time are required, +and with 5σ, ∼ 100000 hours, ten years, are expected (not shown in figures). Thus, fluorescence +detection would require years for observation, even by the LUVOIR-A-like space telescope, and it +is extremely challenging to observe one target. In less optimistic models, namely, the veg-land 0B +model around the Sun in Figure 7(b), the detection of fluorescence signals is even more challenging, +as shown in Figure 11(d). As discussed in Section 3.2, the fluorescence in mod-earth 0B is difficult +to identify. Moreover, cloud coverage obscures the VRE features as well as atmospheric features +on exoplanets (Seager et al. 2005; Tinetti et al. 2006; Kaltenegger et al. 2007). The reflectance in +Figure 12 indicates how clouds suppress the fluorescence signal. Even in the most optimistic model, +the fluorescence in the reflectance is significantly reduced and can hardly be observed. In the mod- +earth model, it is impossible to identify the fluorescence signals. The only possible way to observe +surface vegetation with significant cloud coverage, except for atmospheric gases, would be the VRE +(∼0.1 in reflectance in the optimistic model). Thus, the existence of water clouds that are expected +in Earth-like planets with surface water seems to be critical for fluorescence detection. However, +around TRAPPIST-1, as the relevant argument was shown in Session 4.3.2, we found that the Chl +fluorescence in the K I lines was insensitive to the coverage by Earth clouds, which could be an +advantage in the Chl detection over BChl one. +The fluorescence feature would be poorly determined with 900 hours of exposure time with 1σ +errors, whereas the VRE feature can be identified. Even for a LUVOIR-A-like space telescope, an +enormous observational time would be needed to identify the fluorescence in addition to the VRE +with more confidence for detecting traces of photosynthesis. We also investigated the detectability of +fluorescence by a space telescope with a different diameter. A 6 m space telescope is recommended for +future space missions, according to Astro2020 Decadal Survey. With a 6-m diameter, ∼ 300,000 hours +of observation time are required to identify fluorescence. When we adopt a 30 m space telescope with +1σ errors, the required exposure time is reduced to ∼ 800 hours. Furthermore, one of the background +noises, i.e., the readout noise, can be suppressed with data processing because of increasing reads in +an exposure as implemented for H2RG infrared detectors (e.g., Brandt et al. 2017; Kuzuhara et al. +2018). When the readout noise is assumed to be zero all over the wavelengths, the required observation +times are reduced to ∼ 250,000, ∼ 7,000 and ∼ 500 hours with the 6-, 15-, and 30-m diameters. +4.3.2. Apparent Enhancement in Fluorescence around Ultracool Stars and Possible Detection with +High-Dispersion Spectroscopy +Figure 13 shows the contribution of fluorescence around three host stars. Around TRAPPIST-1 the +apparent enhancement in reflectance induced by fluorescence is significant compared to around the +other two stars because TRAPPIST-1 has strong absorption features spanning the wavelengths of +the fluorescence. Within the TRAPPIST-1 stellar absorption features, reflected light from the planet +is reduced, allowing the fluorescence emission to become a much larger fraction of the outgoing +flux (reflected + fluorescence) at these wavelengths. This is analogous to the methodology of SIF +detection with remote sensing observations and the retrieval processes by determining how much the +fluorescence influences the Fraunhofer lines (Maier et al. 2004). These spectroscopic features may be +widely used for fluorescence detection around ultracool stars. +Figure 13(a) shows that the reflectance is highly enhanced due to the absorption lines of K I in +the stellar spectrum of TRAPPIST-1, which is not affected by water clouds (Figure 12). The degree +of enhancement for each line depends on the atmospheric compositions of an Earth-like planet. Fig- + +20 +Komatsu et al. +Figure 11. Simulated spectrum with the biological fluorescence on a second Earth around a Sun-like star +at 10 pc from the Earth, assuming a LUVOIR-A-like space telescope. (a–c) The results from the veg-only +0B model and (d) Fp/Fs with the veg-land 0B model. (a) Fp/Fs with 9000 hours of observation time. The +solid line shows Fp/Fs and the error bar indicates the noise at each wavelength. (b) A magnification of +Fp/Fs in (a). (c) The SNR in (a). +ure 13(b,c) presents a spiky feature due to absorption of FeH and VO, as commonly observed around +ultracool stars. Therefore, observing the possible fluorescence signal with high spectral resolution +using extremely large ground telescopes would be worthwhile. +5. CONCLUSIONS +In this paper, we explored fluorescence from photosynthesis as a biosignature on an exoplanet for +future observations in great detail and identified the situations in which the signal could be enhanced, +and the regions of the spectrum where fluorescence from chlorophylls and bacteriochlorophylls could +be most detectable for Earth-like planets around different stars. We also described how we could +enhance the possibility to more definitively detect the action of photosynthesis. For direct imaging +observations, however, we found that the detection of fluorescence emissions would be extremely +challenging to observe and especially not feasible for the planned 6m space telescope. More details +are provided as follows. +We considered fluorescence emissions from Chl- and BChl-based vegetation in a clear-sky condition +on an Earth-like planet around the Sun and two M dwarfs (GJ667 C and TRAPPIST-1). Chl- and +BChl-based leaves show a VRE in wavelengths around 700–750 and 1000–1100 nm. The fluorescence +emissions from Chls and BChls occur at wavelengths from 650 to 800 nm and 1000 to 1100 nm, cor- +responding to the longest Q absorption band of each pigment. The two peaks of Chl fluorescence + +1e-9 +1e-10 +1.0 +(a) +(b) +0.8 +2.0 +10 Fflour. +1.5 +5 Fflour. +F +0.4 +O Fflour. +1.0 +0.2 +0.5 +0.0 +0.0 +006 +950 +1000 1050 1100 1150 1200 +980 +1000 +1020 +1040 +1060 +Wavelength [nm] +Wavelength [nm] +1e-10 +3.5 +(C) +d) +3.0 +2.5 +101 +FS 2.0 +SNR + 1.5 +10 Fflour. +5 Fflour. +1.0 +1 Fflour. +0.5 +O Fflour. +100. +0.0 +900 +950 +1000 1050 1100 1150 1200 +900 +950 +1000 1050 1100 1150 1200 +Wavelength [nm] +Wavelength[nmlPhotosynthetic fluorescence on Exoplanets +21 +Figure 12. The effect of cloud on the reflectance with veg-only 0B and 0C models. The models are the +same as the veg-only 0B in Figure 7 and the veg-only 0C in Figure 4 but with cloud coverage. +at 680 and 740 nm arise from the PSII and PSI, respectively. Thus, atmospheric absorption bands, +such as H2O, CH4, O2, and O3, and the VRE could be overlapped with the fluorescence emissions +from Chls and BChls. Chl fluorescence emission from PSI is blended with the steep VRE feature. +Fluorescence emitted from PSII on an Earth-like planet is the most promising feature for observation, +but it may also be reduced by nonphotochemical quenching processes and reabsorption of photons by +surrounding Chls. Conversely, the fluorescence emitted from BChls is not suppressed by the sharp +increase in the reflectance due to the VRE and atmospheric absorption by, for example, water va- +por, except for CH4 absorption around 1000 nm. Therefore, the BChl fluorescence in the wavelength +range of 1000–1100 nm, rather than Chl fluorescence, may be a more promising biosignature from +photosynthetic organisms on a planetary surface. In both cases of Chl- and BChl-based vegetation, +the simultaneous detection of the VRE and fluorescence is significant for identifying photosynthetic +activity on an exoplanet, because we do not know exactly what kind of vegetation exists in the planet +in principal and we need more information for further validation to identify the trace of photosyn- +thesis. If BChl-bearing photosynthetic bacteria inhabit water without any leaf or tree structures, +the fluorescence spectrum is the only surface reflectance feature that can be used to access such +underwater photosynthetic organisms, although the fluorescence signal would be reduced according +to the opacity of overlying liquid water. +Based on our understanding of photosynthesis, the intensity of fluorescence is lower in photosyn- +thetic bacteria compared to land plants. Here, we presented four factors that enhance the fluorescence +emission for possible detection of biological fluorescence on an exoplanet: (1) increase in Chl/BChl +concentration per land area, (2) small overlap of absorption and fluorescence spectrum, (3) low + +veg-only OB +veg-only 0C +0.5 +1.2 +0.4 +1.0 +0.3 +0.8 +Sun +0.6 +0.2 +0.4 +0.1 +0.2 +0.0 +0.Q: +900 +950 1000 1050 1100 1150 1200 +720 +740 +760 +780 +800 +0.5 +1.2 + 0.4 +1.0 +10 Fflour. +nc +J667C +0.8 +5 Fflour. +0.6 +1 Fflour. +efl +0.4 +O Fflour. +R 0.1 +0.2 +0.0 +0.0 +006 +950 1000 1050 1100 1150 1200 +720 +740 +760 +780 +800 +0.5 +1.2 +0.4 +TRAPPIST-1 +1.0 +nc +0.8 +0.6 +0.2 +efl +0.4 +R +0.1 +0.2 +0.0 +0.0 +900 +950 1000 1050 1100 1150 1200 +720 +740 +760 +780 +800 +wavelength (nm) +wavelength (nm)22 +Komatsu et al. +Figure 13. The apparent enhancement of fluorescence in reflectance due to stellar absorption around the +three template stars: (a) veg-only 0C model (Figure 4), (b) veg-only 0B model (Figure 7), and (c) anoxic B +model (Figure 10). +photosynthetic efficiency, and (4) suppression of heat dissipation. This study assumed a linear pho- +toresponse of fluorescence to excitation light intensity. If a planet is on a large elliptical orbit and +the telescope has sufficient sensitivity to temporally resolve changes in fluorescence as a function of +time, the nonlinear photoresponse from the biological fluorescence can be identified. Assuming a +LUVOIR-A-like mission, an enormous duration (around 9000 hours) would be required to detect the +BChl fluorescence emission, whose fluorescence yield is 5–10 times larger than that of vegetation on +Earth in the optimistic cases for an Earth-Sun twin at a distance of 10 pc from the Earth. In addition, +the cloud coverage significantly affects the detection of fluorescence as well as other spectral features +because the cloud more strongly obscures fluorescence emissions than the VRE feature. Interestingly, +the fluorescence in the reflectance was found to be remarkably enhanced in all three cases around +TRAPPIST-1 because of its strong absorption in the stellar atmosphere, like the SIF detection by +remote sensing using Fraunhofer lines. The reflectance excess due to K I absorption and VO/FeH +absorption can be a promising feature for characterizing the fluorescence around ultracool stars in +Chl and BChl cases. Note that Chl fluorescence in K I lines was still prominent with water clouds. +Thus, one of the most important future works would be the mock observation assuming a 30 +m class ground-based telescope to investigate how the apparent enhancement in reflectance due +to stellar absorption could help the fluorescence detection around ultracool stars. In addition, to +better support the future detection of fluorescence emissions on an exoplanet, further studies are +required from various perspectives. For example, planetary spectra for a wide range of atmospheric +and surface conditions consistent with biological fluorescence emission should be estimated and tested + +(a) veg-land oC +(b) veg-land OB +(c) anoxic B +0.08 +0.20 +0.7- +10 Fflour. +5 Fflour. +0.06 +0.15 +0.5 +1 Fflour. +Sun +0.04 +0.10. +0.3 +O Fflour. +0.02 +0.05 +0.1 +0.9 +0.00- +0.00. +766 +767 +768 +769 +770 +771 +1000 +1020 +1040 +1060 +1000 +1020 +1040 +1060 +0.08 +0.20 +0.7. +0.06- +0.15 +J667C +0.04- +0.10. +0.3 +G +0.02 +0.05 +0.1 +0.9 +0.00 +0.00 +767 +770 +766 +768 +769 +771 +1000 +1020 +1040 +1060 +1000 +1020 +1040 +1060 +0.5 +0.08 +0.20 + 0.4 +TRAPPIST-1 +Reflectance +0.06- +0.15 +0.3 +0.04- +0.10. +0.2 +0.02 +0.05- +0.1 +0.9 +0.00 +0.00. +766 +767 +768 +769 +770 +771 +1000 +1020 +1040 +1060 +1000 +1020 +1040 +1060 +wavelength (nm) +wavelength (nm) +wavelength (nm)Photosynthetic fluorescence on Exoplanets +23 +using radiation transfer calculations because our studies considered still-limited conditions. Moreover, +we need to conduct simulations on how the fluorescence is observed on an exoplanet when a global SIF +map data from remote sensing of the Earth are applied. Also, experimental validation of prominent +NIR fluorescence emissions is needed in some species of photosynthetic organisms and conditions. +ACKNOWLEDGMENTS +We would like to thank one anonymous reviewer for constructive comments to improve the paper. +We also thank Tatsuya Miyauchi, Haruki Oshio, Yu Someya, Tomoki Kiyono, and Masanori Takeda +for fruitful discussions at NIES on SIF detection by remote sensing, which led to the draft idea of +this study, and Kouki Hikosaka (Tohoku University) and Hibiki Noda (NIES) for further discussions +and for introducing SIF identification by remote sensing. The data for the LUVOIR noise model was +helpfully provided by Geronimo Villanueva and Ravi Kopparapu (NASA/Goddard). Y.H. and N.N. +were supported by a Grant-in-Aid for Scientific Research on Innovative Areas (JSPS KAKENHI +grant number 18H05439). +PyAstronomy (https://github.com/sczesla/PyAstronomy) was used in +mock observations assuming a space telescope. In several cases, numerical data were extracted from +figures in published papers using WebPlotDigitizer (https://automeris.io/WebPlotDigitizer/). +APPENDIX +A. EMPIRICAL RAYLEIGH SCATTERING +The effect of Rayleigh scattering is implemented empirically as follows (Bucholtz 1995): +τR(λ)=βs(λ)Ts +Ps +� z′ +0 +P(z) +T(z)dz, +(A1) +where τR is the Rayleigh optical depth at altitude z′; T(z) and P(z) are the temperature and pressure +at z, respectively. We adopted the T − P profile in the U.S. standard atmosphere 1976 from 0 to +60 km to compute the Rayleigh scattering cross-section in the atmosphere of an Earth-like planet. +The actual T − P profile in the atmosphere of an Earth-like planet around a star other than the Sun +is quite different from that in the Earth’s atmosphere. Rayleigh scattering, however, has a negligible +effect on the transmittance at wavelengths from 600 to 1100 nm (≈ 6 % in transmittance at 600 nm, +reducing with increasing wavelength, and then < 1 % at 1100 nm for an Earth-like planet around the +Sun, for instance), which is closely related to the fluorescence from Chls and BChls. Ts and Ps are +the temperature and pressure at standard conditions on Earth, respectively (Ts = 288.15 K and Ps += 1013.25 mbars). The total Rayleigh volume-scattering coefficient βs is expressed as: +βs(λ) = Aλ−B−Cλ−D/λ, +(A2) +where the coefficients A, B, C, and D are empirically determined (see Table 3 in Bucholtz (1995)). +B. LUVOIR NOISE MODEL +We implemented a noise model assuming a LUVOIR-A-like mission. The formalism and the pa- +rameters are based on Robinson et al. (2016), but, as shown in Table 2, we updated some parameters + +24 +Komatsu et al. +Parameter +Description +Adopted Value +D +Mirror Diameter +6, 15, 30 m +C +Raw Contrast +10−10 +R +Instrumental spectral resolution +70 +TTele +Accounts for light lost due to contamination +0.95 +and inefficiencies in the main collecting area +Tread +Read-out efficiency +0.75 +TQE +Raw quantum efficiency +0.9 +fpa +Fraction of planetary light that falls within photometric aperture +1 +X +Width of photometric aperture as multiple of λ/D +0.61 arcsec +Nez +Number of Exozodis +4.5 +De− +Dark current (UVIS/NIR) +3E-5/2E-3 e−/s +Re− +Read noise per pixel (UVIS/NIR)a +0/2.5 e− +θIWA +Inner working angle of the coronagraph as multiple of λ/D +3 +λ0 +Diffraction limit at the wavelength +500 nm +Table 2. Parameters for simulations based on a LUVOIR-A-like mission. +aTaken from the Planetary Spectrum Generator for LUVOIR/A-VIS and A-NIR, which is maintained by +NASA (https://psg.gsfc.nasa.gov/instrument.php). +(with several treatments) following Kopparapu et al. (2021) for our simulations with the LUVOIR-A +telescope. +The total noise in the observation Ctotal is calculated by: +Ctotal =Cp + Cs + Cb, +(B3) +where Cp is the number of planet photons, Cs is the stellar photon noise (leakage through the +coronagraph), and Cb is the background noise, which is the sum of zodi Cz, exozodi Cez, dark current +CD, and readout noise CR. The internal thermal noise is ignored because the thermal contribution is +negligible in our wavelengths of interest. Note that the noise in Equation B3 corresponds to variance +rather than the standard deviation. The noise count is expressed as: +Cnoise = +� +Cp + Cs + 2Cb +(B4) +where the double Cb accounts for the on-off observation with and without the planet. The on-off +observation corresponds to the subtraction of point spread functions of a central star. S/N for each +wavelength λ is defined by: +S/N = Cp +Cnoise +. +(B5) +The Fp and Fs are now defined to be the reflected light from a planet and the stellar flux acquired +by the telescope at a wavelength (bin) λ. When observing Fp/Fs, the 1σ error at λ is given as: +σ(λ)= Fp +Fs +1 +S/N. +(B6) + +Photosynthetic fluorescence on Exoplanets +25 +The end-to-end throughput for planetary fluxes is calculated as: +Ttotal =TTeleTcorToptTreadTQE, +(B7) +where TTele is an account for light lost due to contamination and inefficiencies in the main collecting +area, Tread is the read-out efficiency, and TQE is the raw quantum efficiency for the detector. The +coronagraphic Tcor and the optical Topt throughputs are the same as in Figure 9 in Kopparapu et al. +(2021). +We updated the formalism on noise from zodis, exozodis, and readout as follows; In Robinson et al. +(2016), the spectral shape of zodis (exozodis) was assumed to be equal to that of the Sun (the host +star). Instead, we explicitly adopt the normalized reflectance on solar zodis, ˜R⊙,λ, in the model +to better account for the zodical light in a exoplanetary system. We calculate ˜R⊙,λ by tracing the +spectral data from observations of the zodical light (see Figure 8 in Kawara et al. (2017) and Figure +10 in Tsumura et al. (2010)) with the normalization in the V band. Using ˜R⊙,λ, the noise from zodis +is expressed as: +Cz = πλ2D2 +4hcR +F⊙,λ(1au) +F⊙,V (1au) +˜R⊙,λF0,V 10−Mz,V /2.5TtotalΩ∆texp, +(B8) +where F⊙,λ is the solar flux density at λ, F⊙,V is the solar flux density in the V band, h is the +Planck constant, c is the speed of light, Mz,V = 23 mag arcsec−2 is the V -band zodical-light surface +brightness, and ∆texp is the exposure time. The circular photometry aperture size is expressed as +Ω = π(Xλ/D)2. Assuming the exozodis’s reflectance to be the same as ˜R⊙,λ, the noise from exozodis +is written as: +Cez = πλ2D2 +4hcR +�1au +r +�2 Fs,λ(1au) +Fs,V (1au) +Fs,V (1au) +F⊙,V (1au) +˜R⊙,λF0,V Nez10−Mez,V /2.5TtotalΩ∆texp, +(B9) +where Fs,λ is the stellar flux density at λ, Fs,V is the stellar flux density in the V band, and r is the +distance between the planet and the parent star. Mez,V = 22 mag arcsec−2 is the V -band exozodical +light surface brightness. 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M., et al. 2018, Photosynthesis research, 135, +177 +Yao, L., Yang, D., Liu, Y., et al. 2021, A New +Global Solar-induced Chlorophyll Fluorescence +(SIF) Data Product from TanSat +Measurements, Springer + diff --git a/PtE2T4oBgHgl3EQfVgea/content/tmp_files/load_file.txt b/PtE2T4oBgHgl3EQfVgea/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3107382c094f2300709dbb277304fe943c98b386 --- /dev/null +++ b/PtE2T4oBgHgl3EQfVgea/content/tmp_files/load_file.txt @@ -0,0 +1,1571 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf,len=1570 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='03824v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='EP] 10 Jan 2023 Draft version January 11, 2023 Typeset using LATEX preprint style in AASTeX63 Photosynthetic Fluorescence from Earth-Like Planets around Sun-Like and Cool Stars Yu Komatsu,1, 2 Yasunori Hori,1, 2 Masayuki Kuzuhara,1, 2 Makiko Kosugi,1, 2, 3 Kenji Takizawa,1, 3 Norio Narita,4, 1, 5 Masashi Omiya,1, 2 Eunchul Kim,3 Nobuhiko Kusakabe,1, 2 Victoria Meadows,6, 7 and Motohide Tamura1, 2, 8 1Astrobiology Center, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 3National Institute for Basic Biology, 38 Nishigonaka, Myodaiji, Okazaki, Aichi 444-8585, Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 4Komaba Institute for Science, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8902, Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 5Instituto de Astrof´ısica de Canarias (IAC), 38205 La Laguna, Tenerife, Spain 6Department of Astronomy and Astrobiology Program, University of Washington, Box 351580, Seattle, Washington 98195, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 7NASA Nexus for Exoplanet System Science, Virtual Planetary Laboratory Team, Box 351580, University of Washington, Seattle, Washington 98195, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 8Department of Astronomy, Graduate School of Science, The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' (Received August 15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Revised November 12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Accepted November 15, 2022) Submitted to ApJ ABSTRACT Remote sensing of the Earth has demonstrated that photosynthesis is traceable as the vegetation red edge (VRE), which is the steep rise in the reflection spectrum of vegeta- tion, and as solar-induced fluorescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' This study examined the detectability of bio- logical fluorescence from two types of photosynthetic pigments, chlorophylls (Chls) and bacteriochlorophylls (BChls), on Earth-like planets with oxygen-rich/poor and anoxic atmospheres around the Sun and M dwarfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Atmospheric absorption, such as H2O, CH4, O2, and O3, and the VRE obscure the fluorescence emissions from Chls and BChls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' We found that BChl-based fluorescence for wavelengths of 1000–1100 nm, assuming the spectrum of BChl b-bearing purple bacteria, could provide a suitable biosignature but only in the absence of the water cloud coverage or other strong absorbers near 1000 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The Chl fluorescence is weaker for several reasons, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=', spectral blending with the VRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The apparent reflectance excess is greatly increased in both Chl and BChl cases around TRAPPIST-1 due to fluorescence and stellar absorption lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' This could be a promising feature for detecting the fluorescence around ultracool red dwarfs by follow- up ground-based observations with high spectral resolution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' however, it requires a long time around Sun-like stars, even for a LUVOIR-like space mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Moreover, the simul- taneous detection of fluorescence and VRE is key to identifying traces of photosynthesis Corresponding author: Yu Komatsu yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='komatsu@nao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='jp 2 Komatsu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' because absorption, reflectance, and fluorescence are physically connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' For further validation of fluorescence detection, the nonlinear response of biological fluorescence as a function of light intensity could be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Keywords: astrobiology, planets and satellites: atmospheres, planets and satellites: sur- faces, planets and satellites: terrestrial planets 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' INTRODUCTION The ultimate goal of characterizing rocky planets is to identify potential biosignatures, spec- tral fingerprints of atmospheric gases, and surface features produced by biological activities (Des Marais et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Schwieterman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Meadows et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The simultaneous identifi- cation of oxygen, ozone, and methane on rocky habitable planets shows promise as a way to detect Earth-like life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Oxygenic photosynthesis produces a unique feature in the reflection spectrum on a planetary surface, called the vegetation red edge (VRE), as well as biosignature gases (Kiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2007a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The VRE is the steep difference in the reflection spectrum of the surface vegetation around 700 nm due to chlorophyll (Chl) absorption in the visible region and the large reflectance by cell structures in the near-infrared (NIR) region (Gates et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 1965;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Jacquemoud & Baret 1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Remote sensing of the Earth and Earthshine observations provide spectral indices involved in the VRE, such as the NDVI, which is a normalized difference in the reflection spectrum of the Earth between the visible and NIR wavelength regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The Moderate Resolution Imaging Spectroradiometer (MODIS) onboard NASA’s Terra satellite at 16-day intervals at 500 m and 1 km resolutions shows that the NDVI varies from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='05 to nearly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='9, whose upper limit is obtained at a dense forest site during the peak growing season (Huete et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Whereas remote sensing observes local areas on Earth, Earthshine observations provide disk-averaged spectra of the Earth, leading to fruitful insights into exoplanet applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The apparent reflectance change in the Earth’s disk-averaged spectrum due to surface vegetation is less than 2% (Monta˜n´es-Rodr´ıguez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The NDVI calculated from the Earthshine observations varies up to ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='10, depending on different views of the Earth, and is reduced by cloud coverage (Tinetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The application of NDVI to disk-averaged spectra assuming Earth-like exoplanets requires caution because remote sensing observes only local areas on the Earth to map vegetation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' For instance, Livengood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' (2011) found that additional spectral bands to NDVI are required to distinguish between the Earth vegetation and the Moon surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The VRE signals from exoplanets around stars other than a Sun-like star are challenging to predict due to the complexity of photosynthetic mechanisms in different light environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' However, the VRE on exoplanets may still be recognizable as an anomalous time-varying due to seasonal variability of the vegetation, and step-function-like spectroscopic feature at wavelengths different from those on the Earth (Seager et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Tinetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' (2006) proposed that if a three-photon photosynthetic scheme were working on exoplanets around M dwarfs, where there was little or no visible light, then the red edge of vegetation could also be shifted into the NIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' However, according to Takizawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' (2017), even around M dwarfs, the evolution of photosynthesis in water may drive a preference for using visible light rather than NIR, even after organisms colonize land surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Moreover, the light absorption properties of land vegetation could be optimized after long-term adaptive evolution de- pending on stellar irradiations as estimated by Lehmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Anoxygenic photosynthesis as performed by organisms such as purple bacteria, is thought to precede the emergence of oxygenic Photosynthetic fluorescence on Exoplanets 3 photosynthesis, whose global effect was characterized by the great oxidation event (∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='3 billion years ago (Ga)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Sanrom´a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' (2013) discussed the detectability of light reflected from purple bacteria with bacteriochlorophyll (BChl) as a photosynthetic pigment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' They showed that purple bacteria exhibit detectable features, and their VRE peak is redder than higher plants, assuming an Earth- like planet before the rise of oxygen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' In a comprehensive study of different pigment reflectivity, Schwieterman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' (2015) showed that both nonphotosynthetic pigments and photosynthetic pig- ments affect the disk-averaged spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Furthermore, as for false positive detection, the reflectance features of some minerals on the Earth are similar to the VRE ones (Seager et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Schwieterman 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Thus, extracting the VRE signal from reflected light should require knowledge of the surface environment on an exoplanet and high-resolution spectroscopic observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Fluorescence is another photosynthesis-related phenomenon that could also be a remote-sensing biosignature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Fluorescence is one of the de-excitation processes of photosynthetic pigments from the excited states to the ground state, along with intersystem crossing and inner conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Photosyn- thetic organisms on the Earth use Chls or BChls as light-absorbing pigments and electron donors/ac- ceptors in the primary reactions of photosynthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The photon energy captured by Chls/BChls is mainly transferred to the reaction center (RC), which is the pigment-protein complex at the center of the photosystem used for photochemical reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' A part of photon energy is, however, dissipated as heat or emitted as fluorescence from light-harvesting antenna systems, which are pigment-protein complexes surrounding RC that capture light energy and deliver the energy to the RC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Excess photon energy is preferentially removed as heat dissipation, rather than fluorescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' As a result, fluores- cence yield tends to be a smaller percentage of the excess energy and fluctuates with the degree of the excitation energy transfer (EET) between Chls, and heat dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The fluorescence yield of photosynthetic organisms is estimated to be ∼5%, whereas that of free Chls/BChls in organic solvents is ∼ 30% (Grimm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Plants and other oxygenic phototrophs use two different photosystems in sequence, that is, pho- tosystem II (PSII) and photosystem I (PSI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The energy level of the RC of PSII is higher, being equivalent to 680 nm, than that of PSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' In general, Chl fluorescence is mainly emitted from PSII because the excess light energy in PSI is immediately dissipated as heat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Therefore, the fluorescence spectrum of a cell has a peak at 680 nm, and the distribution of fluorescence emission extends to wavelengths up to 780 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Note that fluorescence emissions at 680 nm under highly concentrated Chls conditions, such as a leaf structure, decrease due to reabsorption by peripheral Chls with a red-absorption band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Conversely, the six BChls (BChl a, b, c, d, e, and g) used in non-oxygenic photosynthetic bacteria, such as purple bacteria, green sulfur and nonsulfur bacteria and heliobac- teria (Kiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2007b), mainly absorb far-red light in vivo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The BChl b in purple bacteria has the longest wavelength absorbance (1010 nm) and fluorescence (1050 nm) emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' However, the detailed characteristics of fluorescence from BChls, such as fluorescence yield and its variation in light environments, remain poorly understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' In contrast to the VRE which tracks the vegetation mass in the remote sensing of the Earth, fluo- rescence can be used as an indicator of active photosynthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The fluorescence signal emitted from the global ground vegetation, which is called solar-induced fluorescence (SIF), can be detected by remote sensing from satellites as excess light seen in the absorption of Fraunhofer lines in sunlight reflected from the Earth, which is the apparent increase in the reflectance spectrum due to fluores- cence (Maier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The observation of SIF is fundamentally challenging because the small SIF 4 Komatsu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' signal is overwhelmed by large background signals in the reflected sunlight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Then, high-resolution spectroscopy utilizes specific wavelengths with large solar absorption, which means the low intensity of reflected light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The SIF is observed as the in-filling effect at these wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' This methodology works because a large contrast is ensured between the Sun and the reflected light from the Earth at specific wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Thus, SIF has been observed in absorption bands by the Fourier high-dispersion spectrometers onboard many environmental satellites (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=', GOSAT (Hamazaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2013), GOME-2 (Callies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2000), and GOSAT-2 (Nakajima et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2012)), which produce the time- series SIF map of Earth (Frankenberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' We can extract information on the ground vegetation and atmospheric/surface environment, especially the gross primary production (GPP), from the changes in the fluorescence map by calibrating the remote observations with the results of local ground observations (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Such as the SIF in Earth observations, the detection of photosynthetic fluorescence in a planet around stars will investigate the surface envi- ronment and vegetation conditions on exoplanets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' High-resolution spectroscopy would be inevitable for the exofluorescence detection, and the contrast between a planet and its host star should be high enough at specific wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Biofluorescence, similar to that shown by coral reefs on Earth, has been suggested as a new potential biosignature for exoplanets experiencing strong UV radiation from F stars (O’Malley-James & Kaltenegger 2018) and M stars (O’Malley-James & Kaltenegger 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' It might work if the fluorescence were emitted very efficiently according to gained photons in their habitats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' As mentioned above, photosynthetic pigments are a potential emitter of biofluorescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' However, the yield and detectability of photosynthetic fluorescence on the surface of exoplanets have not yet been examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Finding surface biosignatures on Earth-like exoplanets, including the potential detectability of biofluorescence, would be one of the important goals of future astronomy and may become possible with future space missions such as the Large UV/Optical/IR Surveyor (LUVOIR) or the Habit- able Exoplanet Observatory (HabEx), and next-generation extremely large ground-based telescopes (TMT, ELT, and GMT) observing in reflected light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Thus, it is important to quantitatively evaluate the detectability of any potential surface biosignature using expected specifications of specific future missions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' This study made the first attempt to investigate the detectability of photosynthetic fluorescence on Earth-like exoplanets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The remainder of this paper is structured as follows: Section 2 describes the surface vegetation model for an Earth-like planet in the habitable zone and fluorescence emissions based on the photoresponse of photosynthetic organisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Section 3 shows the expected fluorescence emissions in the reflected light spectra on an Earth-like planet around an M dwarf or the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' In Sec- tion 4, we discuss the physiological conditions of photosynthesis that enhance fluorescence emissions and its unique features for future detection, including false-positive signals and seasonal changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Additionally, we present the detectability of biofluorescence by a future space-based telescope assum- ing the LUVOIR telescope parameters, and the key spectral feature possibly useful for the detection by follow-up observations with high-dispersion spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' In the last section, we summarize our paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' MATERIALS AND METHODS We assume that the radiation from a planetary surface is the sum of the reflected light on the surface and the fluorescence emission from photosynthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The outgoing flux on the surface and at Photosynthetic fluorescence on Exoplanets 5 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' (a) Incident radiation and (b) photon flux density at the top of atmosphere (TOA) of an Earth-like planet around the Sun, GJ667C, and TRAPPIST-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The spectral data of the Sun, GJ667C, and TRAPPIST-1 were obtained from Meftah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' (2018), France et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' (2016), and Lincowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' (2018), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' the top of atmosphere (TOA) is given by: F ↑ surface(λ)=F ↓ surface(λ)R(λ) + Ffluor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' (λ), (1) F ↑ TOA(λ)=F ↑ surface(λ)T(λ), (2) where λ is the wavelength, T(λ) is the atmospheric transmittance (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='1), R(λ) is the surface reflectance of a planet (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='3), F ↑ surface(λ) is the upward flux from a planetary surface, F ↑ TOA(λ) is the reflected flux at the TOA, and Ffluor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' (λ) is the net fluorescence emission from photosynthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' F ↓ surface(λ) = F ↓ TOA(λ)T(λ) is the downward flux from the planetary atmosphere to the surface, where F ↓ TOA is the incident flux from a host star at the TOA (see Figure 1 and Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='1 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' We neglect the effects of thermal emission in all the cases and Rayleigh scattering in most cases, as both processes contribute little radiation to our spectral region of interest (600- 1000 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The transmittance T(λ) in the atmosphere of an Earth-like planet through geological evolution was obtained from Rugheimer & Kaltenegger (2018), which was calculated by a 1D coupled radiative/convective-photochemical model for a planetary atmosphere (see also Pavlov & Kasting 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Kasting & Ackerman 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Segura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Stellar Radiation Two nearby M dwarfs, GJ667 C and TRAPPIST-1, have candidate planets in a habitable zone (HZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' We considered fluorescence emissions from photosynthesis on an Earth-like planet in an HZ around GJ667 C, TRAPPIST-1, and the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' We extracted the incident stellar flux from high-resolution spectral data for the Sun (Meftah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2018), GJ667 C (France et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2016), and TRAPPIST-1 (Lincowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The incident flux F ↓ TOA received by an Earth-like planet around GJ667 C, and TRAPPIST-1 is scaled by the current location of GJ667C c, and TRAPPIST-1e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' GJ667 C, and TRAPPIST-1 are modeled as M1V and M8V stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Figure 1 shows the incident flux received by an Earth-like planet at the TOA around the Sun, GJ667 C, and TRAPPIST-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Fluorescence from Photosynthesis le3 tons/s/m2/μm) lel Sun (un/zw/m) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='0 GJ667C TRAPPIST-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='5 (mmol-pho 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='6 Density 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='4 Flux 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='5 Photon Flux 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='0 0 250 500 750 10001250 15001750 2000 0 250 500 750 10001250150017502000 Wavelength(nm) Wavelength[nm]6 Komatsu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Fluorescence emissions from a planetary surface F ′ fluor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' are expressed as: F ′ fluor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' (λ) = scvπF std fluor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' × f(λ), (3) where cv is the surface coverage of vegetation (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='3), s is the scaling factor from the stan- dard observed fluorescence emission reflecting the photosynthetic activity, and F std fluor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' is the standard fluorescence intensity from vegetation based on field measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The spectral shape of fluores- cence emissions from a photosynthetic organism at wavelength λ is defined by f(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' In this study, F std fluor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='0 (W m−2 µm−1 sr−1) (Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2021) and s = 0, 1, 5, and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The net fluorescence intensity Ffluor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' is calculated by considering acquired photons at the habitat using F ′ fluor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' in Equation (3) as: Ffluor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' (λ) = χ χ0 F ′ fluor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' (λ), (4) χ ≡ � n(λ)σ(λ)dλ, (5) χ0 ≡ � nsun,ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' (λ)σchls(λ)dλ, (6) where χ is the light absorption efficiency, n(λ) is the photon flux density at the planetary surface, and σ(λ) is the absorption coefficient of a photosynthetic pigment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' χ0 represents the standard absorption efficiency on Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The subscript chls on σ(λ) represents chlorophylls (see Chl:abs in Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' nsun,ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' (λ) is the photon flux density on the surface of the Earth from the reference solar spectral irradiance at an air mass of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='5 (National Renewable Energy Laboratory), which corresponds to a typical irradiance for Earth vegetation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' We considered an incident flux from a star under two sky conditions, a clear sky and 60% cloud cover, to estimate the reflectance at the TOA and χ on the ground, in accordance with the setup for the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' We assumed the clear sky condition, if not specified, and the cloud condition appeared only in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='1 (Figure 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' In the cloudy condition, 60% of the radiation is reflected in three kinds of clouds, and 40% of the radiation reaches the ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' For the clouds, we assumed that 40% are low water clouds, 40% are high water clouds, and 20% are high ice clouds (Gao & Kaufman 2003) at 1, 6, and 12 km altitude, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' To model Earth-like conditions, the effect of Rayleigh scattering in a planetary atmosphere was also considered in the cloudy condition using a previously described empirical approach by Bucholtz (1995) (see Appendix A for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Equation (4) indicates that F ′ fluor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' is linearly scaled to the number of incoming photons that are absorbed by chlorophylls at the planetary surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' In other words, chlorophylls can emit strong fluorescence if the spectral shapes of n(λ) and σ(λ) match well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Note that n(λ) is exactly the same as F ↓ surface(λ), and its unit is shown in Figure 1(b) (F ↓ TOA in the figure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' This treatment in Equations (3) and (4) can be applied to the relationship between the incoming photons and the photons emitted as fluorescence on an Earth-like planet around various stars other than the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Figure 2 shows the normalized spectra of fluorescence f(λ) and absorption coefficient σ(λ) for Chls and BChls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The peak wavelength of f(λ) is red-shifted from that of σ(λ), which is called the Stokes shift (Lakowicz 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' There are two absorption bands in the σ(λ) of chlorophylls: the B band (known as the Soret band) in the short-wavelength region and the Q band in the long-wavelength region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The primary fluorescence emission is derived from the Q absorption band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' In this study, Photosynthetic fluorescence on Exoplanets 7 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Fluorescence (f(λ): solid curves) and photoabsorption (σ(λ): dashed curves) spectra for chloro- phylls (Chl: black) and bacteriochlorophylls (BChl: red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The absorption coefficient of chlorophylls in units of cm2 µg−1 was obtained from Feret et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The fluorescence spectrum is expressed by the Gaussian functions given in Frankenberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' (2012) and Guanter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' For bacteriochlorophylls, f(λ) and σ(λ) adopt those of the LH1–RC complex of a bacteriochlorophyll b containing purple photosynthetic bac- teria (Magdaong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The nondimensional absorption spectrum for BChl is normalized at the peak value in the longest absorption band, the Q band, of the Chl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Two fluorescence spectra are normalized at their peak values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' we modeled f(λ) for Chls as the superposition of two Gaussian distributions (Frankenberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Guanter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2010) with means of 680 nm (PSII) and 740 nm (PSI and PSII).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' σ(λ) for Chl uses the model vegetation with chlorophylls (σchls(λ)) (Feret et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' We obtained f(λ) and σ(λ) for BChls from the spectral data for the LH1–RC complex, the supramolecular complex of the light-harvesting core antenna (LH1), and the RC in a bacteriochlorophyll b containing purple photosynthetic bacteria (see Figure 3 in Magdaong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2016)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Note that we used only σ(λ) in the Q band for calculating χ and χ0 because free Chls and BChl–protein complexes in each solution affect each spectrum in the B band to different degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Surface Vegetation To determine the detectability of vegetation fluorescence, we use two leaf models for our exper- iments: one which assumes the reflectance spectrum and fluorescence of standard chlorophyll and another that uses a scaled version of the spectrum of bacteriochlorophyll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The reflectance of a planet is expressed as R(λ) = � i ciri(λ), where i denotes the surface type, ci is the fraction of the surface coverage of type i, and ri the reflectance of type i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' We obtained the reflection spectra for various surface types including vegetation, ocean, and coast from the USGS Digital Spectral Library and the ASTER Spectral Library (Baldridge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The detailed compositions used in this paper are summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The reflectance of the surface vegetation rv is estimated from radiation transfer calculations for a modeled leaf (Jacquemoud & Baret 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Feret et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2008), using σ(λ) over all the wavelengths shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Figure 3 shows the reflectance of a Chl-based leaf (“stan- Absorption coefficient (cm2/μg) Normalized fluorescence 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='0 400 600 800 1000 1200 Wavelength (nm) Chl: abs Chl: fl BChl: abs BChl: fl8 Komatsu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The reflectance of vegetation estimated from radiation transfer calculations for two leaf models: Chl (“standard”) and BChl (“hypothetical”) (Jacquemoud & Baret 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Feret et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The light absorption spectrum for Chls and BChls uses σ(λ) in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' dard”) and a BChl-based leaf (“hypothetical”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' In the latter case, we assumed the vegetation on a different planet has a photosynthetic pigment whose optical property is the same as BChl exhibiting the VRE in the longer wavelength region as shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' As the input to the radiative transfer calculations, we used the absorption spectra of Chl (Feret et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2008) and BChl (Magdaong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The unitless absorption spectrum for BChl is normalized at the peak in that for Chl, unlike the calculations of χ and χ0 in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' As shown in Figure 3, both Chl- and BChl-based leaves show a large reflectance (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=', the VRE) in the wavelength ranges around 700–750 and 1000–1100 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The green bump around 500 nm is observed in the reflectance for Chl, and larger and broader bumps are observed from ∼500 to 950 nm for BChls by the larger difference in the wavelength between the B and Q bands than observed for Chl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Like many kinds of photosynthetic organisms, the organisms with BChl could have acquired accessory pigments such as carotenoids (Cars) that absorb photons with wavelengths between the B and Q bands of Chl (Cogdell 1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The effective light absorption by accessory pigments can suppress the increase in reflectance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' With or without accessory pigments, the bump for BChl does not affect fluorescence emissions in the wavelength (see Figures 7, 8, and 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The low reflectance from ∼500 to 700 nm (1000 nm), due to the light absorption by Chls (BChls), affects the reflectance of the planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The degree of reduction in the overall planetary reflectance varies depending on the surface coverage by vegetation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' RESULTS We considered three fluorescence cases on an Earth-like planet at different stages of atmospheric evolution around the Sun, GJ667 C, and TRAPPIST-1 for different surface biosignatures: Earth-like (Chl) vegetation, hypothetical BChl-based vegetation, and biological fluorescence without any surface vegetation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Our models for the surface compositions, vegetation, fluorescence types, and atmospheric compositions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=', transmittance, are summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Mod-earth corresponds to the surface condition for the Modern Earth, leading to a lesser contribution of fluorescence emissions than in the other two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The veg-only models are considered optimistic conditions for fluorescence emissions where vegetation covers the whole planetary surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The veg-land models, with 70 % ocean, 2% coast, and 28% land covered with the vegetation, lie between the mod-earth and veg-only models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' As mentioned in Section 2, we considered two leaf models for land vegetation: Chl-based vegetation and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='5 Reflectance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='2 Chl 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='1 BChl 500 1000 1500 2000 Wavelength (nm)Photosynthetic fluorescence on Exoplanets 9 BChl-based vegetation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' For the atmospheric compositions of an Earth-like planet, we adopted the Modern Earth model at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='0 Ga (oxygen-rich atmosphere), the Paleoproterozoic Earth model at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='0 Ga (oxygen-poor atmosphere), and the Archean Earth model at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='9 Ga (anoxic atmosphere) (see Table 1 in Rugheimer & Kaltenegger 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' As an extreme case, we assumed the presence of photosynthetic bacteria with BChl spread over the land and ocean on an Archean-Earth-like planet with no surface vegetation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' We assumed a clear sky for all atmospheric conditions in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Model name Surface compositions Surface vegetation Fluorescence type T(λ) cv veg-only 0C Chl surf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Chl fluor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='0 Ga veg-only 2C 100% vegetation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='0 Ga 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='00 veg-only 0B BChl surf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' BChl fluor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='0 Ga veg-only 2B 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='0 Ga veg-land 0C Chl surf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Chl fluor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='0 Ga veg-land 2C 70% ocean, 2% coast 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='0 Ga 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='28 veg-land 0B and 28% vegetation BChl surf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' BChl fluor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='0 Ga veg-land 2B 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='0 Ga mod-earth 0C Chl surf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Chl fluor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='0 Ga mod-earth 2C 70% ocean, 2% coast 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='0 Ga 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='168 mod-earth 0B and 28 % mixed land BChl surf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' BChl fluor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='0 Ga mod-earth 2B (incl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='8% vegetation) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='0 Ga anoxic B 70% ocean, 2% coast and BChl fluor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='9 Ga 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='72 28% mixed land at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='9 Ga Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Surface composition, vegetation, its fluorescence types, and atmospheric transmittance (T(λ)) for all the cases in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Mixed land is composed of 60% vegetation (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='8% in total), 15% snow, 9% granite, 9% basalt, and 7% sand (Baldridge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2009);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' mixed land at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='9 Ga means the land model of the Archean Earth at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='9 Ga, which is composed of 35% basalt, 40% granite, 15% snow, and 10% sand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Chl surf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' and BChl surf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' correspond to reflection spectra of Chl and BChl in Figure 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The spectral shapes of fluorescence emissions f(λ) for Chl fluor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' and BChl fluor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' correspond to the fluorescence spectra of Chl and BChl in Figure 2, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' their intensities Fflour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' are scaled in Equations (3) and (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' cv is given by the relationship between the surface coverage of vegetation and the fluorescence emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' s={0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='0, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='0, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' We obtained T(λ) at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='0, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='9 Ga from Rugheimer & Kaltenegger (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Case-1: Planets with Earth-Like Vegetation In case-1, Earth-like vegetation (Chl) emits fluorescence on the surface of an Earth-like planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The fluorescence emissions from chlorophyll are visible at the wavelengths from 650 to 800 nm, as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' To determine the contribution of fluorescence from planets, the reflectance is defined as F ↑ TOA(λ)/F ↓ TOA(λ) and calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Figures 4 and 5 show the reflectance of an Earth-like planet with the Modern Earth’s atmosphere (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='0 Ga) and an oxygen-poor atmosphere (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='0 Ga), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The O2, O3, CH4, and H2O absorption features in the atmosphere are imprinted in the reflectivity in the visible–NIR wavelengths from 600–800 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The oxygen-poor-atmosphere models show less conspicuous patterns in the reflectance profile in the 700 to 750 nm wavelength region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The reflec- tivity between 600 and 700 nm is nearly constant but increases with decreasing surface coverage of 10 Komatsu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' vegetation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The VRE is observed as the steep rise in the reflectance from 700 to 750 nm (also see Figure 3), whereas the reflectance excess due to fluorescence is quite small, even in optimistic condi- tions (veg-only models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Note that the red curve with 1Ffluor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' around the Sun in the mod-earth model (Figure 4), corresponding to the modern earth fluorescence, is hardly seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Around TRAPPIST-1, however, sharp increase in the reflectance around 770 nm is due to the strong absorption of potassium in the stellar atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' As a result, we observed similar features in the light reflected from an Earth-like planet with different atmospheric compositions around TRAPPIST-1 (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='2 for further discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Figure 6 shows the reflectance excess due to fluorescence emissions on an Earth-like planet with the Modern Earth’s atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Atmospheric absorptions, such as H2O, O2, and O3, weaken the Gaussian features in the fluorescence emissions from an Earth-like planet around the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The fluorescence from chlorophylls around 740 nm is less pronounced for a planet around M dwarfs than one around the Sun because of weaker radiation flux in the wavelength region of 700–750 nm (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' In addition, a sudden increase in reflectance due to the VRE obscures the fluorescence emission around 740 nm (see Figures 4 and 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' As a result, the Chl fluorescence around 680 nm emitted from PSII on an Earth-like planet would be the most promising feature for detection (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Note that nonphotochemical quenching processes can decrease the fluorescence intensity around 680 nm, and the fluorescence emission is further reduced by the reabsorption of photons within the canopy (Porcar-Castell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Case-2: Planets with Bacteriochlorophylls-Based Vegetation In case-2, BChl-based vegetation, as the major photosynthetic pigment, covers the surface of a planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The BChls are assumed to emit the same degree of fluorescence intensity as the Earth’s vegetation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' As shown in Figure 2, fluorescence from BChls occurs in the wavelength range from 1000 to 1100 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' In contrast to case-1, fluorescence emissions with 5 and 10Ffluor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' show strong features around 1050 nm in almost all conditions in Figures 7 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Identifying the fluorescence on the Earth’s vegetation level (≲ Ffluor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=') is still challenging even in the optimistic case, that is, (a) veg-only 0B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The reflectivity between 1000 and 1050 nm becomes slightly higher for mod-earth models with less surface vegetation coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' As shown in Figure 9, the BChl organisms efficiently absorb photons and emit fluorescence with less absorption and scattering in the planetary atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The fluorescence emissions from BChls that we assumed are invulnerable to blending with the steep increase in the reflectance by the VRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' As a result, we found a more significant fluorescence contribution to the reflected light in case-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Atmospheric properties, such as chemical compositions and cloud coverage, change the fluorescence profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The water absorption is weak for wavelengths from 1000 to 1100 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' If the major absorption bands of a photosynthetic pigment lie in wavelengths longer or shorter than 1000–1100 nm, the pres- ence of water vapor in the atmosphere complicates the detection of fluorescence emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' A strong absorption due to CH4 in an oxygen-poor atmosphere also hides fluorescence near 1000 nm (see the GJ667C models in Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The BChl organisms bearing BChl b and their Stokes shift are ideal for detecting fluorescence in wavelengths longer than the characteristic wavelength of fluorescence from Chls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Thus, fluorescence in the wavelength range of 1000 -1100 nm could be a suitable biosignature for photosynthetic organisms, such as bacteriochlorophylls, on planetary surfaces unless they coexist with strong absorbers near 1000 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Photosynthetic fluorescence on Exoplanets 11 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Reflectance of an Earth-like planet with the Modern Earth’s atmosphere (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='0Ga) around the Sun, GJ667C, and TRAPPIST-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The three colors represent the reflected light from a planet with Ffluor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' (s = 1: red), 5Ffluor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' (s = 5: blue), and 10Ffluor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' (s = 10: green), where Ffluor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' is the fluorescence emission from chlorophylls observed on the Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' No-fluorescence emission models are also indicated by gray lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' We assumed Earth-like vegetation (chlorophylls) covers the planetary surface (see Table 1 for model details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The reflectance is defined here as F ↑ TOA(λ)/F ↓ TOA(λ), where F ↑ TOA(λ) is the light reflected from the ground at the top of atmosphere (TOA), and F ↓ TOA(λ) is the flux at TOA induced by stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' For each case around TRAPPIST-1, the reflectance with a logarithmic scale is also shown as the inset plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The VRE with a sharp rise in the reflectance is observed in the wavelength range from 1050 to 1100 nm in case-2, as shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Reflectance excess due to BChl fluorescence is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='01–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='05 for the Modern Earth atmosphere models (see Figure 9), whereas that due to the VRE is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='4–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='5 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='1–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='15) for veg-only models (veg-land and mod-earth models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Bacteriochrolophylls’ fluorescence causes a slight increase in reflectance around 1000 -1100 nm compared to the VRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Such nonprominent fluorescence emission with a Gaussian shape in the wavelength different from the VRE feature can be extracted from the reflectance profile using data processing such as principal component analysis (PCA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Photosynthetic organisms different from those around the Sun are expected to exhibit VRE and fluorescence features in different wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Thus, not only spectral features due to atmospheric molecules but also the simultaneous detection of the VRE and the fluorescence will help identify traces of photosynthesis on an exoplanet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Probably, when we found a possible signal of VRE, the fluorescence would be useful for further validation, because the VRE signal is stronger than the fluorescence one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Case-3: Anoxic World (without VRE) (a) veg-only 0C (b) veg-land oC (c) mod-earth 0C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='8 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='0Q 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='00 600 650 700 750 800 600 650 700 750 800 009 650 700 750 800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='2010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='2010 TRAPPIST-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='1510- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='15 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='0Q 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='0Q 600 650 700 750 800 600 650 700 750 800 600 650 700 750 800 wavelength (nm) wavelength (nm) wavelength (nm)12 Komatsu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The same as Figure 4, but for an Earth-like planet with an oxygen-poor atmosphere (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='0 Ga).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Reflectance excess due to chlorophyll fluorescence emissions on an Earth-like planet with the Modern Earth’s atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' (a) veg-only 0C (b) veg-land 0C (c) mod-earth 0C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='10 10 Fflour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='03 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='5 ince 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='15 TRAPPIST-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='4 Reflectar 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='00 900 950 1000 1050 1100 1150 1200 006 950 1000 1050 1100 1150 1200 006 950 1000 1050 1100 1150 1200 wavelength (nm) wavelength (nm) wavelength (nm)14 Komatsu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Reflectance excess due to bacteriochlorophyll fluorescence emissions on an Earth-like planet with the Modern Earth’s atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' In case-3, an Earth-like planet has the same reduced atmosphere as the Archean Earth at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='9 Ga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Anoxic bacteria with photosynthetic pigments such as bacteriochlorophylls may spread over the surface of a planet with a CO2-rich atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Anoxic bacteria are assumed to live in the ocean and coast (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=', cv = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='72) and emit only fluorescence whose intensity is comparable to the standard emission from land plants, without the distinct reflectance of a vegetation surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Fluorescence emissions from anoxic bacteria adopt those from bacteriochlorophylls on the Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Figure 10 shows the reflectance of an Archean-Earth-like planet with BChl-based bacteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' In the reflection spectra, a strong water absorption appears around 950 and 1150 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The relatively high reflectance across the wavelength range is mainly from the light reflected by the land.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' We observe fluorescence emissions in the wavelength range between 1000 and 1100 nm owing to the lack of light reflected from BChl- bearing oceanic bacteria, including the VRE feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Intense absorption in the stellar atmosphere enhances the apparent reflectance of a planet around TRAPPIST-1 (see also Figures 4 and 5, and Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' DISCUSSION This study demonstrated reflectance with photosynthetic fluorescence on an Earth-like planet around the Sun and two M dwarfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' This section reviews the biological processes of photosynthe- sis and then considers the future detection of biofluorescence on an exoplanet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='1, we discuss the possible physiological conditions that enhance the fluorescence emissions on a planet based on our understanding of Chl fluorescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='2, we discuss the possible false positive or negative detection of fluorescence (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='1), and the potential usage of the nonlinear photore- (a) veg-only OB (b) veg-land OB (c) mod-earth 0B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='04 10 Fflour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' nce 5 Fflour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 0.' metadata={'source': 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+page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='03 J667C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='02 G 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='00 900 1000 1100 1200 900 1000 1100 1200 900 1000 1100 1200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='04 Difference in TRAPPIST-1 ince 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='03- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='10 Reflectal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='02- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='05 R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='01- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='00 900 1000 1100 1200 900 1000 1100 1200 900 1000 1100 1200 wavelength (nm) wavelength (nm) wavelength (nm)Photosynthetic fluorescence on Exoplanets 15 Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The reflectance of an Earth-like planet with an anoxic atmosphere and no land vegetation (anoxic B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' anoxic B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='15 Reflectance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='10 Sun 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='0Q 900 1000 1100 1200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='15 Reflectance 10 Fflour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' GJ667C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='10 5 Fflour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 1 Fflour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='05 O Fflour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='00 006 1000 1100 1200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='15 Reflectance TRAPPIST-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='00 006 1000 1100 1200 wavelength (nm)16 Komatsu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' sponse in fluorescence yield to excitation light intensity to distinguish between biofluorescence and the false positive/negative signals of fluorescence (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Finally, in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='3, we show the fluorescence detection with telescopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' We present the detectability of fluorescence from an Earth twin around a Sun-like star u sing the noise model for a LUVOIR-A-like mission (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='1), and the remarkable enhancement in the reflectance due to the absorption lines of stars, which could be a promising feature for detection by high-dispersion spectroscopy, especially around ultracool stars (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Possible Physiological Conditions for Supporting Fluorescence Detection This study adopted the typical fluorescence spectrum of Chl-containing plants and LH1–RC purified from BChl b-bearing purple bacteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The fluorescence spectrum of the LH1–RC complex suspended in buffer solution was measured under laboratory conditions with a low concentration of LH1–RC in the solution to avoid the reabsorption of fluorescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Cells having LH1-RC in vivo would result in an ∼ 50 nm shift in the spectral peak wavelength toward longer wavelengths under dense conditions, because the reabsorption of fluorescence reduces the shorter-wavelength part of fluorescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' A red- shifted fluorescence spectrum should still be observable because it is located within the atmospheric window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' For the fluorescence intensity of vascular plants on the ground, we referred to the standard value (Ffluor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=') for the fluorescence model on exoplanets in our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The possible detection of fluorescence emissions on exoplanets would require ≳ 5Ffluor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' with BChl (see Figures 7, 8, and 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' There are four potential factors that increase the fluorescence yield in photosynthetic organisms from the biophysical viewpoint of photosynthetic studies on existing phototrophs on the Earth: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Increasing Chl/BChl concentration per land area A high concentration of Chls and BChls enhances their fluorescence intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' In general, the Chl/BChl concentration in a cell increases for capturing as many photons as possible under low light conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Fluorescence increases linearly with Chl/BChl concentration when cell density is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' In contrast, the fluorescence intensity reaches a saturation level in highly dense environments due to the reabsorption of fluorescence by cells (Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Small spectral overlap between absorption and fluorescence The large separation between the main absorption band and its fluorescence band increases the fluorescence intensity of concentrated cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' In photosynthetic organisms, the excitation energy is transferred between Chls, and the Chl fluorescence tends to be emitted from long- wavelength Chls (LWC), which has the reddest absorption band in a photosystem because the excess excitation energy is easily trapped at the lowest energy level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' A redshift in the peak wavelength of fluorescence and a blueshift in absorption, which can be caused by the modification of the vibronic interactions of pigments between surrounding proteins and solvent, reduce the spectral overlap between fluorescence and absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The fluorescence emission from LWCs is red-shifted to over 50 nm from that of bulk Chls in some conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Although most plants have a small amount of LWCs in PSII and the Chl fluorescence is absorbed well under high Chl concentrations, far-red absorbable LWC contributing to PSII has been reported in some eukaryote algae (Fujita & Ohki 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Wilhelm & Jakob 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Kotabov´a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Wolf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Kosugi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' These algae show a significant fluorescence emission at far-red-light wavelengths (700–800 nm) at room temperature, and some of them decrease the overlap (Fujita & Ohki 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Kosugi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Photosynthetic fluorescence on Exoplanets 17 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Low photosynthetic efficiency Photon loss in photosynthetic processes reduces the photon yield of fluorescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Excitation yield in PSII has increased throughout the evolutionary processes of photosystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' For ex- ample, the increase in light use efficiency in oxygenic photosynthesis on Earth was achieved by changing the light-harvesting antenna protein from the membrane superficial phycobili- some in cyanobacteria to the light-harvesting Chl binding protein in eukaryotic algae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Fur- thermore, the subsequent modification of LHCs achieved a higher photosynthetic quantum yield in the evolution process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The maximum excitation yield in PSII of vascular plants is estimated to be ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='9, whereas that of green algae and cyanobacteria is ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='8 and ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='6, respectively (Schuurmans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Suppose phototrophs on an exoplanet are in the early stage of evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' In that case, the expected fluorescence yield may be high to compensate for the low efficiency of photon yields in primitive photosynthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Suppression of heat dissipation Photon loss by the heat dissipation in photosynthetic pigments suppresses the photon yield of fluorescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Heat dissipation occurs in the vibrational relaxation of excited pigment molecules, Chls, or accessory pigments such as carotenoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Additionally, light-dependent protection mech- anisms to dissipate the excess light energy as heat are inherent in all the cyanobacteria, algae, and plants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The efficiency of heat dissipation largely depends on the molecular configuration and the environment of pigments binding to proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The energy conversion rate from light to heat in photosystems is crucial in estimating photosynthetic fluorescence on other planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Therefore, the fluorescence yield in photosynthetic pigments should fluctuate over time due to pho- tosynthetic activity and heat dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Further Identification for Confirming Photosynthetic Fluorescence 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Potential false positive/negative of biological fluorescence detection from exoplanets Photosynthetic pigments on an exoplanet may be different from those on Earth, and the wavelength relevant to fluorescence emission from exovegetation remains to be unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' A possible fluorescence signal on other planets can be a false positive or negative detection of biological activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Poten- tial main sources causing false positive/negative could be surface reflectance or fluorescence from minerals on exoplanets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Both Chl and BChl fluorescence in our study can be contaminated by mineral fluorescence, but it is not plausible to expect the fluorescent minerals to cover a fraction of a planetary surface comparable to Earth’s vegetation as far as our knowledge of the Earth’s environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Recently, solar-induced mineral luminescence (SML) has been extracted from SIF data obtained by remote sensing of the Earth (K¨ohler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' They revealed that about 10% of non-vegetated areas are weakly luminescent and speculated that luminescence came from some spots covered by carbonate with Mn2+ and was comparable to SIF (or Chl fluorescence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' However, those areas are negligible on the planetary scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' On the other hand, mineral fluorescence could pollute, to an extent, fluorescence in near-infrared, which includes the BChl fluorescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' For in- stance, silicate (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=', pyroxene and olivine) shows a prominent absorption around 1000 nm caused by Fe2+ (Bishop et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Klima et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Sunshine & Pieters 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Its fluorescence could appear in a slightly longer wavelength from the absorption, whose energy corresponds to the Stokes shift, like other near-infrared fluorescent materials (Jackson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Selvaggio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' While there are 18 Komatsu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' a variety of fluorescent minerals (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=', fluorite, calcite, corundum), we do not deny the possibility that the unexpectedly strong mineral fluorescence could be observed on exotic planets such as a carbide exoplanet (Allen-Sutter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2020) whose surface could be covered by diamond with lattice defects, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=', due to nitrogen-vacancy center (Schirhagl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' To understand potential fluorescence fea- tures from surface components of an exoplanet, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=', rocks and minerals, characterizing atmospheric features is helpful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Besides, as mentioned so far, the simultaneous detection of vegetation reflectance (VRE) and fluorescence features could help identify photosynthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Nonlinear photoresponse in photosynthesis Photosynthetic organisms regulate metabolic processes to maximize the use of available photons under light conditions and emit biological fluorescence by converting light energy via photochemical reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The nonlinear response of the fluorescence yield to the excitation light intensity would be a clue to finding the presence of photosynthetic organisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' If a planet is in an elliptical orbit, the incident flux received by the planet from its host star varies with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Fluorescence emissions from nonbiological processes increase with incident light intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' In contrast, a saturation level of the fluorescence intensity from biological activities, such as photosynthesis, exists because the quantum yields of Chl fluorescence vary according to the light environment and atmospheric CO2 concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The quantum yields of Chl fluorescence are primarily involved in the reduction states of electron acceptors of photosystems for electron transports and excitation energy quenching by photoprotection mechanisms (see Genty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Krause & Weis 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Baker 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' A sudden intense light can induce the reduction in the electron acceptors of PSII, where oxidation of water to generate O2 occurs as a primary step in photosynthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The presence of photoprotection mechanisms also modulates the quantum yields of Chl fluorescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' When dark- or dim-light-adapted leaves are suddenly irradiated with intense light, Chl fluorescence quantum yields rapidly increase by up to five times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Accordingly, the relationship between fluorescence yield and excitation light intensity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=', the number of absorbed photons) provides a hint to explore the origin of fluorescence on a planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Detectability of Biological Fluorescence by Future Telescopes 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The Earth-Sun System as an Earth Twin in a LUVOIR-A-Like Mission We investigated the detectability of fluorescence from an Earth twin around a Sun-like star at 10 pc from the Earth, assuming a LUVOIR-A-like space telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Figure 11 presents the simulated spectra of a second Earth around a Sun-like star at 10 pc with the biological fluorescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' We applied the noise model used in Robinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' (2016) and Kopparapu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' (2021), which accounts for planet photons, stellar photon noise, and background noise, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=', zodi, exozodi, read-out, and dark current noises with the throughput assuming the LUVOIR-A telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The parameters and the formalism used in this paper are presented in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Figures 11(a–c) show the results of the most optimistic model for the fluorescence signal (veg-only 0B) from the Earth-Sun system observed from 10 pc with a 15 m space telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The original data are the same as those of the Sun in Figure 7(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' In Figure 11(a), Fp/Fs observed at the telescope for each wavelength bin is shown as solid lines, with the random noise as the 1σ error bars for each bin, in 9000 hours of exposure time, where Fp is the reflected light from the planet and Fs is the starlight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Figure 11(b) depicts a magnification of the spectrum in Figure 11(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Some error bars are outside the solid line, but the spectral feature of fluorescence emission is recognizable for each case in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Figure 11(c) shows the SNR with the same observation time as that in Figure 11(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The difference between 0 and 5 Ffluor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' is larger Photosynthetic fluorescence on Exoplanets 19 than 1σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' To detect the fluorescence with 3σ error, ∼ 50000 hours of exposure time are required, and with 5σ, ∼ 100000 hours, ten years, are expected (not shown in figures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Thus, fluorescence detection would require years for observation, even by the LUVOIR-A-like space telescope, and it is extremely challenging to observe one target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' In less optimistic models, namely, the veg-land 0B model around the Sun in Figure 7(b), the detection of fluorescence signals is even more challenging, as shown in Figure 11(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' As discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='2, the fluorescence in mod-earth 0B is difficult to identify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Moreover, cloud coverage obscures the VRE features as well as atmospheric features on exoplanets (Seager et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Tinetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Kaltenegger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The reflectance in Figure 12 indicates how clouds suppress the fluorescence signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Even in the most optimistic model, the fluorescence in the reflectance is significantly reduced and can hardly be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' In the mod- earth model, it is impossible to identify the fluorescence signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The only possible way to observe surface vegetation with significant cloud coverage, except for atmospheric gases, would be the VRE (∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='1 in reflectance in the optimistic model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Thus, the existence of water clouds that are expected in Earth-like planets with surface water seems to be critical for fluorescence detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' However, around TRAPPIST-1, as the relevant argument was shown in Session 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='2, we found that the Chl fluorescence in the K I lines was insensitive to the coverage by Earth clouds, which could be an advantage in the Chl detection over BChl one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The fluorescence feature would be poorly determined with 900 hours of exposure time with 1σ errors, whereas the VRE feature can be identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Even for a LUVOIR-A-like space telescope, an enormous observational time would be needed to identify the fluorescence in addition to the VRE with more confidence for detecting traces of photosynthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' We also investigated the detectability of fluorescence by a space telescope with a different diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' A 6 m space telescope is recommended for future space missions, according to Astro2020 Decadal Survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' With a 6-m diameter, ∼ 300,000 hours of observation time are required to identify fluorescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' When we adopt a 30 m space telescope with 1σ errors, the required exposure time is reduced to ∼ 800 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Furthermore, one of the background noises, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=', the readout noise, can be suppressed with data processing because of increasing reads in an exposure as implemented for H2RG infrared detectors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=', Brandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Kuzuhara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' When the readout noise is assumed to be zero all over the wavelengths, the required observation times are reduced to ∼ 250,000, ∼ 7,000 and ∼ 500 hours with the 6-, 15-, and 30-m diameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Apparent Enhancement in Fluorescence around Ultracool Stars and Possible Detection with High-Dispersion Spectroscopy Figure 13 shows the contribution of fluorescence around three host stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Around TRAPPIST-1 the apparent enhancement in reflectance induced by fluorescence is significant compared to around the other two stars because TRAPPIST-1 has strong absorption features spanning the wavelengths of the fluorescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Within the TRAPPIST-1 stellar absorption features, reflected light from the planet is reduced, allowing the fluorescence emission to become a much larger fraction of the outgoing flux (reflected + fluorescence) at these wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' This is analogous to the methodology of SIF detection with remote sensing observations and the retrieval processes by determining how much the fluorescence influences the Fraunhofer lines (Maier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' These spectroscopic features may be widely used for fluorescence detection around ultracool stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Figure 13(a) shows that the reflectance is highly enhanced due to the absorption lines of K I in the stellar spectrum of TRAPPIST-1, which is not affected by water clouds (Figure 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The degree of enhancement for each line depends on the atmospheric compositions of an Earth-like planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Fig- 20 Komatsu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Simulated spectrum with the biological fluorescence on a second Earth around a Sun-like star at 10 pc from the Earth, assuming a LUVOIR-A-like space telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' (a–c) The results from the veg-only 0B model and (d) Fp/Fs with the veg-land 0B model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' (a) Fp/Fs with 9000 hours of observation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The solid line shows Fp/Fs and the error bar indicates the noise at each wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' (b) A magnification of Fp/Fs in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' (c) The SNR in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' ure 13(b,c) presents a spiky feature due to absorption of FeH and VO, as commonly observed around ultracool stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Therefore, observing the possible fluorescence signal with high spectral resolution using extremely large ground telescopes would be worthwhile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' CONCLUSIONS In this paper, we explored fluorescence from photosynthesis as a biosignature on an exoplanet for future observations in great detail and identified the situations in which the signal could be enhanced, and the regions of the spectrum where fluorescence from chlorophylls and bacteriochlorophylls could be most detectable for Earth-like planets around different stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' We also described how we could enhance the possibility to more definitively detect the action of photosynthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' For direct imaging observations, however, we found that the detection of fluorescence emissions would be extremely challenging to observe and especially not feasible for the planned 6m space telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' More details are provided as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' We considered fluorescence emissions from Chl- and BChl-based vegetation in a clear-sky condition on an Earth-like planet around the Sun and two M dwarfs (GJ667 C and TRAPPIST-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Chl- and BChl-based leaves show a VRE in wavelengths around 700–750 and 1000–1100 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The fluorescence emissions from Chls and BChls occur at wavelengths from 650 to 800 nm and 1000 to 1100 nm, cor- responding to the longest Q absorption band of each pigment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The two peaks of Chl fluorescence 1e-9 1e-10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='0 (a) (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='0 10 Fflour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='5 5 Fflour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='4 O Fflour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='0 006 950 1000 1050 1100 1150 1200 980 1000 1020 1040 1060 Wavelength [nm] Wavelength [nm] 1e-10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='5 (C) d) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='5 101 FS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='0 SNR 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='5 10 Fflour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 5 Fflour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='0 1 Fflour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='5 O Fflour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='0 900 950 1000 1050 1100 1150 1200 900 950 1000 1050 1100 1150 1200 Wavelength [nm] Wavelength[nmlPhotosynthetic fluorescence on Exoplanets 21 Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The effect of cloud on the reflectance with veg-only 0B and 0C models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The models are the same as the veg-only 0B in Figure 7 and the veg-only 0C in Figure 4 but with cloud coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' at 680 and 740 nm arise from the PSII and PSI, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Thus, atmospheric absorption bands, such as H2O, CH4, O2, and O3, and the VRE could be overlapped with the fluorescence emissions from Chls and BChls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Chl fluorescence emission from PSI is blended with the steep VRE feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Fluorescence emitted from PSII on an Earth-like planet is the most promising feature for observation, but it may also be reduced by nonphotochemical quenching processes and reabsorption of photons by surrounding Chls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Conversely, the fluorescence emitted from BChls is not suppressed by the sharp increase in the reflectance due to the VRE and atmospheric absorption by, for example, water va- por, except for CH4 absorption around 1000 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Therefore, the BChl fluorescence in the wavelength range of 1000–1100 nm, rather than Chl fluorescence, may be a more promising biosignature from photosynthetic organisms on a planetary surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' In both cases of Chl- and BChl-based vegetation, the simultaneous detection of the VRE and fluorescence is significant for identifying photosynthetic activity on an exoplanet, because we do not know exactly what kind of vegetation exists in the planet in principal and we need more information for further validation to identify the trace of photosyn- thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' If BChl-bearing photosynthetic bacteria inhabit water without any leaf or tree structures, the fluorescence spectrum is the only surface reflectance feature that can be used to access such underwater photosynthetic organisms, although the fluorescence signal would be reduced according to the opacity of overlying liquid water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Based on our understanding of photosynthesis, the intensity of fluorescence is lower in photosyn- thetic bacteria compared to land plants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Here, we presented four factors that enhance the fluorescence emission for possible detection of biological fluorescence on an exoplanet: (1) increase in Chl/BChl concentration per land area, (2) small overlap of absorption and fluorescence spectrum, (3) low veg-only OB veg-only 0C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='0 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='0 900 950 1000 1050 1100 1150 1200 720 740 760 780 800 wavelength (nm) wavelength (nm)22 Komatsu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The apparent enhancement of fluorescence in reflectance due to stellar absorption around the three template stars: (a) veg-only 0C model (Figure 4), (b) veg-only 0B model (Figure 7), and (c) anoxic B model (Figure 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' photosynthetic efficiency, and (4) suppression of heat dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' This study assumed a linear pho- toresponse of fluorescence to excitation light intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' If a planet is on a large elliptical orbit and the telescope has sufficient sensitivity to temporally resolve changes in fluorescence as a function of time, the nonlinear photoresponse from the biological fluorescence can be identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Assuming a LUVOIR-A-like mission, an enormous duration (around 9000 hours) would be required to detect the BChl fluorescence emission, whose fluorescence yield is 5–10 times larger than that of vegetation on Earth in the optimistic cases for an Earth-Sun twin at a distance of 10 pc from the Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' In addition, the cloud coverage significantly affects the detection of fluorescence as well as other spectral features because the cloud more strongly obscures fluorescence emissions than the VRE feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Interestingly, the fluorescence in the reflectance was found to be remarkably enhanced in all three cases around TRAPPIST-1 because of its strong absorption in the stellar atmosphere, like the SIF detection by remote sensing using Fraunhofer lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The reflectance excess due to K I absorption and VO/FeH absorption can be a promising feature for characterizing the fluorescence around ultracool stars in Chl and BChl cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Note that Chl fluorescence in K I lines was still prominent with water clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Thus, one of the most important future works would be the mock observation assuming a 30 m class ground-based telescope to investigate how the apparent enhancement in reflectance due to stellar absorption could help the fluorescence detection around ultracool stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' In addition, to better support the future detection of fluorescence emissions on an exoplanet, further studies are required from various perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' For example, planetary spectra for a wide range of atmospheric and surface conditions consistent with biological fluorescence emission should be estimated and tested (a) veg-land oC (b) veg-land OB (c) anoxic B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='7- 10 Fflour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 5 Fflour.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' 766 767 768 769 770 771 1000 1020 1040 1060 1000 1020 1040 1060 wavelength (nm) wavelength (nm) wavelength (nm)Photosynthetic fluorescence on Exoplanets 23 using radiation transfer calculations because our studies considered still-limited conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Moreover, we need to conduct simulations on how the fluorescence is observed on an exoplanet when a global SIF map data from remote sensing of the Earth are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Also, experimental validation of prominent NIR fluorescence emissions is needed in some species of photosynthetic organisms and conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' ACKNOWLEDGMENTS We would like to thank one anonymous reviewer for constructive comments to improve the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' We also thank Tatsuya Miyauchi, Haruki Oshio, Yu Someya, Tomoki Kiyono, and Masanori Takeda for fruitful discussions at NIES on SIF detection by remote sensing, which led to the draft idea of this study, and Kouki Hikosaka (Tohoku University) and Hibiki Noda (NIES) for further discussions and for introducing SIF identification by remote sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The data for the LUVOIR noise model was helpfully provided by Geronimo Villanueva and Ravi Kopparapu (NASA/Goddard).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' were supported by a Grant-in-Aid for Scientific Research on Innovative Areas (JSPS KAKENHI grant number 18H05439).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' PyAstronomy (https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='com/sczesla/PyAstronomy) was used in mock observations assuming a space telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' In several cases, numerical data were extracted from figures in published papers using WebPlotDigitizer (https://automeris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='io/WebPlotDigitizer/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' EMPIRICAL RAYLEIGH SCATTERING The effect of Rayleigh scattering is implemented empirically as follows (Bucholtz 1995): τR(λ)=βs(λ)Ts Ps � z′ 0 P(z) T(z)dz, (A1) where τR is the Rayleigh optical depth at altitude z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' T(z) and P(z) are the temperature and pressure at z, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' We adopted the T − P profile in the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' standard atmosphere 1976 from 0 to 60 km to compute the Rayleigh scattering cross-section in the atmosphere of an Earth-like planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The actual T − P profile in the atmosphere of an Earth-like planet around a star other than the Sun is quite different from that in the Earth’s atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Rayleigh scattering, however, has a negligible effect on the transmittance at wavelengths from 600 to 1100 nm (≈ 6 % in transmittance at 600 nm, reducing with increasing wavelength, and then < 1 % at 1100 nm for an Earth-like planet around the Sun, for instance), which is closely related to the fluorescence from Chls and BChls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Ts and Ps are the temperature and pressure at standard conditions on Earth, respectively (Ts = 288.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='15 K and Ps = 1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='25 mbars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The total Rayleigh volume-scattering coefficient βs is expressed as: βs(λ) = Aλ−B−Cλ−D/λ, (A2) where the coefficients A, B, C, and D are empirically determined (see Table 3 in Bucholtz (1995)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' LUVOIR NOISE MODEL We implemented a noise model assuming a LUVOIR-A-like mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The formalism and the pa- rameters are based on Robinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' (2016), but, as shown in Table 2, we updated some parameters 24 Komatsu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Parameter Description Adopted Value D Mirror Diameter 6, 15, 30 m C Raw Contrast 10−10 R Instrumental spectral resolution 70 TTele Accounts for light lost due to contamination 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='95 and inefficiencies in the main collecting area Tread Read-out efficiency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='75 TQE Raw quantum efficiency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='9 fpa Fraction of planetary light that falls within photometric aperture 1 X Width of photometric aperture as multiple of λ/D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='61 arcsec Nez Number of Exozodis 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='5 De− Dark current (UVIS/NIR) 3E-5/2E-3 e−/s Re− Read noise per pixel (UVIS/NIR)a 0/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='5 e− θIWA Inner working angle of the coronagraph as multiple of λ/D 3 λ0 Diffraction limit at the wavelength 500 nm Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Parameters for simulations based on a LUVOIR-A-like mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' aTaken from the Planetary Spectrum Generator for LUVOIR/A-VIS and A-NIR, which is maintained by NASA (https://psg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='gov/instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='php).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' (with several treatments) following Kopparapu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' (2021) for our simulations with the LUVOIR-A telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The total noise in the observation Ctotal is calculated by: Ctotal =Cp + Cs + Cb, (B3) where Cp is the number of planet photons, Cs is the stellar photon noise (leakage through the coronagraph), and Cb is the background noise, which is the sum of zodi Cz, exozodi Cez, dark current CD, and readout noise CR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The internal thermal noise is ignored because the thermal contribution is negligible in our wavelengths of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Note that the noise in Equation B3 corresponds to variance rather than the standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The noise count is expressed as: Cnoise = � Cp + Cs + 2Cb (B4) where the double Cb accounts for the on-off observation with and without the planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The on-off observation corresponds to the subtraction of point spread functions of a central star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' S/N for each wavelength λ is defined by: S/N = Cp Cnoise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' (B5) The Fp and Fs are now defined to be the reflected light from a planet and the stellar flux acquired by the telescope at a wavelength (bin) λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' When observing Fp/Fs, the 1σ error at λ is given as: σ(λ)= Fp Fs 1 S/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' (B6) Photosynthetic fluorescence on Exoplanets 25 The end-to-end throughput for planetary fluxes is calculated as: Ttotal =TTeleTcorToptTreadTQE, (B7) where TTele is an account for light lost due to contamination and inefficiencies in the main collecting area, Tread is the read-out efficiency, and TQE is the raw quantum efficiency for the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The coronagraphic Tcor and the optical Topt throughputs are the same as in Figure 9 in Kopparapu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' We updated the formalism on noise from zodis, exozodis, and readout as follows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' In Robinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' (2016), the spectral shape of zodis (exozodis) was assumed to be equal to that of the Sun (the host star).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Instead, we explicitly adopt the normalized reflectance on solar zodis, ˜R⊙,λ, in the model to better account for the zodical light in a exoplanetary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' We calculate ˜R⊙,λ by tracing the spectral data from observations of the zodical light (see Figure 8 in Kawara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' (2017) and Figure 10 in Tsumura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' (2010)) with the normalization in the V band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Using ˜R⊙,λ, the noise from zodis is expressed as: Cz = πλ2D2 4hcR F⊙,λ(1au) F⊙,V (1au) ˜R⊙,λF0,V 10−Mz,V /2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='5TtotalΩ∆texp, (B8) where F⊙,λ is the solar flux density at λ, F⊙,V is the solar flux density in the V band, h is the Planck constant, c is the speed of light, Mz,V = 23 mag arcsec−2 is the V -band zodical-light surface brightness, and ∆texp is the exposure time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' The circular photometry aperture size is expressed as Ω = π(Xλ/D)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Assuming the exozodis’s reflectance to be the same as ˜R⊙,λ, the noise from exozodis is written as: Cez = πλ2D2 4hcR �1au r �2 Fs,λ(1au) Fs,V (1au) Fs,V (1au) F⊙,V (1au) ˜R⊙,λF0,V Nez10−Mez,V /2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content='5TtotalΩ∆texp, (B9) where Fs,λ is the stellar flux density at λ, Fs,V is the stellar flux density in the V band, and r is the distance between the planet and the parent star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Mez,V = 22 mag arcsec−2 is the V -band exozodical light surface brightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' Even if the original treatment of exozodical light is adopted, our results do not significantly vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' We calculate the read-out noise (CR) to be CR = NpixNreadR2 e− instead of CR = NpixNreadRe− in Robinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' (2016) to more realistically incorporate the noise propagation, where Npix is the number of contribution pixels, Nread is the number of reads at each observation, and Re− is the read noise count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=' REFERENCES Allen-Sutter, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE2T4oBgHgl3EQfVgea/content/2301.03824v1.pdf'} +page_content=', Garhart, E.' metadata={'source': 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a/Q9E4T4oBgHgl3EQflA2r/content/tmp_files/2301.05156v1.pdf.txt b/Q9E4T4oBgHgl3EQflA2r/content/tmp_files/2301.05156v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b5199be370776c3ebf37f8d1af63e7476d7134da --- /dev/null +++ b/Q9E4T4oBgHgl3EQflA2r/content/tmp_files/2301.05156v1.pdf.txt @@ -0,0 +1,731 @@ +Hadronic observables from master-field simulations +Marco Cè,𝑎,∗ Mattia Bruno,𝑏 John Bulava,𝑐 Anthony Francis,𝑑 Patrick Fritzsch,𝑒 +Jeremy R. Green,𝑐 Maxwell T. Hansen 𝑓 and Antonio Rago𝑔,ℎ +𝑎Albert Einstein Center for Fundamental Physics (AEC) and Institut für Theoretische Physik, Universität +Bern, Sidlerstrasse 5, 3012 Bern, Switzerland +𝑏Dipartimento di Fisica “Giuseppe Occhialini”, Università degli Studi di Milano-Bicocca and INFN - +Sezione di Milano Bicocca, Piazza della Scienza 3, 20126 Milan, Italy +𝑐Deutsches Elektronen-Synchrotron DESY, Platanenallee 6, 15738 Zeuthen, Germany +𝑑Institute of Physics, National Yang Ming Chiao Tung University, Hsinchu, Taiwan 30010 +𝑒School of Mathematics and Hamilton Mathematics Institute, Trinity College Dublin, Dublin 2, Ireland +𝑓 Higgs Centre for Theoretical Physics, School of Physics and Astronomy, The University of Edinburgh, +Edinburgh EH9 3FD, United Kingdom +𝑔IMADA and CP3, University of Southern Denmark, Odense, Denmark +ℎDepartment of Theoretical Physics, CERN, 1211 Geneva 23, Switzerland +E-mail: marcoce@itp.unibe.ch +Substantial progress has been made recently in the generation of master-field ensembles. This +has to be paired with efficient techniques to compute observables on gauge field configurations +with a large volume. Here we present the results of the computation of hadronic observables, +including hadron masses and meson decay constants, on large-volume and master-field ensembles +with physical volumes of up to (18 fm)4 and 𝑚 𝜋𝐿 up to 25, simulated using 𝑁f = 2 + 1 stabilized +Wilson fermions. We obtain sub-percent determinations from single gauge configurations with the +combined use of position-space techniques, volume averages and master-field error estimation. +The 39th International Symposium on Lattice Field Theory, +8th-13th August, 2022, +Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, Germany +∗Speaker +© Copyright owned by the author(s) under the terms of the Creative Commons +Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). +https://pos.sissa.it/ +arXiv:2301.05156v1 [hep-lat] 12 Jan 2023 + +Hadronic observables from master-field simulations +Marco Cè +1. +Introduction +Gauge-field configurations in a lattice theory with a mass gap have the stochastic locality property, +that is, gauge-invariant local fields at large physical separations are stochastically independent. +The master-field paradigm introduced by Lüscher [1] proposes to use stochastic locality to obtain +observable estimates from a single or at most a few representative gauge-field configurations on very +large lattices, making use of the invariance under translations of the theory and of volume averages. +As a first application of this paradigm, stochastic locality has been used to compute the +topological susceptibility at 𝑇 > 𝑇𝑐 in master-field simulations of SU(3) Yang–Mills theory [2]. In +a theory with fermions such as QCD, numerical simulations are performed after integrating out +fermions exactly. Hadronic observables in QCD are expressed in terms of contractions of quark +propagators whose locality is not manifest. Moreover, the sheer size of the lattices requires stabilising +measures that have been studied in ref. [3]. These include a slight modification of the standard +𝑂(𝑎)-improved lattice Dirac operator, replacing the HMC with the stochastic molecular dynamics +(SMD) algorithm, employing quadruple-precision lattice sums and uniform-norm stopping criteria +for the Dirac equation solver. Recent progress in master-field simulations has been presented at the +Lattice 2021 conference [4, 5] and at this conference [6]. In these proceedings we further develop +the position-space techniques introduced in ref. [5], by presenting an estimator for position-space +correlators that scales efficiently with the volume. +Estimation of observables on master fields is explained in details in ref. [1]. In summary, the +expectation value ⟨O(𝑥)⟩ of a local field O(𝑥) is obtained averaging over translations +⟪O(𝑥)⟫ = 1 +𝑉 +∑︁ +𝑧 +O(𝑥 + 𝑧), +⟨O(𝑥)⟩ = ⟪O(𝑥)⟫ + 𝑂 +� +𝑉−1/2� +, +(1) +with the variance of this estimator given by +𝜎2 +⟪O⟫(𝑥) = +� +[⟪O(𝑥)⟫ − ⟨O(𝑥)⟩]2� += 1 +𝑉 +∑︁ +𝑦 +⟨O(𝑦)O(0)⟩𝑐 += 1 +𝑉 +������ +∑︁ +|𝑦|≤𝑅 +⟨O(𝑦)O(0)⟩𝑐 + 𝑂 +� +e−𝑚𝑅������� += 1 +𝑉 +������ +∑︁ +|𝑦|≤𝑅 +⟪O(𝑦)O(0)⟫𝑐 + 𝑂 +� +e−𝑚𝑅� ++ 𝑂 +� +𝑉−1/2������� +, +(2) +where in the second line we first used the fact that the connected correlator of the local field O(𝑥) +decays exponentially with spacetime separation, and then we applied again translation averages. +2. +Position-space correlators +In this work we focus on correlation functions in position space +𝐶𝑃𝑃(𝑥) → 𝑐2 +𝑃 +4𝜋2 +𝑚 𝜋 +|𝑥| 𝐾1(𝑚 𝜋|𝑥|), +(3a) +𝐶𝐴𝑃,𝜇(𝑥) → 𝑐𝐴𝑐𝑃 +4𝜋2 +𝑥𝜇 +|𝑥| +𝑚 𝜋 +|𝑥| 𝐾2(𝑚 𝜋|𝑥|), +(3b) +𝐶𝐴𝐴,𝜇𝜈(𝑥) → +𝑐2 +𝐴 +4𝜋2 +� +−𝛿𝜇𝜈 +1 +𝑥2 𝐾2(𝑚 𝜋|𝑥|) + 𝑥𝜇𝑥𝜈 +𝑥2 +�𝑚 𝜋 +|𝑥| 𝐾1(𝑚 𝜋|𝑥|) + 4 +𝑥2 𝐾2(𝑚 𝜋|𝑥|) +�� +, +(3c) +𝐶𝑁 𝑁 (𝑥) → +𝑐2 +𝑁 +4𝜋2 +𝑚2 +𝑁 +|𝑥| +� +𝐾1(𝑚𝑁 |𝑥|) + /𝑥 +|𝑥| 𝐾2(𝑚𝑁 |𝑥|) +� +, +(3d) +2 + +Hadronic observables from master-field simulations +Marco Cè +Table 1: Parameters of the master-field lattices used in this study (with 𝑎 ≈ 0.094 fm and 𝑚 𝜋 ≈ 270 MeV, +see also ref. [4]), together with information on the statistics used in the observable computation as explained +in section 3. +𝐿/𝑎 +𝐿 [fm] +𝑚 𝜋𝐿 +𝑛cnfg +𝑏/𝑎 +|𝐺| +𝑛shift +𝑏shift/𝑎 +𝑛point +A +96 +9 +12.5 +5 +48 +8 +512 +12 +4096 +B +192 +18 +25 +2 +48 +128 +32 +24 +4096 +where the subscript indicates the two-point function of either pseudoscalar densities 𝑃 = ¯𝑢𝛾5𝑑, axial +current 𝐴𝜇 = ¯𝑢𝛾𝜇𝛾5𝑑 or nucleon spinor 𝑁 = 𝜖𝑎𝑏𝑐(𝑢𝑇 +𝑎𝐶𝛾5𝑑𝑏)𝑢𝑐, as a function of the source-sink +separation 𝑥. +In eqs. (3), the asympotic behaviour for 𝑥 → ∞ of these correlators is given assuming the +symmetries of the continuum theory in an infinite volume. From position-space correlators one +can extract simple hadronic observables, including the masses 𝑚 𝜋 and 𝑚𝑁 and the decay constant +𝑓𝜋 = 𝑐𝐴/𝑚 𝜋, as demonstrated in ref. [5]. +Once computed on the lattice as discussed in the following section, these correlators as a +function of the four-dimensional source-sink separation 𝑥 include lattice discretization effects that +break the rotational symmetry and depend on the direction of 𝑥. In this study, we limit ourselves to +the radial correlators ˚𝐶(𝑟) introduced in ref. [5] that are averaged over 𝑆3(𝑟) = {𝑥 ∈ R4 : |𝑥| = 𝑟}, +the 3-sphere of radius 𝑟, and by construction depend only on the radial coordinate 𝑟 = |𝑥|. While +˚𝐶𝑃𝑃(𝑟) = 𝐶𝑃𝑃(𝑥), for the 𝐴𝑃-correlator 𝐶𝐴𝑃,𝜇(𝑥) we contract the open 𝜇 index with the only +available four-vector 𝑥𝜇 to obtain a scalar, ˚𝐶𝐴𝑃(𝑟) = 𝑥𝜇𝐶𝐴𝑃,𝜇(𝑥) → 𝑐𝐴𝑐𝑃 +4𝜋2 𝑚 𝜋𝐾2(𝑚 𝜋𝑟). In the case +of 𝐶𝐴𝐴,𝜇𝜈 there are two ways to obtain a scalar, +˚𝐶 (1) +𝐴𝐴(𝑟) = 𝛿𝜇𝜈𝐶𝐴𝐴,𝜇𝜈(𝑥), +˚𝐶 (2) +𝐴𝐴(𝑟) = 𝑥𝜇𝑥𝜈𝐶𝐴𝐴,𝜇𝜈(𝑥), +(4) +and similarly for the nucleon correlator that is a spinor, with /𝑥 = 𝛾𝜇𝑥𝜇, +˚𝐶 (1) +𝑁 𝑁 (𝑟) ≡ tr 𝐶𝑁 𝑁 (𝑥), +˚𝐶 (2) +𝑁 𝑁 (𝑟) ≡ tr /𝑥𝐶𝑁 𝑁 (𝑥). +(5) +On the lattice, an estimator of these radial correlators is given by +˚𝐶(𝑟) = +1 +r4(𝑟2) +∑︁ +|𝑥 |=𝑟 +𝐶(𝑥) +(6) +where r4 is defined in ref. [5]. +We note that the symmetry of 𝑆3(𝑟) is broken not only by 𝑎 ≠ 0 but also by the finite size of the +hypercubic box and by the fact that we choose antiperiodic (instead of periodic) boundary conditions +in one of the four dimensions for quarks. However, as we show in section 5 these boundary effects +are not visible at the current level of precision on the master-field lattices in table 1 considered here, +differently from what we observed on smaller volumes [5]. +3. +Grid of point sources estimator +The simplest way to compute the correlators introduced in section 2 numerically is to solve the +Dirac equation on a point source, that is, a source spinor that is supported on a single lattice point, +and subsequently perform the suitable contractions of spinor and space-time indeces. A consequence +3 + +Hadronic observables from master-field simulations +Marco Cè +of this naive strategy applied on gauge-field configurations with a large volume is that the effort for +each correlator point source scales proportionally with the volume, which is clearly not optimal. +Indeed, most of the resources are spent in computing the correlator at a distance from the source of +multiple correlation lengths, which has an exponentially suppressed contribution to the physics and +in most of the cases is completely dominated by noise. +Instead, we would like to exploit stochastic locality to define estimators that scale efficiently +with the volume and are suitable for master-field applications. Taking as an example the radial +correlators ˚𝐶(𝑟) introduced in section 2, let us assume that we are interested in physics that can be +extracted from correlators up to a maximum radial source-sink separation 𝑟max. Ref. [1] sketches a +decomposition of the lattice in space-time domains, or blocks, that are physically large, such that all +the lattice points within an 𝑟max distance from a source point at the centre of each block are within +the same block. This implies a block size 𝑏 > 2𝑟max. Solving the Dirac equation in each block, +imposing Dirichlet boundary conditions at the block boundary of the gauge field, one can decouple +the computational cost of the estimator from the volume of the global lattice. However, this method +introduces boundary effects that can be large for sink points close to the boundaries [1, 7], see also +refs. [8, 9]. We leave the exploration of this direction for future work, and we focus here on a simpler +approach that does not require a dedicated correction computation. +We introduce a set of lattice points 𝐺 that are separated (on average) by a physical distance +constant in the volume, such that the number of points |𝐺| ∝ 𝑉, that is, it grows proportionally with +the volume. On these point we introduce stochastic sources that satisfy +� +𝜂𝑖(𝑥)𝜂† +𝑗(𝑦) +� +𝜂 = 𝛿𝑖 𝑗𝛿𝑥𝑦𝐼 +for 𝑥, 𝑦 ∈ 𝐺, +(7) +where 𝐼 is the identity matrix in spin and colour space. By contracting at the sink with stochastic +noise corresponding to each coordinate 𝑦 ∈ 𝐺 one obtains |𝐺| ∝ 𝑉 samples of the quark propagator, +one for each 𝑦 ∈ 𝐺, from a single global-lattice inversion that is 𝑂(𝑉) computationally. Each sample +has a spurious contribution of stochastic nature from source points 𝑥 ≠ 𝑦, which is suppressed +by averaging the quark propagator over a number of sources 𝑛src and does not contribute to the +expectation value. Mesonic two-point functions that contract two quark propagators require 𝑛src ≥ 2 +to obtain an unbiased estimator, that in the case of the pseudoscalar-density two-point function reads +𝐶𝐺 +𝑃𝑃(𝑥; 𝑦) = +1 +𝑛src(𝑛src − 1) +∑︁ +𝑖≠𝑗 +Re +� +𝜓† +𝑖 (𝑥 + 𝑦)𝜓 𝑗(𝑥 + 𝑦)𝜂† +𝑗(𝑦)𝜂𝑖(𝑦) +� +, +(8) +where 𝜓𝑖(𝑥) = � +𝑦 𝐷−1(𝑥; 𝑦)𝜂𝑖(𝑦) and the double sum over 𝑖 ≠ 𝑗 can be computed in 𝑂(𝑛src) cost. +In this approach, since |𝐺| ∝ 𝑉, efficient scaling of the solutions of the Dirac equation is +achieved. Moreover, 𝐺 implicitly realises a domain decomposition by labelling each lattice point +with the closest 𝑦 ∈ 𝐺.1 Eq. (8) is in principle valid for any 𝑥 and 𝑦, but if only (𝑥, 𝑦) pairs that are +in the same domain are considered then one can compute efficiently all the |𝐺| ∝ 𝑉 contributions +with a single 𝑂(𝑉) pass over the whole lattice. This realises the optimal volume scaling for the +contractions too. It also lowers the required 𝑛src since the “correct” source 𝑦 is always the closest to +the sink 𝑥 and spurious contributions are further suppressed by the longer source-sink separation.2 +1Up to points equidistant from two or more 𝑦 ∈ 𝐺 that require additional conditions to be assigned to a domain. +2These spurious contributions are only stochastic and do not modify the expectation value, although we note that they +can have different quantum numbers and decay slower than the correlator being estimated. +4 + +Hadronic observables from master-field simulations +Marco Cè +𝑏 +𝑟max = +√ +2𝑏/2 +𝑟max = +√ +2𝑏/2 +𝑦 ∈ 𝐺 +𝑦 ∈ 𝐺 +𝑥 +Figure 1: Sketch of the estimator with a grid of point sources over a two-dimensional window of the lattice. +The set 𝐺 of source points +∈ 𝐺 is a regular grid with spacing 𝑏 and even point only. A mesonic two-point +function is evaluated at sink point 𝑥 that is in the domain defined by 𝑦 ∈ 𝐺 and within a distance 𝑟max from 𝑦. +One of the spurious contributions from the “wrong” source is shown in light grey. +Moreover, it implies 𝑟max = min𝑥,𝑦∈𝐺 |𝑥 − 𝑦|/2, that is, the minimum of the semidistance of points +in 𝐺. Therefore, 𝐺 has to be sparse enough for correlators at the relevant radial separations 𝑟 ≤ 𝑟max +to be accessible. +We study this setup on two sets of a few master fields whose parameters are given in table 1. +The master fields in both sets are hypercubic boxes with equal extent in each dimension denoted +by 𝐿, such that the volume is 𝑉 = 𝐿4. The 𝐿 = 192𝑎 master fields denoted by B (𝑛cnfg = 2) have +exactly 16 times, twice in each dimension, the volume of the ones with 𝐿 = 96𝑎 in set A (𝑛cnfg = 5) +and otherwise identical parameters, and we can thus define equivalent 𝐺s on both sets and study the +volume scaling. We employ U(1) noise that satisfies eq. (7). The simplest choice for 𝐺 is a regular +grid with spacing 𝑏, which matches the domain decomposition proposed in ref. [1], with 𝑏 = 48𝑎 +being a suitable choice in our case. However, the definition of 𝐺 is more flexible. In this work, we +employ a grid with only even (or equivalently odd) points, which results in 𝑟max = +√ +2𝑏/2 ≃ 33.94𝑎 +instead of 𝑏/2 = 24𝑎, at the cost of halving the number of points on the grid.3 The total number +of points is thus |𝐺| = (𝐿/𝑏)4/2 that evaluates to 8 and 128 for A and B respectively. We fix +𝑛src = 2 and with the current precision we do not observe deviations from the expected behaviour, +especially at 𝑟 close to 𝑟max, that can be attributed to spurious contributions. Further optimisation +such as systematically and exactly removing the closer spurious contributions, e.g. with hierarchical +probing [11], are not explored here. +The statistics obtained with a single source, e.g. eight points on each master field in A, is limited +by the need of balancing the density of 𝐺 with a lower limit on the 𝑟max suitable to extract long-range +physics. To increase the statistics we simply propose to recompute eq. (8) on 𝑛shift sources, each +time shifting 𝐺 to have a distinct support. This is done four times for each direction in the case of A +and twice for each direction in B. An extra factor of two is obtained by pairing each even-only 𝐺 +with the corresponding odd-only, leading to 𝑛shift = 512 and 32 for A and B respectively. Combined +with |𝐺|, the final result is the same number of source points 𝑛point = 4096 for both volumes, on +3This results in a doubled |𝐺|𝑟4max/𝑉 density. Indeed, it corresponds to a 𝐷4 lattice (or equivalently 𝐹4 lattice) that +has the densest known packing of equal spheres in four dimensions [10]. +5 + +Hadronic observables from master-field simulations +Marco Cè +a regular grid with spacing 𝑏shift = 12𝑎 and 24𝑎 for A and B respectively. Ignoring that on the A +lattices source points are on average twice as close and thus potentially more correlated than on B, +in our setup we have same statistics for each gauge field configuration for both A and B. Crucially, +thanks to the optimal volume scaling of the stochastic grid correlator, this matching statistic has +been obtained at an equivalent computational cost. +4. +Master-field errors +The estimator in section 3 applied to the radial correlator leads to a collection of up to 4096 +correlators for each master-field configuration on a regular grid of source points with spacing +𝑏shift = 𝐿/8. Applying stochastic locality, the expectation value +� ˚𝐶(𝑟) +� is given up to volume- +suppressed corrections by the translation average +� ˚𝐶(𝑟) +� += ⟪ ˚𝐶(𝑟)⟫ + 𝑂 +� +𝑉−1/2� += 1 +𝑉 +∑︁ +𝑦∈𝐺 +˚𝐶(𝑟; 𝑦) + 𝑂 +� +𝑉−1/2� +(9) +where the 𝑦 in ˚𝐶(𝑟; 𝑦) denotes the source point. The error of this estimator can be estimated applying +eq. (2) with O(𝑦) = ˚𝐶(𝑟; 𝑦) +� +[⟪ ˚𝐶(𝑟)⟫ − +� ˚𝐶(𝑟) +� +]2� += 1 +𝑉 +������ +∑︁ +|𝑦|≤𝑅 +⟪ ˚𝐶(𝑟; 𝑦) ˚𝐶(𝑟; 0)⟫𝑐 + 𝑂 +� +e−𝑚𝑅� ++ 𝑂 +� +𝑉−1/2������� +, +(10) +where again the sum over the source coordinates 𝑦 is performed over the grid of point sources. +Finding the optimal 𝑅 to truncate the sum in the r.h.s. has a clear analogy with the well-known Γ +method introduced by Wolff to deal with autocorrelation in Monte Carlo time and estimate an error +with less errors [12], and leads to a generalisation of the Madras–Sokal formula for the statistical +error of the error [13, 14]. This can be implemented in a resource efficient way by computing +the correlation between grid points with higher-dimensional fast Fourier transforms. The optimal +𝑅 depends on the observable. In particular, since each value of the correlator radial source-sink +separation 𝑟 defines a distinct observable with different spacetime support, 𝑅 is a function of 𝑟. +Alternatively, one can apply a four-dimensional binning of the point sources in the grid into +blocks. For instance, blocks of size (24𝑎)4 bin 16 point sources on A and only one point source on +B according to the spacing 𝑏shift in table 1, while blocks of size (48𝑎)4 bin 256 and 16 point sources +respectively. We tested these two bin sizes and observed that this leads to a stable error estimate. In +the following, we show results obtained in the more conservative case, that is, with blocks of size +(48𝑎)4. +We note that master-field error estimation can be combined with standard methods based on +an ensemble of gauge field configurations, e.g. with a five-dimensional variant of the Γ method in +spacetime coordinates and Monte Carlo time. Explorations in this direction can be found in ref. [15]. +5. +Numerical results +We computed 𝑚 𝜋, 𝑚𝑁 and 𝑓𝜋 using position-space correlators on the sets of master fields +whose parameters are listed in table 1. The results for these hadronic observables are listed in table 2. +We employed the technique already studied in ref. [5] to extract the pion mass 𝑚 𝜋 from the +long-distance behaviour in eq. (3a) of the position-space correlator ˚𝐶𝑃𝑃(𝑟). In those proceedings +6 + +Hadronic observables from master-field simulations +Marco Cè +Table 2: Numerical results for hadronic observable with errors estimated à la master field. +𝐿/𝑎 +𝑎𝑚 𝜋 +𝑎𝑚𝑁 +𝑎 𝑓 bare +𝜋 +A +96 +0.126 28(33) +0.500(6) +0.0890(3) +B +192 +0.126 01(19) +0.487(8) +0.0885(4) +5 +10 +15 +20 +25 +30 +r/a +0.08 +0.10 +0.12 +0.14 +0.16 +0.18 +0.20 +ameff +covariant +one-state fit +two-state fit +5 +10 +15 +20 +25 +30 +r/a +0.08 +0.10 +0.12 +0.14 +0.16 +0.18 +0.20 +ameff +covariant +one-state fit +two-state fit +Figure 2: Effective mass of the ˚𝐶𝑃𝑃(𝑟) correlator as a function of 𝑟 for master fields in set A (left plot) and +set B (right plot). On top of the data points with master-field errors shown in blue, we show the results of a +one-state fit in a green band and of a two-states fit in a red band. The thickness of the bands is the statistical +error. +the technique was applied to correlators computed with point sources on an ensemble of gauge field +configurations with a (6 fm)3 space volume, performing a standard error estimation. Here we have a +larger volume that allows us to use the grid of point sources as described in section 3 and estimate +the error à la master field, see section 4. On top of the same number of samples 𝑛point = 4096 for +each configuration, we have 5 configurations in set A and 2 in set B. This means that we have a larger +statistics for the 𝐿 = 96𝑎 master fields from which we expect a ≈ 1.58 reduction of the error. +The effective mass4 of ˚𝐶𝑃𝑃(𝑟) is shown in the two plots in figure 2. For each set, two fits are +performed: a “one-state” fit having 𝑐𝑃 and 𝑚 𝜋 as free parameters, and a “two-states” one with an +added “excited state” term 𝑎1(𝑚1/𝑟)𝐾1(𝑚1/𝑟) with two extra free parameters 𝑎1 and 𝑚1 > 𝑚 𝜋. We +choose appropriate values for the smaller 𝑟 of the correlator data that enter the fit, with different +choices for one-state and two-states fits. Instead, all the data up to largest available 𝑟 = 𝑟max enter +the fit, since we do not observe any boundary effect that constrains us otherwise. The two fits on +each set give compatible results and the corresponding effective mass is shown in figure 2. +From the one-state fits we obtain the results in table 2, which show a good agreement between +the two sets. Contrary to the expectation based on 𝑛cnfg, the error is 40 % smaller on set B. A +possible explanation for this fact is the 𝑏shift = 12𝑎 of the samples of set A, halved with respect to set +B, which can lead to a reduced effective number of samples due to stronger correlations in space. +Similarly, we extract 𝑚𝑁 from the two contractions in eq. (5) of the position-space nucleon +correlator in eq. (3d) as done in ref. [5], but employing the techniques of sections 3 and 4. The results in +table 2 are from the one-state fits to ˚𝐶 (1) +𝑁 𝑁 (𝑟) with the free parameters 𝑐𝑁 and 𝑚𝑁 , and are compatible +4See eq. (10) in ref. [5] for the definition of the effective mass of the radial correlator. +7 + +Hadronic observables from master-field simulations +Marco Cè +5 +10 +15 +20 +25 +30 +r/a +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +ameff +covariant, trNN +covariant, tr/xNN +one-state fit, trNN +one-state fit, tr/xNN +two-state fit, trNN +two-state fit, tr/xNN +5 +10 +15 +20 +25 +30 +r/a +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +ameff +covariant, trNN +covariant, tr/xNN +one-state fit, trNN +one-state fit, tr/xNN +two-state fit, trNN +two-state fit, tr/xNN +Figure 3: Effective mass of the ˚𝐶 (𝑖) +𝑁 𝑁 (𝑟) correlators as a function of 𝑟 for master fields in set A (left plot) and +set B (right plot), where 𝑖 = 1 corresponds to the tr 𝑁𝑁 contraction and 𝑖 = 2 to the tr /𝑥𝑁𝑁 one. On top of the +data points with master-field errors shown in blue and orange for 𝑖 = 1 and 2 respectively, we show the results +of a one-state fit in green and brown bands and of a two-states fit in red and purple bands. The thickness of the +bands is the statistical error. +with the results of two-states fits with the replacement ˚𝐶𝑁 𝑁 (𝑟) → ˚𝐶𝑁 𝑁 (𝑟)[1+𝑎1(𝑚 𝜋/𝑟)𝐾1(𝑚 𝜋𝑟)] +where 𝑎1 is an extra free parameter and 𝑚 𝜋 is fixed. The fit to ˚𝐶 (2) +𝑁 𝑁 (𝑟) shows similar results, +although with a slightly larger central value that can be attributed to different discretization effects. +The effective masses corresponding to data and fits are shown in figure 3. In the case of 𝑚𝑁 , we +observe a larger error on set B, compatible with the lower statistics and showing no indication of +correlation-in-space effects. +We also extract the pion decay constant 𝑓 bare +𝜋 +, where the bare indicates that we do not include the +axial-current renormalization factor, from a combined fit of the four correlators ˚𝐶𝑃𝑃, ˚𝐶𝐴𝑃, ˚𝐶 (1) +𝐴𝐴 and +˚𝐶 (2) +𝐴𝐴. As fit function we employ the long-distance behaviours derived from eqs. 3, which depends +on the free parameters 𝑐𝑃, 𝑐𝐴 and 𝑚 𝜋. As shown from the plots of the ratio between data and fit +functions in figure 4, ˚𝐶𝐴𝑃 approaches the asymptotic behaviour at a smaller value of 𝑟, followed +by ˚𝐶𝑃𝑃 and ˚𝐶 (2) +𝐴𝐴. +˚𝐶 (1) +𝐴𝐴 converges to the asymptotic behaviour at a much larger 𝑟, with the ratio +being initially negative and changing sign around 𝑟 ≈ 14𝑎. The values of 𝑚 𝜋 obtained from these +combined fits are consistent with the previous fits to only the ˚𝐶𝑃𝑃 correlators. The decay constant is +then given by 𝑓 bare +𝜋 += 𝑐𝐴/𝑚 𝜋 and shown in table 2. Like in the case of 𝑚𝑁 , the values on set A and +B are compatible, with a slightly larger error for set B that is consistent with the lower number of +master field configurations. +6. +Conclusions +We have shown that position-space correlators can be used to extract hadron masses and decay +constants with short-distance and cut-off effects under control. Crucially, the statistical error can +be estimated à la master field, obtaining an efficient scaling of the computational effort with the +increased volume. +In this work we studied sphere-averaged radial correlators, but potentially more information is +encoded in correlators as function of four-dimensional coordinates. This requires understanding +effects that break rotational symmetry at finite lattice spacing and is an interesting topic for further +studies. +8 + +Hadronic observables from master-field simulations +Marco Cè +0.020 +0.022 +0.024 +0.026 +|cP|2 +|cP|2 fit +˚CPP +0.0016 +0.0018 +0.0020 +|cAc† +P| +|cAc† +P| fit +˚CAP +10 +20 +30 +r/a +0.00011 +0.00012 +0.00013 +0.00014 +|cA|2 +|cA|2 fit +˚C(1) +AA +10 +20 +30 +r/a +0.00011 +0.00012 +0.00013 +0.00014 +|cA|2 +|cA|2 fit +˚C(2) +AA +0.020 +0.022 +0.024 +0.026 +|cP|2 +|cP|2 fit +˚CPP +0.0016 +0.0018 +0.0020 +|cAc† +P| +|cAc† +P| fit +˚CAP +10 +20 +30 +r/a +0.00011 +0.00012 +0.00013 +0.00014 +|cA|2 +|cA|2 fit +˚C(1) +AA +10 +20 +30 +r/a +0.00011 +0.00012 +0.00013 +0.00014 +|cA|2 +|cA|2 fit +˚C(2) +AA +Figure 4: Plots of the ratio between correlator data and their fitted long-distance behaviours for master fields +in set A (top row) and set B (bottom row). The amplitude in the denominator is set to one, so that the actual +amplitude for each correlator is shown on the vertical axis. In each row, four plots are shown for ˚𝐶𝑃𝑃, ˚𝐶 (1) +𝐴𝐴 +(left column), ˚𝐶𝐴𝑃 and ˚𝐶 (2) +𝐴𝐴 (right column), with the correlator data with master field errors shown in blue. +The amplitude parameters of the corresponding fit function, which are functions of 𝑐𝑃 and 𝑐𝐴, are shown in +an orange horizontal line with a pale orange error band. +Position-space methods find applications in computations of quantities that go beyond the +simple hadronic quantities considered here, such as for example the hadronic vacuum polarisation +contribution to the anomalous magnetic moment of the muon [16, 17], including the so-called +window contribution [18]. The estimators presented here provide a straightforward path to the +computation of this quantities in the master-field paradigm. +Acknowledgements: The research of MB is funded through the MUR program for young researchers “Rita +Levi Montalcini”. AF acknowledges support by the Ministry of Science and Technology Taiwan (MOST) +under grant 111-2112-M-A49-018-MY2. JRG acknowledges support from the Simons Foundation through the +Simons Bridge for Postdoctoral Fellowships scheme. MTH is supported by UKRI Future Leader Fellowship +MR/T019956/1 and in part by UK STFC grant ST/P000630/1. This work was performed using the DiRAC +Data Intensive service at Leicester, operated by the University of Leicester IT Services, which forms part +of the STFC DiRAC HPC Facility (www.dirac.ac.uk). The equipment was funded by BEIS capital +funding via STFC capital grants ST/K000373/1 and ST/R002363/1 and STFC DiRAC Operations grant +ST/R001014/1. DiRAC is part of the National e-Infrastructure. We acknowledge PRACE for awarding +us access to SuperMUC-NG at GCS@LRZ, Germany, where some computations were performed Many +9 + +Hadronic observables from master-field simulations +Marco Cè +simulations were performed on a dedicated HPC cluster at CERN. We gratefully acknowledge the computer +resources and the technical support provided by these institutions. +References +[1] M. Lüscher, Stochastic locality and master-field simulations of very large lattices, EPJ Web +Conf. 175 (2018) 01002 [1707.09758]. +[2] L. Giusti and M. Lüscher, Topological susceptibility at 𝑇 > 𝑇c from master-field simulations of +the SU(3) gauge theory, Eur. Phys. J. C 79 (2019) 207 [1812.02062]. +[3] A. Francis, P. Fritzsch, M. Lüscher and A. Rago, Master-field simulations of 𝑂(𝑎)-improved +lattice QCD: Algorithms, stability and exactness, Comput. Phys. Commun. 255 (2020) 107355 +[1911.04533]. +[4] P. Fritzsch, J. Bulava, M. Cè, A. Francis, M. Lüscher and A. 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Parrino, Coordinate-space calculation of the window +observable for the hadronic vacuum polarization contribution to (𝑔 − 2)𝜇, 2211.15581. +10 + diff --git a/Q9E4T4oBgHgl3EQflA2r/content/tmp_files/load_file.txt b/Q9E4T4oBgHgl3EQflA2r/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6aebc9a2665a5dc61f953156ecef9ccb3cf534f4 --- /dev/null +++ b/Q9E4T4oBgHgl3EQflA2r/content/tmp_files/load_file.txt @@ -0,0 +1,427 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf,len=426 +page_content='Hadronic observables from master-field simulations Marco Cè,𝑎,∗ Mattia Bruno,𝑏 John Bulava,𝑐 Anthony Francis,𝑑 Patrick Fritzsch,𝑒 Jeremy R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' Green,𝑐 Maxwell T.' metadata={'source': 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Dublin 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' Ireland 𝑓 Higgs Centre for Theoretical Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' School of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' The University of Edinburgh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' Edinburgh EH9 3FD,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' United Kingdom 𝑔IMADA and CP3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' University of Southern Denmark,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' Odense,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' Denmark ℎDepartment of Theoretical Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' CERN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' 1211 Geneva 23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' Switzerland E-mail: marcoce@itp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='unibe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='ch Substantial progress has been made recently in the generation of master-field ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' This has to be paired with efficient techniques to compute observables on gauge field configurations with a large volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' Here we present the results of the computation of hadronic observables, including hadron masses and meson decay constants, on large-volume and master-field ensembles with physical volumes of up to (18 fm)4 and 𝑚 𝜋𝐿 up to 25, simulated using 𝑁f = 2 + 1 stabilized Wilson fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' We obtain sub-percent determinations from single gauge configurations with the combined use of position-space techniques, volume averages and master-field error estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' The 39th International Symposium on Lattice Field Theory, 8th-13th August, 2022, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, Germany ∗Speaker © Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='0 International License (CC BY-NC-ND 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' https://pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='sissa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='it/ arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='05156v1 [hep-lat] 12 Jan 2023 Hadronic observables from master-field simulations Marco Cè 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' Introduction Gauge-field configurations in a lattice theory with a mass gap have the stochastic locality property, that is, gauge-invariant local fields at large physical separations are stochastically independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' The master-field paradigm introduced by Lüscher [1] proposes to use stochastic locality to obtain observable estimates from a single or at most a few representative gauge-field configurations on very large lattices, making use of the invariance under translations of the theory and of volume averages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' As a first application of this paradigm, stochastic locality has been used to compute the topological susceptibility at 𝑇 > 𝑇𝑐 in master-field simulations of SU(3) Yang–Mills theory [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' In a theory with fermions such as QCD, numerical simulations are performed after integrating out fermions exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' Hadronic observables in QCD are expressed in terms of contractions of quark propagators whose locality is not manifest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' Moreover, the sheer size of the lattices requires stabilising measures that have been studied in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' These include a slight modification of the standard 𝑂(𝑎)-improved lattice Dirac operator, replacing the HMC with the stochastic molecular dynamics (SMD) algorithm, employing quadruple-precision lattice sums and uniform-norm stopping criteria for the Dirac equation solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' Recent progress in master-field simulations has been presented at the Lattice 2021 conference [4, 5] and at this conference [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' In these proceedings we further develop the position-space techniques introduced in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' [5], by presenting an estimator for position-space correlators that scales efficiently with the volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' Estimation of observables on master fields is explained in details in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' In summary,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' the expectation value ⟨O(𝑥)⟩ of a local field O(𝑥) is obtained averaging over translations ⟪O(𝑥)⟫ = 1 𝑉 ∑︁ 𝑧 O(𝑥 + 𝑧),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' ⟨O(𝑥)⟩ = ⟪O(𝑥)⟫ + 𝑂 � 𝑉−1/2� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' (1) with the variance of this estimator given by 𝜎2 ⟪O⟫(𝑥) = � [⟪O(𝑥)⟫ − ⟨O(𝑥)⟩]2� = 1 𝑉 ∑︁ 𝑦 ⟨O(𝑦)O(0)⟩𝑐 = 1 𝑉 ������ ∑︁ |𝑦|≤𝑅 ⟨O(𝑦)O(0)⟩𝑐 + 𝑂 � e−𝑚𝑅������� = 1 𝑉 ������ ∑︁ |𝑦|≤𝑅 ⟪O(𝑦)O(0)⟫𝑐 + 𝑂 � e−𝑚𝑅� + 𝑂 � 𝑉−1/2������� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' (2) where in the second line we first used the fact that the connected correlator of the local field O(𝑥) decays exponentially with spacetime separation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' and then we applied again translation averages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' Position-space correlators In this work we focus on correlation functions in position space 𝐶𝑃𝑃(𝑥) → 𝑐2 𝑃 4𝜋2 𝑚 𝜋 |𝑥| 𝐾1(𝑚 𝜋|𝑥|),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' (3a) 𝐶𝐴𝑃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='𝜇(𝑥) → 𝑐𝐴𝑐𝑃 4𝜋2 𝑥𝜇 |𝑥| 𝑚 𝜋 |𝑥| 𝐾2(𝑚 𝜋|𝑥|),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' (3b) 𝐶𝐴𝐴,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='𝜇𝜈(𝑥) → 𝑐2 𝐴 4𝜋2 � −𝛿𝜇𝜈 1 𝑥2 𝐾2(𝑚 𝜋|𝑥|) + 𝑥𝜇𝑥𝜈 𝑥2 �𝑚 𝜋 |𝑥| 𝐾1(𝑚 𝜋|𝑥|) + 4 𝑥2 𝐾2(𝑚 𝜋|𝑥|) �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' (3c) 𝐶𝑁 𝑁 (𝑥) → 𝑐2 𝑁 4𝜋2 𝑚2 𝑁 |𝑥| � 𝐾1(𝑚𝑁 |𝑥|) + /𝑥 |𝑥| 𝐾2(𝑚𝑁 |𝑥|) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' (3d) 2 Hadronic observables from master-field simulations Marco Cè Table 1: Parameters of the master-field lattices used in this study (with 𝑎 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='094 fm and 𝑚 𝜋 ≈ 270 MeV, see also ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' [4]), together with information on the statistics used in the observable computation as explained in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' 𝐿/𝑎 𝐿 [fm] 𝑚 𝜋𝐿 𝑛cnfg 𝑏/𝑎 |𝐺| 𝑛shift 𝑏shift/𝑎 𝑛point A 96 9 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='5 5 48 8 512 12 4096 B 192 18 25 2 48 128 32 24 4096 where the subscript indicates the two-point function of either pseudoscalar densities 𝑃 = ¯𝑢𝛾5𝑑, axial current 𝐴𝜇 = ¯𝑢𝛾𝜇𝛾5𝑑 or nucleon spinor 𝑁 = 𝜖𝑎𝑏𝑐(𝑢𝑇 𝑎𝐶𝛾5𝑑𝑏)𝑢𝑐, as a function of the source-sink separation 𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' In eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' (3), the asympotic behaviour for 𝑥 → ∞ of these correlators is given assuming the symmetries of the continuum theory in an infinite volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' From position-space correlators one can extract simple hadronic observables, including the masses 𝑚 𝜋 and 𝑚𝑁 and the decay constant 𝑓𝜋 = 𝑐𝐴/𝑚 𝜋, as demonstrated in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' Once computed on the lattice as discussed in the following section, these correlators as a function of the four-dimensional source-sink separation 𝑥 include lattice discretization effects that break the rotational symmetry and depend on the direction of 𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' In this study, we limit ourselves to the radial correlators ˚𝐶(𝑟) introduced in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' [5] that are averaged over 𝑆3(𝑟) = {𝑥 ∈ R4 : |𝑥| = 𝑟}, the 3-sphere of radius 𝑟, and by construction depend only on the radial coordinate 𝑟 = |𝑥|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' While ˚𝐶𝑃𝑃(𝑟) = 𝐶𝑃𝑃(𝑥), for the 𝐴𝑃-correlator 𝐶𝐴𝑃,𝜇(𝑥) we contract the open 𝜇 index with the only available four-vector 𝑥𝜇 to obtain a scalar, ˚𝐶𝐴𝑃(𝑟) = 𝑥𝜇𝐶𝐴𝑃,𝜇(𝑥) → 𝑐𝐴𝑐𝑃 4𝜋2 𝑚 𝜋𝐾2(𝑚 𝜋𝑟).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' In the case of 𝐶𝐴𝐴,𝜇𝜈 there are two ways to obtain a scalar, ˚𝐶 (1) 𝐴𝐴(𝑟) = 𝛿𝜇𝜈𝐶𝐴𝐴,𝜇𝜈(𝑥), ˚𝐶 (2) 𝐴𝐴(𝑟) = 𝑥𝜇𝑥𝜈𝐶𝐴𝐴,𝜇𝜈(𝑥), (4) and similarly for the nucleon correlator that is a spinor, with /𝑥 = 𝛾𝜇𝑥𝜇, ˚𝐶 (1) 𝑁 𝑁 (𝑟) ≡ tr 𝐶𝑁 𝑁 (𝑥), ˚𝐶 (2) 𝑁 𝑁 (𝑟) ≡ tr /𝑥𝐶𝑁 𝑁 (𝑥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' (5) On the lattice, an estimator of these radial correlators is given by ˚𝐶(𝑟) = 1 r4(𝑟2) ∑︁ |𝑥 |=𝑟 𝐶(𝑥) (6) where r4 is defined in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' We note that the symmetry of 𝑆3(𝑟) is broken not only by 𝑎 ≠ 0 but also by the finite size of the hypercubic box and by the fact that we choose antiperiodic (instead of periodic) boundary conditions in one of the four dimensions for quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' However, as we show in section 5 these boundary effects are not visible at the current level of precision on the master-field lattices in table 1 considered here, differently from what we observed on smaller volumes [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' Grid of point sources estimator The simplest way to compute the correlators introduced in section 2 numerically is to solve the Dirac equation on a point source, that is, a source spinor that is supported on a single lattice point, and subsequently perform the suitable contractions of spinor and space-time indeces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' A consequence 3 Hadronic observables from master-field simulations Marco Cè of this naive strategy applied on gauge-field configurations with a large volume is that the effort for each correlator point source scales proportionally with the volume, which is clearly not optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' Indeed, most of the resources are spent in computing the correlator at a distance from the source of multiple correlation lengths, which has an exponentially suppressed contribution to the physics and in most of the cases is completely dominated by noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' Instead, we would like to exploit stochastic locality to define estimators that scale efficiently with the volume and are suitable for master-field applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' Taking as an example the radial correlators ˚𝐶(𝑟) introduced in section 2, let us assume that we are interested in physics that can be extracted from correlators up to a maximum radial source-sink separation 𝑟max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' [1] sketches a decomposition of the lattice in space-time domains, or blocks, that are physically large, such that all the lattice points within an 𝑟max distance from a source point at the centre of each block are within the same block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' This implies a block size 𝑏 > 2𝑟max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' Solving the Dirac equation in each block, imposing Dirichlet boundary conditions at the block boundary of the gauge field, one can decouple the computational cost of the estimator from the volume of the global lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' However, this method introduces boundary effects that can be large for sink points close to the boundaries [1, 7], see also refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' We leave the exploration of this direction for future work, and we focus here on a simpler approach that does not require a dedicated correction computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' We introduce a set of lattice points 𝐺 that are separated (on average) by a physical distance constant in the volume, such that the number of points |𝐺| ∝ 𝑉, that is, it grows proportionally with the volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' On these point we introduce stochastic sources that satisfy � 𝜂𝑖(𝑥)𝜂† 𝑗(𝑦) � 𝜂 = 𝛿𝑖 𝑗𝛿𝑥𝑦𝐼 for 𝑥, 𝑦 ∈ 𝐺, (7) where 𝐼 is the identity matrix in spin and colour space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' By contracting at the sink with stochastic noise corresponding to each coordinate 𝑦 ∈ 𝐺 one obtains |𝐺| ∝ 𝑉 samples of the quark propagator, one for each 𝑦 ∈ 𝐺, from a single global-lattice inversion that is 𝑂(𝑉) computationally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' Each sample has a spurious contribution of stochastic nature from source points 𝑥 ≠ 𝑦, which is suppressed by averaging the quark propagator over a number of sources 𝑛src and does not contribute to the expectation value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' Mesonic two-point functions that contract two quark propagators require 𝑛src ≥ 2 to obtain an unbiased estimator, that in the case of the pseudoscalar-density two-point function reads 𝐶𝐺 𝑃𝑃(𝑥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' 𝑦) = 1 𝑛src(𝑛src − 1) ∑︁ 𝑖≠𝑗 Re � 𝜓† 𝑖 (𝑥 + 𝑦)𝜓 𝑗(𝑥 + 𝑦)𝜂† 𝑗(𝑦)𝜂𝑖(𝑦) � , (8) where 𝜓𝑖(𝑥) = � 𝑦 𝐷−1(𝑥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' 𝑦)𝜂𝑖(𝑦) and the double sum over 𝑖 ≠ 𝑗 can be computed in 𝑂(𝑛src) cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' In this approach, since |𝐺| ∝ 𝑉, efficient scaling of the solutions of the Dirac equation is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' Moreover, 𝐺 implicitly realises a domain decomposition by labelling each lattice point with the closest 𝑦 ∈ 𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='1 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' (8) is in principle valid for any 𝑥 and 𝑦, but if only (𝑥, 𝑦) pairs that are in the same domain are considered then one can compute efficiently all the |𝐺| ∝ 𝑉 contributions with a single 𝑂(𝑉) pass over the whole lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' This realises the optimal volume scaling for the contractions too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' It also lowers the required 𝑛src since the “correct” source 𝑦 is always the closest to the sink 𝑥 and spurious contributions are further suppressed by the longer source-sink separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='2 1Up to points equidistant from two or more 𝑦 ∈ 𝐺 that require additional conditions to be assigned to a domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' 2These spurious contributions are only stochastic and do not modify the expectation value, although we note that they can have different quantum numbers and decay slower than the correlator being estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' 4 Hadronic observables from master-field simulations Marco Cè 𝑏 𝑟max = √ 2𝑏/2 𝑟max = √ 2𝑏/2 𝑦 ∈ 𝐺 𝑦 ∈ 𝐺 𝑥 Figure 1: Sketch of the estimator with a grid of point sources over a two-dimensional window of the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' The set 𝐺 of source points ∈ 𝐺 is a regular grid with spacing 𝑏 and even point only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' A mesonic two-point function is evaluated at sink point 𝑥 that is in the domain defined by 𝑦 ∈ 𝐺 and within a distance 𝑟max from 𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' One of the spurious contributions from the “wrong” source is shown in light grey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' Moreover, it implies 𝑟max = min𝑥,𝑦∈𝐺 |𝑥 − 𝑦|/2, that is, the minimum of the semidistance of points in 𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' Therefore, 𝐺 has to be sparse enough for correlators at the relevant radial separations 𝑟 ≤ 𝑟max to be accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' We study this setup on two sets of a few master fields whose parameters are given in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' The master fields in both sets are hypercubic boxes with equal extent in each dimension denoted by 𝐿, such that the volume is 𝑉 = 𝐿4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' The 𝐿 = 192𝑎 master fields denoted by B (𝑛cnfg = 2) have exactly 16 times, twice in each dimension, the volume of the ones with 𝐿 = 96𝑎 in set A (𝑛cnfg = 5) and otherwise identical parameters, and we can thus define equivalent 𝐺s on both sets and study the volume scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' We employ U(1) noise that satisfies eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' The simplest choice for 𝐺 is a regular grid with spacing 𝑏, which matches the domain decomposition proposed in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' [1], with 𝑏 = 48𝑎 being a suitable choice in our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' However, the definition of 𝐺 is more flexible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' In this work, we employ a grid with only even (or equivalently odd) points, which results in 𝑟max = √ 2𝑏/2 ≃ 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='94𝑎 instead of 𝑏/2 = 24𝑎, at the cost of halving the number of points on the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='3 The total number of points is thus |𝐺| = (𝐿/𝑏)4/2 that evaluates to 8 and 128 for A and B respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' We fix 𝑛src = 2 and with the current precision we do not observe deviations from the expected behaviour, especially at 𝑟 close to 𝑟max, that can be attributed to spurious contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' Further optimisation such as systematically and exactly removing the closer spurious contributions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' with hierarchical probing [11], are not explored here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' The statistics obtained with a single source, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' eight points on each master field in A, is limited by the need of balancing the density of 𝐺 with a lower limit on the 𝑟max suitable to extract long-range physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' To increase the statistics we simply propose to recompute eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' (8) on 𝑛shift sources, each time shifting 𝐺 to have a distinct support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' This is done four times for each direction in the case of A and twice for each direction in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' An extra factor of two is obtained by pairing each even-only 𝐺 with the corresponding odd-only, leading to 𝑛shift = 512 and 32 for A and B respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' Combined with |𝐺|, the final result is the same number of source points 𝑛point = 4096 for both volumes, on 3This results in a doubled |𝐺|𝑟4max/𝑉 density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' Indeed, it corresponds to a 𝐷4 lattice (or equivalently 𝐹4 lattice) that has the densest known packing of equal spheres in four dimensions [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' 5 Hadronic observables from master-field simulations Marco Cè a regular grid with spacing 𝑏shift = 12𝑎 and 24𝑎 for A and B respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' Ignoring that on the A lattices source points are on average twice as close and thus potentially more correlated than on B, in our setup we have same statistics for each gauge field configuration for both A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' Crucially, thanks to the optimal volume scaling of the stochastic grid correlator, this matching statistic has been obtained at an equivalent computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' Master-field errors The estimator in section 3 applied to the radial correlator leads to a collection of up to 4096 correlators for each master-field configuration on a regular grid of source points with spacing 𝑏shift = 𝐿/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' Applying stochastic locality, the expectation value � ˚𝐶(𝑟) � is given up to volume- suppressed corrections by the translation average � ˚𝐶(𝑟) � = ⟪ ˚𝐶(𝑟)⟫ + 𝑂 � 𝑉−1/2� = 1 𝑉 ∑︁ 𝑦∈𝐺 ˚𝐶(𝑟;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' 𝑦) + 𝑂 � 𝑉−1/2� (9) where the 𝑦 in ˚𝐶(𝑟;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' 𝑦) denotes the source point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' The error of this estimator can be estimated applying eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' (2) with O(𝑦) = ˚𝐶(𝑟;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' 𝑦) � [⟪ ˚𝐶(𝑟)⟫ − � ˚𝐶(𝑟) � ]2� = 1 𝑉 ������ ∑︁ |𝑦|≤𝑅 ⟪ ˚𝐶(𝑟;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' 𝑦) ˚𝐶(𝑟;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' 0)⟫𝑐 + 𝑂 � e−𝑚𝑅� + 𝑂 � 𝑉−1/2������� , (10) where again the sum over the source coordinates 𝑦 is performed over the grid of point sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' Finding the optimal 𝑅 to truncate the sum in the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' has a clear analogy with the well-known Γ method introduced by Wolff to deal with autocorrelation in Monte Carlo time and estimate an error with less errors [12], and leads to a generalisation of the Madras–Sokal formula for the statistical error of the error [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' This can be implemented in a resource efficient way by computing the correlation between grid points with higher-dimensional fast Fourier transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' The optimal 𝑅 depends on the observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' In particular, since each value of the correlator radial source-sink separation 𝑟 defines a distinct observable with different spacetime support, 𝑅 is a function of 𝑟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' Alternatively, one can apply a four-dimensional binning of the point sources in the grid into blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' For instance, blocks of size (24𝑎)4 bin 16 point sources on A and only one point source on B according to the spacing 𝑏shift in table 1, while blocks of size (48𝑎)4 bin 256 and 16 point sources respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' We tested these two bin sizes and observed that this leads to a stable error estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' In the following, we show results obtained in the more conservative case, that is, with blocks of size (48𝑎)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' We note that master-field error estimation can be combined with standard methods based on an ensemble of gauge field configurations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' with a five-dimensional variant of the Γ method in spacetime coordinates and Monte Carlo time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' Explorations in this direction can be found in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' Numerical results We computed 𝑚 𝜋, 𝑚𝑁 and 𝑓𝜋 using position-space correlators on the sets of master fields whose parameters are listed in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' The results for these hadronic observables are listed in table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' We employed the technique already studied in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' [5] to extract the pion mass 𝑚 𝜋 from the long-distance behaviour in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' (3a) of the position-space correlator ˚𝐶𝑃𝑃(𝑟).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' In those proceedings 6 Hadronic observables from master-field simulations Marco Cè Table 2: Numerical results for hadronic observable with errors estimated à la master field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' 𝐿/𝑎 𝑎𝑚 𝜋 𝑎𝑚𝑁 𝑎 𝑓 bare 𝜋 A 96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='126 28(33) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='500(6) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='0890(3) B 192 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='126 01(19) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='487(8) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='0885(4) 5 10 15 20 25 30 r/a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='20 ameff covariant one-state fit two-state fit 5 10 15 20 25 30 r/a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='20 ameff covariant one-state fit two-state fit Figure 2: Effective mass of the ˚𝐶𝑃𝑃(𝑟) correlator as a function of 𝑟 for master fields in set A (left plot) and set B (right plot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' On top of the data points with master-field errors shown in blue, we show the results of a one-state fit in a green band and of a two-states fit in a red band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' The thickness of the bands is the statistical error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' the technique was applied to correlators computed with point sources on an ensemble of gauge field configurations with a (6 fm)3 space volume, performing a standard error estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' Here we have a larger volume that allows us to use the grid of point sources as described in section 3 and estimate the error à la master field, see section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' On top of the same number of samples 𝑛point = 4096 for each configuration, we have 5 configurations in set A and 2 in set B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' This means that we have a larger statistics for the 𝐿 = 96𝑎 master fields from which we expect a ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='58 reduction of the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' The effective mass4 of ˚𝐶𝑃𝑃(𝑟) is shown in the two plots in figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' For each set, two fits are performed: a “one-state” fit having 𝑐𝑃 and 𝑚 𝜋 as free parameters, and a “two-states” one with an added “excited state” term 𝑎1(𝑚1/𝑟)𝐾1(𝑚1/𝑟) with two extra free parameters 𝑎1 and 𝑚1 > 𝑚 𝜋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' We choose appropriate values for the smaller 𝑟 of the correlator data that enter the fit, with different choices for one-state and two-states fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' Instead, all the data up to largest available 𝑟 = 𝑟max enter the fit, since we do not observe any boundary effect that constrains us otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' The two fits on each set give compatible results and the corresponding effective mass is shown in figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' From the one-state fits we obtain the results in table 2, which show a good agreement between the two sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' Contrary to the expectation based on 𝑛cnfg, the error is 40 % smaller on set B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' A possible explanation for this fact is the 𝑏shift = 12𝑎 of the samples of set A, halved with respect to set B, which can lead to a reduced effective number of samples due to stronger correlations in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' Similarly, we extract 𝑚𝑁 from the two contractions in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' (5) of the position-space nucleon correlator in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' (3d) as done in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' [5], but employing the techniques of sections 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' The results in table 2 are from the one-state fits to ˚𝐶 (1) 𝑁 𝑁 (𝑟) with the free parameters 𝑐𝑁 and 𝑚𝑁 , and are compatible 4See eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' (10) in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' [5] for the definition of the effective mass of the radial correlator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' 7 Hadronic observables from master-field simulations Marco Cè 5 10 15 20 25 30 r/a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='8 ameff covariant, trNN covariant, tr/xNN one-state fit, trNN one-state fit, tr/xNN two-state fit, trNN two-state fit, tr/xNN 5 10 15 20 25 30 r/a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='8 ameff covariant, trNN covariant, tr/xNN one-state fit, trNN one-state fit, tr/xNN two-state fit, trNN two-state fit, tr/xNN Figure 3: Effective mass of the ˚𝐶 (𝑖) 𝑁 𝑁 (𝑟) correlators as a function of 𝑟 for master fields in set A (left plot) and set B (right plot), where 𝑖 = 1 corresponds to the tr 𝑁𝑁 contraction and 𝑖 = 2 to the tr /𝑥𝑁𝑁 one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' On top of the data points with master-field errors shown in blue and orange for 𝑖 = 1 and 2 respectively, we show the results of a one-state fit in green and brown bands and of a two-states fit in red and purple bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' The thickness of the bands is the statistical error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' with the results of two-states fits with the replacement ˚𝐶𝑁 𝑁 (𝑟) → ˚𝐶𝑁 𝑁 (𝑟)[1+𝑎1(𝑚 𝜋/𝑟)𝐾1(𝑚 𝜋𝑟)] where 𝑎1 is an extra free parameter and 𝑚 𝜋 is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' The fit to ˚𝐶 (2) 𝑁 𝑁 (𝑟) shows similar results, although with a slightly larger central value that can be attributed to different discretization effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' The effective masses corresponding to data and fits are shown in figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' In the case of 𝑚𝑁 , we observe a larger error on set B, compatible with the lower statistics and showing no indication of correlation-in-space effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' We also extract the pion decay constant 𝑓 bare 𝜋 , where the bare indicates that we do not include the axial-current renormalization factor, from a combined fit of the four correlators ˚𝐶𝑃𝑃, ˚𝐶𝐴𝑃, ˚𝐶 (1) 𝐴𝐴 and ˚𝐶 (2) 𝐴𝐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' As fit function we employ the long-distance behaviours derived from eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' 3, which depends on the free parameters 𝑐𝑃, 𝑐𝐴 and 𝑚 𝜋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' As shown from the plots of the ratio between data and fit functions in figure 4, ˚𝐶𝐴𝑃 approaches the asymptotic behaviour at a smaller value of 𝑟, followed by ˚𝐶𝑃𝑃 and ˚𝐶 (2) 𝐴𝐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' ˚𝐶 (1) 𝐴𝐴 converges to the asymptotic behaviour at a much larger 𝑟, with the ratio being initially negative and changing sign around 𝑟 ≈ 14𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' The values of 𝑚 𝜋 obtained from these combined fits are consistent with the previous fits to only the ˚𝐶𝑃𝑃 correlators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' The decay constant is then given by 𝑓 bare 𝜋 = 𝑐𝐴/𝑚 𝜋 and shown in table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' Like in the case of 𝑚𝑁 , the values on set A and B are compatible, with a slightly larger error for set B that is consistent with the lower number of master field configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' Conclusions We have shown that position-space correlators can be used to extract hadron masses and decay constants with short-distance and cut-off effects under control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' Crucially, the statistical error can be estimated à la master field, obtaining an efficient scaling of the computational effort with the increased volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' In this work we studied sphere-averaged radial correlators, but potentially more information is encoded in correlators as function of four-dimensional coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' This requires understanding effects that break rotational symmetry at finite lattice spacing and is an interesting topic for further studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' 8 Hadronic observables from master-field simulations Marco Cè 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='020 0.' metadata={'source': 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+page_content='0018 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='0020 |cAc† P| |cAc† P| fit ˚CAP 10 20 30 r/a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='00011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='00012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='00013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='00014 |cA|2 |cA|2 fit ˚C(1) AA 10 20 30 r/a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='00011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='00012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='00013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='00014 |cA|2 |cA|2 fit ˚C(2) AA Figure 4: Plots of the ratio between correlator data and their fitted long-distance behaviours for master fields in set A (top row) and set B (bottom row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' The amplitude in the denominator is set to one, so that the actual amplitude for each correlator is shown on the vertical axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' In each row, four plots are shown for ˚𝐶𝑃𝑃, ˚𝐶 (1) 𝐴𝐴 (left column), ˚𝐶𝐴𝑃 and ˚𝐶 (2) 𝐴𝐴 (right column), with the correlator data with master field errors shown in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' The amplitude parameters of the corresponding fit function, which are functions of 𝑐𝑃 and 𝑐𝐴, are shown in an orange horizontal line with a pale orange error band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' Position-space methods find applications in computations of quantities that go beyond the simple hadronic quantities considered here, such as for example the hadronic vacuum polarisation contribution to the anomalous magnetic moment of the muon [16, 17], including the so-called window contribution [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' The estimators presented here provide a straightforward path to the computation of this quantities in the master-field paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' Acknowledgements: The research of MB is funded through the MUR program for young researchers “Rita Levi Montalcini”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' AF acknowledges support by the Ministry of Science and Technology Taiwan (MOST) under grant 111-2112-M-A49-018-MY2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' JRG acknowledges support from the Simons Foundation through the Simons Bridge for Postdoctoral Fellowships scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' MTH is supported by UKRI Future Leader Fellowship MR/T019956/1 and in part by UK STFC grant ST/P000630/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' This work was performed using the DiRAC Data Intensive service at Leicester, operated by the University of Leicester IT Services, which forms part of the STFC DiRAC HPC Facility (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='dirac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content='uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' The equipment was funded by BEIS capital funding via STFC capital grants ST/K000373/1 and ST/R002363/1 and STFC DiRAC Operations grant ST/R001014/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' DiRAC is part of the National e-Infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' We acknowledge PRACE for awarding us access to SuperMUC-NG at GCS@LRZ, Germany, where some computations were performed Many 9 Hadronic observables from master-field simulations Marco Cè simulations were performed on a dedicated HPC cluster at CERN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'} +page_content=' We gratefully acknowledge the computer 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a/SNE3T4oBgHgl3EQfDglb/content/tmp_files/2301.04287v1.pdf.txt b/SNE3T4oBgHgl3EQfDglb/content/tmp_files/2301.04287v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e7b9af99da8d8cdc974b4af8ae430f9d84be3201 --- /dev/null +++ b/SNE3T4oBgHgl3EQfDglb/content/tmp_files/2301.04287v1.pdf.txt @@ -0,0 +1,1844 @@ +arXiv:2301.04287v1 [math.NT] 11 Jan 2023 +ON INVERTED KLOOSTERMAN SUMS OVER FINITE FIELDS +XIN LIN AND DAQING WAN +ABSTRACT. The classical n-variable Kloosterman sums over finite fields are well understood by +Deligne’s theorem from complex point of view and by Sperber’s theorem from p-adic point of view. +In this paper, we study the complex and p-adic estimates of inverted n-variable Kloosterman sums, +addressing a question of N. Katz (1995). We shall give two complex estimates. The first one is +elementary based on Gauss sums. The second estimate is deeper, depending on the cohomological +results of Adolphson-Sperber, Denef-Loeser and Fu for twisted toric exponential sums. This deeper +result assumes that the characteristic p does not divide n + 1. Combining with Dwork’s p-adic +theory, we also determine the exact p-adic valuations for zeros and poles of the L-function associated +to inverted n-variable Kloosterman sums in the case p ≡ 1 mod (n + 1). As we shall see, the +inverted n-variable Kloosterman sum is more complicated than the classical n-variable Kloosterman +sum in all aspects in the sense that our understanding is less complete, partly because the Hodge +numbers are now mostly 2 instead of 1. +1. INTRODUCTION +Let Fq be the finite field of q elements with characteristic p. Let ψ : Fq → C∗ be a nontrivial +additive character and let χ1, . . . , χm : F∗ +q → C∗ be multiplicative characters. A classical problem +in number theory is to give a good estimate for the mixed character sum +� +xi∈F∗q +χ1(f1) · · · χm(fm)ψ(f), +(1.1) +where f, f1, . . . , fm ∈ Fq +� +x±1 +1 , . . . , x±1 +n +� +are Laurent polynomials. Reducing m if necessary, we +may assume that all the χi’s are non-trivial. Using the Gauss sum +G(χ) = +� +x∈F∗q +χ(x)ψ(x), +one obtains the well known relation +� +xi,yj∈F∗q +χ1(y1) · · · χm(ym)ψ (f + y1f1 + · · · + ymfm) += G(χ1) · · · G(χm) +� +xi∈F∗q +χ1(f1) · · · χm(fm)ψ(f). +As the Gauss sums are well-understood, the study of (1.1) is reduced to the study of the following +type of twisted toric exponential sum +� +xi∈F∗q +χ1(x1) · · · χn(xn)ψ (f(x1, · · · , xn)) , +(1.2) +where some of the χi’s may be trivial. This type of twisted toric exponential sum has been studied +extensively in the literature, most notably by Adolphson-Sperber [AS87a, AS89, AS90, AS93] via +Dwork’s p-adic cohomology, and by Denef-Loeser [DL91] and Fu [Fu09,Fu16] via Grothendieck’s +ℓ-adic cohomology. For the complex estimate, both approaches depend on Deligne’s theorem on +the Weil conjectures. A sharp estimate is obtained when f is non-degenerate with respect to its +Newton polyhedron ∆(f). For arbitrary f, the sum is still far from well understood. +2020 Mathematics Subject Classification. 11T23, 11L05, 11L07, 11S40. +Key words and phrases. Inverted Kloosterman sums, Exponential sums, L-function, Finite field. +1 + +2 +XIN LIN AND DAQING WAN +An important example of toric exponential sums is the classical n-variable Kloosterman sum, +where χ1 = · · · = χn = 1 and +f(x1, · · · , xn) = x1 + · · · + xn + +b +x1 · · · xn +, b ∈ F∗ +q. +In this case, the complex weights were determined by Deligne’s well known theorem, and the p- +adic slopes were determined by Sperber’s theorem [Spe80]. It should be noted that for twisted +n-variable Kloosterman sum (when some of the χi’s are non-trivial), the p-adic slopes are not +completely determined in general, except in the case n = 1 for which Adolphson-Sperber [AS87b] +obtained the generic p-adic slopes. If we invert the above Laurent polynomial and consider the +following rational function +f(x1, · · · , xn) = +1 +x1 + · · · + xn + +b +x1···xn +, b ∈ F∗ +q, +which is no longer a Laurent polynomial, we are led to the so-called inverted Kloosterman sum. +The study of such sums goes back to N. Katz [Kat95]. +More precisely, in this paper, we study the following twisted inverted n-variable Kloosterman +sum defined by +Sn(χ, b) = +� +x1···xn+1=b, xi∈F∗q +x1+···+xn+1̸=0 +χ1(x1) · · · χn+1(xn+1)ψ +� +1 +x1 + · · · + xn+1 +� += +� +x1+···+xn+ +b +x1···xn ̸=0 +xi∈F∗q +χ1(x1) · · · χn(xn)χn+1 +� +b +x1 · · · xn +� +ψ +� +1 +x1 + · · · + xn + +b +x1···xn +� +, +where b ∈ F∗ +q and n ≥ 1. When n = 1, Katz [Kat95] obtained a sharp upper bound for S1(χ, b). +This result along with the papers of Angel [Ang96] and Evans [Eva95] proves that finite upper +half plane graphs are Ramanujan in characteristic 2. In [Kat95], Katz raised the question: what +can be said for Sn(χ, b) when n ≥ 1? The aim of this paper is to study these inverted n-variable +Kloosterman sums from both complex and p-adic point of views. As we shall see, this class of sums +is very interesting as various new features and additional difficulties arise. +Remark. The exact sum introduced in [Kat95] is the following related sum, +Tn(χ, b) = +� +x1···xn+1=1 +x1+···+xn+1̸=0 +χ1(x1) · · · χn+1(xn+1)ψ +� +b +x1 + · · · + xn+1 +� +. +Upon the change of variables xi → bxi, one sees that +Tn(χ, b) = Sn(χ, +1 +bn+1 )χ1 · · · χn+1(b). +Thus, the two families of sums Sn(χ, b) and Tn(χ, b) are essentially equivalent. We work with +Sn(χ, b) as it is the closer inverted analogue of the classical Kloosterman sum. +For complex estimate of Sn(χ, b), the best one can hope for would be a square root cancellation +in the sum Sn(χ, b), i.e. +Sn(χ, b) = On(q +n +2 ). +As we shall see, this is not true for n > 2 when χ1 = · · · = χn+1, in which case, Sn(χ, b) has the +main term −qn−1χ1(b) whose exponent n − 1 is larger than the exponent n/2. +We shall give two different estimates for Sn(χ, b). The first estimate is based on an elementary +method via Gauss sums which already shows the new feature of a non-trivial main term when +χ1 = · · · = χn+1. We obtain the following simple estimate for Sn(χ, b). + +ON INVERTED KLOOSTERMAN SUMS OVER FINITE FIELDS +3 +Theorem 1.1. Notations as above. If χ1 = · · · = χn+1, we have +����Sn(χ, b) + (q − 1)n +q +χ1(b) +���� ≤ q +n+1 +2 . +If χi ̸= χj for some i ̸= j, we have +|Sn(χ, b)| ≤ q +n+1 +2 . +For large q, the error term q(n+1)/2 is not the optimal square root cancellation yet. To obtain the +deeper square root cancellation with error term On(qn/2), we reduce Sn(χ, b) to a certain twisted +toric exponential sum S∗ +k(χ, f) which can be handled by the results of Adolphson-Sperber, Denef- +Loeser and Fu. We check that the related Laurent polynomial f is non-degenerate if p does not +divide n + 1. This gives our second estimate. +Theorem 1.2. Notations as above. Suppose p ∤ (n + 1). If χ1 = · · · = χn+1, we have +|Sn(χ, b) + (q − 1)n − (−1)n +q +χ1(b)| ≤ (2n + 1)q +n +2 . +If χi ̸= χj for some i ̸= j, we have +|Sn(χ, b)| ≤ 2(n + 1)q +n +2 . +It is clear that the estimate in Theorem 1.2 is better than the estimate in Theorem 1.1 if q > +4(n + 1)2. It is expected that Theorem 1.2 (with possibly a better constant) remains true when p +divides n + 1. But in this singular case, the above toric sum results do not apply and one would +need a different approach. +For p-adic slopes, we focus on the simpler untwisted case. When χ1 = · · · = χn+1, we study +the p-adic valuations for the reciprocal roots and poles of the generating L-function of the inverted +n-variable Kloosterman sum. We show that the Newton polygon agrees with the Hodge polygon +if and only if p ≡ 1 mod n + 1. As a consequence, this completely determines the p-adic slope +sequence when p ≡ 1 mod n + 1. This is described more precisely below. +Our approach is to reduce the generating L-function to a certain untwisted toric L-function. To +construct the relationship between the two L-functions, we need to consider the inverted Klooster- +man sum defined over every finite extension Fqk. Suppose χ1 = · · · = χn+1, the inverted n-variable +Kloosterman sum over Fqk is defined by +Sk,n(b) = +� +x1+···+xn+ +b +x1···xn ̸=0 +xi∈F∗ +qk +ψ +� +Trk +� +1 +x1 + · · · + xn + +b +x1···xn +�� +, +where b ∈ F∗ +q and Trk : Fqk → Fq is the trace map. The generating L-function of Sk,n(b) is defined +by +Ln(b, T) = exp +� ∞ +� +k=1 +Sk,n(b)T k +k +� +. +Applying some systematic results available for the related toric L-function, we obtain the complex +and p-adic absolute values for all the reciprocal roots and poles of Ln(b, T) under given restrictions +on p. +Theorem 1.3. Suppose p ∤ (n + 1). The L-function is a rational function of the following form: +Ln(b, T)(−1)n+1 = (1 − T)(n+1) +n +� +j=2 +� +1 − qj−1T +�(n +j)(−1)j−1 2n +� +i=1 +(1 − αiT). +As complex numbers, the reciprocal roots αi satisfy |αi| = q +n +2 for all 1 ≤ i ≤ 2n. + +4 +XIN LIN AND DAQING WAN +If p ≡ 1 mod (n + 1), viewing the αi’s as p-adic numbers, the slope sequence {vq(αi)}2n +i=1 in +increasing order is given by +{0, 1, 1, 2, 2, . . . , n − 1, n − 1, n}. +Our results show that for the inverted n-variable Kloosterman sum, if p does not divide n + 1, +the primitive middle cohomology has dimension 2n, pure of weight n and with Hodge numbers +{1, 2, 2, · · · , 2, 1}. This is in contrast to the classical n-variable Kloosterman sum, where the middle +cohomology has dimension n + 1, pure of weight n and with Hodge numbers {1, 1, 1, · · · , 1, 1}. +The larger Hodge numbers suggest that new features and additional difficulties would likely arise +in studying inverted n-variable Kloosterman sums. +As a corollary of the first part in Theorem 1.3, we get a slightly better bound for Sk,n(b). +Corollary 1.4. If p ∤ (n + 1), for all integers k ≥ 1, we have +|Sk,n(b) + (qk − 1)n − (−1)n(qk + 1) +qk +| ≤ 2nq +nk +2 . +When k = 1, the exponential sum Sk,n(b) reduces to Sn(χ, b) with χ1 = · · · = χn+1. Explicitly, +the estimate in Corollary 1.4 is better than the estimate in the first case of Theorem 1.2. We remark +that the condition p ≡ 1 mod (n + 1) in the second part of Theorem 1.3 is necessary and sufficient +for the same conclusion to hold. Thus, if p ̸≡ 1 mod (n + 1), the slope sequence will be strictly +different, but we do not know the exact slope sequence in this case. +The rest of this paper is organized as follows. In section 2, we review some technical methods +including Adolphson-Sperber’s theorems and Dwork’s theory on toric exponentials sums. In section +3, we use these methods to prove the main results. +We end the introduction by mentioning several recent references [FW21] [Li21] [YZ22] [CL22] +[LC22] which applied some of the related toric techniques in treating different classes of non-toric +n variable exponential sums arising from analytic number theory. +2. PRELIMINARIES ON TORIC EXPONENTIAL SUMS +2.1. Rationality of the toric L-function. Let f ∈ Fq +� +x±1 +1 , . . . , x±1 +n +� +be a Laurent polynomial and +its associated twisted toric exponential sum is defined to be +S∗ +k(χ, f) = +� +xi∈F∗ +qk +χ1(Nk(x1)) · · · χn(Nk(xn))ψ(Trk(f)), +(2.1) +where Trk : Fqk → Fq is the trace map, Nk : Fqk → Fq is the norm map, χ1, . . . , χn : F∗ +q → C∗ are +multiplicative characters and ψ : Fq → C∗ is a nontrivial additive character. A classical problem in +number theory is to estimate the absolute values of S∗ +k(χ, f). +A well known theorem of Dwork-Bombieri-Grothendieck [Dwo60,Bom66,Gro68] says that the +generating L-function of S∗ +k(χ, f) is a rational function: +L∗(χ, f, T) = exp +� ∞ +� +k=1 +S∗ +k(χ, f)T k +k +� += +�d1 +i=1(1 − αiT) +�d2 +j=1(1 − βjT) +, +where all the reciprocal zeros and poles are non-zero algebraic integers. In particular, when all of χi +are trivial characters, the untwisted toric exponential sum and the associated L-function are denoted +by S∗ +k(f) and L∗(f, T) respectively. +Through logarithmic derivatives, we have +S∗ +k(χ, f) = +d2 +� +j=1 +βk +j − +d1 +� +i=1 +αk +i , +k ∈ Z≥1. +(2.2) +Thus, the estimate of S∗ +k(χ, f) is reduced to understanding all the absolute values of the reciprocal +zeros αi and poles βj. Deligne’s theorem on Riemann hypothesis [Del80] describes the bounds for + +ON INVERTED KLOOSTERMAN SUMS OVER FINITE FIELDS +5 +all the absolute values of αi and βj in general. The complex absolute values of reciprocal zeros and +poles satisfy +|αi| = qui/2, |βj| = qvj/2, ui ∈ Z ∩ [0, 2n], vj ∈ Z ∩ [0, 2n]. +For non-archimedean absolute values, Deligne proved that |αi|ℓ = |βj|ℓ = 1 when ℓ is a prime and +ℓ ̸= p. For p-adic absolute values, one has +|αi|p = q−ri, |βj|p = q−sj, ri ∈ Q ∩ [0, n], sj ∈ Q ∩ [0, n]. +The integer ui (resp. vj) is called the weight of αi (resp. βj) and the rational number ri (resp. sj) +is called the slope of αi (resp. βj). +In the past few decades, there has been tremendous interest in determining the weights and slopes +of the generating L-functions. Without any further condition on the Laurent polynomial f, it is even +hard to determine the number of reciprocal roots and poles. Most of the existing work about the +weights and slopes relies on a suitable smoothness condition. For toric exponential sums, this +usually means the non-degenerate condition, see below for the precise definition. +Let +(2.3) +f(x1, . . . xn) = +J +� +j=1 +ajxVj +be a Laurent polynomial with aj ∈ F∗ +q and Vj = (v1j, . . . , vnj) ∈ Zn (1 ≤ j ≤ J). The Newton +polyhedron of f, ∆(f), is defined to be the convex closure in Rn generated by the origin and the +lattice points Vj (1 ≤ j ≤ J). For δ ⊂ ∆(f), let the Laurent polynomial +f δ = +� +Vj∈δ +ajxVj +be the restriction of f to δ. +Definition 2.1 (non-degenerate). A Laurent polynomial f is called non-degenerate if for each closed +face δ of ∆(f) of arbitrary dimension which doesn’t contain the origin, the partial derivatives +�∂f δ +∂x1 +, . . . , ∂f δ +∂xn +� +have no common zeros with x1 . . . xn ̸= 0 over the algebraic closure of Fq. +When f is non-degenerate, Adolphson-Sperber [AS89] proved that the untwisted toric L-function +L∗(f, T)(−1)n−1 is a polynomial and improved the bound for weight. +Theorem 2.2 ( [AS89]). For any non-degenerate f ∈ Fq[x±1 +1 , . . . , x±1 +n ], the associated L-function +L∗(f, T)(−1)n−1 is a polynomial of degree n! Vol(∆(f)). Namely, +L∗(f, T)(−1)n−1 = +n! Vol(∆(f)) +� +i=1 +(1 − αiT), αi ̸= 0. +For any multiplicative characters χi (nontrivial or trivial), the degree and weights of the twisted +toric L-function are studied and completed by Adolphson-Sperber [AS91, AS93], [DL91]and Fu +[Fu09] under the non-degeneracy assumption. These results lead to the following bound for the +twisted toric exponential sum. +Theorem 2.3 ( [DL91] [AS93] [Fu09]). Let f ∈ Fq +� +x±1 +1 , . . . , x±1 +n +� +be a Laurent polynomial with +∆ = ∆(f). If f is non-degenerate, one has +|S∗ +k(χ1, . . . , χn, f)| ≤ n! Vol(∆)q +nk +2 . +For the slopes of the L-function, the situation is somewhat simpler in the untwisted case, oth- +erwise, even the description of Adolphson-Sperber’s “Hodge lower bound" is a little cumbersome. +Thus, the definitions and theories discussed in the following subsections focus on the untwisted +L-function. + +6 +XIN LIN AND DAQING WAN +2.2. Newton polygon and Hodge polygon. To determine the q-adic slopes of its reciprocal roots, +we introduce the q-adic Newton polygon. +Definition 2.4 (Newton polygon). Let L(T) = �n +i=0 aiT i ∈ 1+TQp[T], where Qp is the algebraic +closure of Qp. The q-adic Newton polygon of L(T) is defined to be the lower convex closure of the +set of points {(k, ordq(ak)) |k = 0, 1, . . . , n} in R2. +Lemma 2.5 ( [Kob84]). Notations as above. Let L(T) = (1−α1T) . . . (1−αnT) be the factoriza- +tion of L(T) in terms of reciprocal roots αi ∈ Qp. Let λi = ordq αi. If λ is the slope of the q-adic +Newton polygon of L(T) with horizontal length l, then precisely l of the λi are equal to λ. +The q-adic Newton polygon of L∗(f, T)(−1)n−1 is denoted as NP(f). Lemma 2.5 relates NP(f) +to the q-adic valuation of reciprocal roots of toric L-functions. The definition of NP(f) relies on +the coefficients of L-function, which makes it hard to compute directly. When f is non-degenerate, +Adolphson and Sperber proved that L∗(f, T)(−1)n−1 is a polynomial and NP(f) has a topological +lower bound called Hodge polygon, which is easier to determine. Thus, we shall compute Hodge +polygon and consider when the Newton polygon coincides with this lower bound. +Let ∆ be an n-dimensional integral polytope containing the origin in Rn. For u ∈ Rn, the +weight function w(u) represents the smallest non-negative real number c such that u ∈ c∆. Denote +w(u) = ∞ if such c doesn’t exist. Assume δ is a co-dimension 1 face of ∆ not containing the +origin. Let D(δ) be the least common multiple of the denominators of the coefficients in the linear +equation defining δ, normalized to have constant term 1. We define the denominator of ∆ to be the +least common multiple of all such D(δ) given by: +D = D(∆) = lcmδD(δ), +where δ runs over all the co-dimension 1 faces of ∆ that don’t contain the origin. It’s easy to check +w(Zn) ⊆ +1 +D(∆)Z≥0 ∪ {+∞}. +For a non-negative integer k, let +(2.4) +W∆(k) = # +� +u ∈ Zn|w(u) = k +D +� +be the number of lattice points in Zn with weight k/D. Its generating function is known to be a +rational function of the following form +∞ +� +k=0 +W∆(k)tk/D = +�nD +k=0 H∆(k)tk/D +(1 − t)n +. +This leads to +Definition 2.6 (Hodge number). Let ∆ be an n-dimensional integral polytope containing the origin +in Rn. For a non-negative integer k, the k-th Hodge number of ∆ is defined to be +H∆(k) = +n +� +i=0 +(−1)i +�n +i +� +W∆(k − iD). +(2.5) +It is known that +H∆(k) = 0, +if +k > nD. +Based on the Hodge numbers, we define the Hodge polygon of a given polyhedron ∆ ∈ Rn as +follows. +Definition 2.7 (Hodge polygon). The Hodge polygon HP(∆) of ∆ is the lower convex polygon in +R2 with vertices (0,0) and +Qk = +� +k +� +m=0 +H∆(m), 1 +D +k +� +m=0 +mH∆(m) +� +, +k = 0, 1, . . . , nD, + +ON INVERTED KLOOSTERMAN SUMS OVER FINITE FIELDS +7 +where H∆(k) is the k-th Hodge number of ∆, k = 0, 1, . . . , nD. +That is, HP(∆) is a polygon starting from origin (0,0) with a slope k/D side of horizontal length +H∆(k) for k = 0, 1, . . . , nD. The vertex Qk is called a break point if H∆(k + 1) ̸= 0 where +k = 1, 2, . . . , nD − 1. +Note that the horizontal length H∆(k) is the number of lattice points of weight k/D in a certain +fundamental domain corresponding to a basis of the p-adic cohomology space used to compute +the L-function. By a theorem of Adolphson-Sperber, the Hodge polygon is a lower bound of the +corresponding Newton polygon. +Theorem 2.8 ( [AS89]). For every prime p and non-degenerate Laurent polynomial f with ∆(f) = +∆ ⊂ Rn, we have +NP(f) ≥ HP(∆), +where NP(f) is the q-adic Newton polygon of L∗(f, T)(−1)n−1. Furthermore, the endpoints of +NP(f) and NP(∆) coincide. +Definition 2.9 (ordinary). A Laurent polynomial f is called ordinary if NP(f) = HP(∆). +It is clear that the ordinary property of a Laurent polynomial depends on its Newton polyhedron +∆ and on the coefficients of f(x). Applying the facial decomposition theorem [Wan93], we reduce +the ordinary property of f to its smaller pieces which are easier to deal with. +Theorem 2.10 (Facial decomposition theorem [Wan93]). Let f be a non-degenerate Laurent poly- +nomial over Fq. Assume ∆ = ∆(f) is n-dimensional and δ1, . . . , δh are all the co-dimension 1 +faces of ∆ which don’t contain the origin. Let f δi denote the restriction of f to δi. Then f is +ordinary if and only if f δi is ordinary for 1 ≤ i ≤ h. +2.3. Boundary decomposition theorems. Before describing the boundary decomposition, we ex- +press the L-function in terms of the Fredholm determinant of an infinite Frobenius matrix via +Dwork’s trace formula. +2.3.1. Dwork’s trace formula. Let Qp be the field of p-adic numbers and Ω be the completion of +Qp. A fixed primitive p-th root of unity in Ω is denoted as ζp. Let π be the element of Qp(ζp) +satisfies +∞ +� +m=0 +πpm +pm = 0, π ≡ ζp − 1 +mod (ζp − 1)2, +and +ordp π = +1 +p − 1. +Then, π is a uniformizer of Qp(π) and thus Qp(π) = Qp(ζp). Let Ep(t) be the Artin-Hasse +exponential series, +Ep(t) = exp +� ∞ +� +m=0 +tpm +pm +� += +∞ +� +m=0 +λmtm ∈ Zp[[x]]. +In Dwork’s terminology, a splitting function θ(t) is defined to be +θ(t) = Ep(πt) = +∞ +� +m=0 +λmπmtm. +A Laurent polynomial f ∈ Fq[x±1 +1 , . . . , x±1 +n ] is written as +f = +J +� +j=1 +¯ajxVj, +where Vj ∈ Zn and ¯aj ∈ F∗ +q. Let aj be the Teichmüller lifting of ¯aj in Ω satisfying aq +j = aj. Let +F(f, x) = +J +� +j=1 +θ(ajxVj) = +� +r∈Zn +Fr(f)xr. + +8 +XIN LIN AND DAQING WAN +The coefficients are given by +Fr(f) = +� +u +( +J +� +j=1 +λujauj +j )πu1+···+uJ, +r ∈ Zn, +where the sum is over all the solutions of the following linear system +J +� +j=1 +ujVj = r +with +uj ∈ Z≥0, +and λm is m-th coefficient of the Artin-Hasse exponential series Ep(t). +Assume ∆ = ∆(f). Let L(∆) = Zn ∩ C(∆) be the set of lattice points in the closed cone +generated by origin and ∆. For a given point r ∈ Rn, define the weight function to be +w(r) := inf +⃗u + + + +J +� +j=1 +uj| +J +� +j=1 +ujVj = r, +uj ∈ R≥0 + + + . +The infinite semilinear Frobenius matrix A1(f) is the following matrix whose rows and columns +are indexed by the lattice points in L(∆) with respect to the weights: +A1(f) = (ar,s(f)) = (Fps−r(f)πw(r)−w(s)), +where r, s ∈ L(∆). The infinite linear Frobenius matrix Aa(f) is defined to be +Aa(f) = A1(f)Aτ +1(f) · · · Aτ a−1 +1 +(f), +where τ is the absolute Frobenius automorphism. +Dwork’s trace formula can be expressed in terms of the matrix Aa(f) as follows, see [Wan04] +Theorem 2.11. We have +(2.6) +L∗(f, T)(−1)n−1 = +n +� +i=0 +det(I − TqiAa(f))(−1)i(n +i). +Equivalently, +(2.7) +det(I − TAa(f)) = +∞ +� +i=0 +� +L∗(f, qiT)(−1)n−1�(n+i−1 +i +) . +Now it suffices to understand the determinant det(I−TAa(f)). Based on the fact that ordp Fr(f) ≥ +w(r) +p−1 , we have the following estimate +ordp(ar,s(f)) ≥ w(ps − r) + w(r) − w(s) +p − 1 +≥ w(s). +Let ξ be an element in Ω satisfying ξD = πp−1. Then A1(f) can be written in a block form, +A1(f) = + + + + + + + + +A00 +ξA01 +· · · +ξiA0i +· · · +A10 +ξA11 +· · · +ξiA1i +· · · +... +... +... +... +Ai0 +ξAi1 +· · · +ξiAii +· · · +... +... +... +... + + + + + + + + +, +where the block Aii is a p-adic integral W∆(i)×W∆(i) matrix. This implies that the q-adic Newton +polygon of det(I −TA1(f)) has a natural lower bound which can be identified with the chain level +version of the Hodge polygon. + +ON INVERTED KLOOSTERMAN SUMS OVER FINITE FIELDS +9 +Definition 2.12. Let P(∆) be the polygon in R2 with vertices (0, 0) and +Pk = +� +k +� +m=0 +W∆(m), 1 +D +k +� +m=0 +mW∆(m) +� +, +k = 0, 1, 2, . . . +The chain level version of Adolphson-Sperber’s lower bound and the ordinary property are as +follows. +Proposition 2.13 ( [AS87a]). The q-adic Newton polygon of det(I − TAa(f)) lies above P(∆). +Proposition 2.14 ( [Wan04]). Notations as above. Assume f is non-degenerate with ∆ = ∆(f). +Then NP(f) = HP(∆) if and only if the q-adic Newton polygon of det(I − TAa(f)) coincides +with its lower bound P(∆). +2.3.2. Boundary decomposition. Let f ∈ Fq[x±1 +1 , . . . , x±1 +n ] with ∆ = ∆(f), where ∆ is an n- +dimensional integral convex polyhedron in Rn containing the origin. Let C(∆) be the cone gener- +ated by ∆ in Rn. +Definition 2.15. The boundary decomposition +B(∆) = { the interior of a closed face in C(∆) containing the origin} +is the unique interior decomposition of C(∆) into a disjoint union of relatively open cones. +If the origin is a vertex of ∆, then it is the unique 0-dimensional open cone in B(∆). Recall that +A1(f) = (ar,s(f)) is the infinite semilinear Frobenius matrix whose rows and columns are indexed +by the lattice points in L(∆). For Σ ∈ B(∆), we define A1(Σ, f) to be the submatrix of A1(f) +with r, s ∈ Σ. Let f Σ be the restriction of f to the closure of Σ. Then A1(Σ, f Σ) denotes the +submatrix of A1(f Σ) with r, s ∈ Σ. +Let B(∆) = {Σ0, . . . , Σh} such that dim(Σi) ≤ dim(Σi+1), i = 0, . . . , h − 1. Define Bij = +(ar,s(f)) with r ∈ Σi and s ∈ Σj (0 ≤ i, j ≤ h). After a permutation of basis vectors, the infinite +semilinear Frobenius matrix can be written as +(2.8) +A1(f) = + + + + + +B00 +B01 +· · · +B0h +B10 +B11 +· · · +B1h +... +... +... +... +Bh0 +Bh1 +· · · +Bhh + + + + + , +where Bij = 0 for i > j. Then det(I −TA1(f)) = �h +i=0 det(I −TBii) and we have the boundary +decomposition theorem. +Theorem 2.16 (Boundary decomposition [Wan93]). Let f ∈ Fq[x±1 +1 , . . . , x±1 +n ] with ∆ = ∆(f). +Then we have the following factorization +det(I − TA1(f)) = +� +Σ∈B(∆) +det +� +I − TA1(Σ, f Σ) +� +. +2.4. Diagonal local theory. In this subsection, we introduce some non-degenerate and ordinary +criteria when the Laurent polynomial is diagonal. +Definition 2.17. A Laurent polynomial f ∈ Fq[x±1 +1 , . . . , x±1 +n ] is called diagonal if f has exactly n +non-constant terms and ∆(f) is an n-dimensional simplex in Rn. +Let f be a diagonal Laurent polynomial over Fq. Write +f(x1, x2, . . . xn) = +n +� +j=1 +ajxVj, +where aj ∈ F∗ +q and Vj = (v1j, . . . , vnj) ∈ Zn for 1 ≤ j ≤ n. Let ∆ = ∆(f). The vertex matrix of +∆ is defined to be +M(∆) = (V1, . . . , Vn), + +10 +XIN LIN AND DAQING WAN +where the i-th column is the i-th exponent of f. Since f is diagonal, M(∆) is invertible. +Proposition 2.18. Suppose f ∈ Fq[x±1 +1 , . . . , x±1 +n ] is diagonal with ∆ = ∆(f). Then f is non- +degenerate if and only if p is relatively prime to det(M(∆)). +Let S(∆) be the solution set of the following linear system +M(∆) + + + + + +r1 +r2 +... +rn + + + + + ≡ 0 (mod1), +ri ∈ Q ∩ [0, 1). +It’s easy to prove that S(∆) is an abelian group and its order is given by +|det M(∆)| = n! Vol(∆). +(2.9) +Let Sp(∆) denote the prime to p part of S(∆). It is an abelian subgroup of order equal to the +prime to p factor of det M(∆). In particular, Sp(∆) = S(∆) if p is relatively prime to det M(∆). +By the Stickelberger theorem for Gauss sums, we have the following ordinary criterion for a non- +degenerate Laurent polynomial [Wan04]. +Proposition 2.19. A diagonal Laurent polynomial f is ordinary at p if and only if the norm function +|r| = r1 + · · · + rn on Sp(∆) is stable under the p-action: That is, for each r ∈ Sp(∆), we have +|r| = |{pr}|, where {pr} is the class of pr in Sp(∆). +3. PROOF OF THE MAIN THEOREMS +We prove the main theorems in this section. +3.1. Proof of Theorem 1.1. Recall that for integer n ≥ 1, the twisted inverted n-variable Kloost- +erman sum is defined to be +Sn(χ, b) = +� +x1···xn+1=b +x1+···+xn+1̸=0 +χ1(x1) · · · χn+1(xn+1)ψ +� +1 +x1 + · · · + xn+1 +� +, +where b ∈ F∗ +q, ψ : Fq → C∗ is a nontrivial additive character and χ1, . . . , χn+1 : F∗ +q → C∗ are +multiplicative characters. Let χ : F∗ +q → C∗ denote a multiplicative character. By the orthogonality +of characters, we have +Sn(χ, b) = +1 +q(q − 1) +� +λ,xi∈F∗q +� +u∈Fq +ψ (u (x1 + · · · + xn+1 − λ)) χ1(x1) · · · χn+1(xn+1) +× ψ +�1 +λ +� � +χ +χ +�x1 · · · χn+1 +b +� += +1 +q(q − 1) +� +λ∈F∗q +� +xi∈F∗q +χ1(x1) · · · χn+1(xn+1)ψ +� 1 +λ +� � +χ +χ +�x1 · · · χn+1 +b +� ++ +1 +q(q − 1) +� +λ∈F∗q +� +xi∈F∗q +� +u∈F∗q +ψ (u (x1 + · · · + xn+1 − λ)) +× χ1(x1) · · · χn+1(xn+1)ψ +� 1 +λ +� � +χ +χ +�x1 · · · χn+1 +b +� +=S1 + S2. +(3.1) +Then +S1 = +1 +q(q − 1) +� +λ∈F∗q +ψ +�1 +λ +� � +χ +χ−1(b) +� +xi∈F∗q +(χχ1) (x1) · · · (χχn+1) (xn+1) + +ON INVERTED KLOOSTERMAN SUMS OVER FINITE FIELDS +11 += +1 +q(q − 1) +� +λ∈F∗q +ψ +�1 +λ +� � +χ +χ−1(b) +n+1 +� +i=1 + + � +xi∈F∗q +(χχi) (xi) + + += + + + +−(q − 1)n +q +χ1(b), +if χ1 = · · · = χn+1, +0, +otherwise. +(3.2) +If χ is trivial, the Gauss sum G(χ) = −1. If χ is non-trivial, |G(χ)| = √q. Then +S2 = +1 +q(q − 1) +� +λ,u∈F∗q +� +χ +χ−1(b) +� +xi∈F∗q +(χχ1) (x1)ψ(ux1) · · · (χχn+1) (xn+1)ψ(uxn+1) +× ψ(−uλ)ψ +� 1 +λ +� += +1 +q(q − 1) +� +λ,u∈F∗q +� +χ +χ−1(b)χn+1χ1 · · · χn+1(u)ψ(−uλ)ψ +�1 +λ +� +G(χχ1) · · · G(χχn+1) += +1 +q(q − 1) +� +χ +χ−1(b) + + � +λ∈F∗q +χn+1χ1 · · · χn+1 +� +− 1 +λ +� +ψ +� 1 +λ +� + G(χn+1χ1 · · · χn+1) +× G(χχ1) · · · G(χχn+1) += +1 +q(q − 1) +� +χ +χ−1(b)χn+1χ1 · · · χn+1(−1)G(χn+1χ1 · · · χn+1)G(χn+1χ1 · · · χn+1) +× G(χχ1) · · · G(χχn+1). +(3.3) +Since |G(χ)| ≤ √q, it follows that |S2| ≤ q +n+1 +2 . Combining (3.1) and (3.2), we can deduce the +following bounds. +����Sn(χ, b) + (q − 1)n +q +χ1(b) +���� ≤ q +n+1 +2 , +if χ1 = · · · = χn+1, +and +|Sn(χ, b)| ≤ q +n+1 +2 , +if χi ̸= χj for some i ̸= j. +This proves Theorem 1.1. +3.2. Proof of Theorem 1.2. The twisted inverted Kloosterman sum Sn(χ, b) has the expression +Sn(χ, b) = +� +x1+···+xn+ +b +x1···xn ̸=0 +xi∈F∗q +χ1(x1) · · · χn(xn)χn+1 +� +b +x1 · · · xn +� +× ψ +� +1 +x1 + · · · + xn + +b +x1···xn +� += +� +z +� +x1+···+xn+ +b +x1···xn +� +=1 +z, xi∈F∗q +χn+1(b) (χ1χn+1)(x1) · · · (χnχn+1)(xn)ψ (z) += 1 +q +� +z, xi∈F∗q +y∈Fq +χn+1(b) (χ1χn+1)(x1) · · · (χnχn+1)(xn) +× ψ +� +z + y +� +1 − z +� +x1 + · · · + xn + +b +x1 · · · xn +��� + +12 +XIN LIN AND DAQING WAN += χn+1(b) +q + + � +z, xi∈F∗q +(χ1χn+1)(x1) · · · (χnχn+1)(xn)ψ (z) + En(χ, b) + + += + + + + + +−(q − 1)n +q +χ1(b) + 1 +qχ1(b)En(χ, b), +if χ1 = · · · = χn+1, +1 +qχn+1(b)En(χ, b), +if χi ̸= χj for some i ̸= j, +(3.4) +where +En(χ, b) = +� +y,z,xi∈F∗q +(χ1χn+1)(x1) · · · (χnχn+1)(xn) +× ψ +� +z + y +� +1 − z +� +x1 + · · · + xn + +b +x1 · · · xn +��� +. +In order to prove Theorem 1.2, it suffices to estimate En(χ, b). +Let f ∈ Fq[x±1 +1 , . . . , x±1 +n+2] be the Laurent polynomial defined by +f(x1, · · · , xn+2) = xn+1 +� +1 − xn+2 +� +x1 + · · · + xn + +b +x1 · · · xn +�� ++ xn+2. +As defined in (2.1), En(χ, b) is the twisted toric exponential sum associated to f. Let ∆ = ∆(f) +denote the Newton polyhedron corresponding to f. Clearly, dim ∆ = n + 2 and ∆ has n + 4 +vertices in Rn+2: V0 = (0, · · · , 0)(the origin), V1 = (1, 0, · · · , 0, 1, 1), V2 = (0, 1, · · · , 0, 1, 1), +. . . , Vn = (0, 0, · · · , 1, 1, 1), Vn+1 = (−1, · · · , −1, 1, 1), Vn+2 = (0, · · · , 0, 1, 0) and Vn+3 = +(0, · · · , 0, 0, 1). Furthermore, ∆ has exactly 2 co-dimension 1 faces not containing the origin. +Explicitly, they are +δ1 : xn+1 = 1 +and +δ2 : xn+2 = 1. +Vertices V1, . . . , Vn+2 determine the face δ1 and vertices V1, . . . , Vn+1, Vn+3 determine the face δ2. +Let M(δi) be the vertex matrix of δi, we have +M(δ1) = + + + + + + + + + +1 +0 +· · · +0 +−1 +0 +0 +1 +· · · +0 +−1 +0 +... +... +... +... +... +... +0 +0 +· · · +1 +−1 +0 +1 +1 +· · · +1 +1 +1 +1 +1 +· · · +1 +1 +0 + + + + + + + + + +, +M(δ2) = + + + + + + + + + +1 +0 +· · · +0 +−1 +0 +0 +1 +· · · +0 +−1 +0 +... +... +... +... +... +... +0 +0 +· · · +1 +−1 +0 +1 +1 +· · · +1 +1 +0 +1 +1 +· · · +1 +1 +1 + + + + + + + + + +. +(3.5) +Explicitly, each f δi is diagonal for i = 1, 2. The restriction of f to δi is defined by +f δi = +� +Vj∈δi +ajxVj. +Proposition 3.1. +(i). The denominator D = 1. +(ii). f is non-degenerate if and only if p ∤ (n + 1). +(iii). Vol(∆) = 2n + 2 +(n + 2)!. +Proof. The denominator D = 1 can be deduced immediately from the equation of δi. Since δ1 and +δ2 are the co-dimension 1 faces of ∆(f) not containing the origin, it suffices to prove f δ1 and f δ2 +are non-degenerate. By Proposition 2.18, f δi is non-degenerate if and only if p is relatively prime +to det(M(δi)). By formula (3.5), +det(M(δ1)) = −(n + 1) +and +det(M(δ2)) = n + 1. +(3.6) +This proves (ii). + +ON INVERTED KLOOSTERMAN SUMS OVER FINITE FIELDS +13 +V0 +V1 +V2 +V3 +V4 +FIGURE 1. ∆ for n = 1 +Let ∆i be the polytope generated by δi and the origin. The facial decomposition of ∆ implies +that +Vol(∆) = Vol(∆1) + Vol(∆2). +By formula (2.9) and (3.6), we obtain (iii). +□ +Combining Theorem 2.3 with Proposition 3.1, if p ∤ (n + 1), we have +|En(χ, b)| ≤ (n + 2)! Vol(∆) · q +n+2 +2 += 2(n + 1)q +n+2 +2 , +(3.7) +where p is the characteristic of Fq. Putting (3.4) and (3.7) together, we then obtain the following +bounds when p ∤ (n + 1). +|Sn(χ, b) + (q − 1)n +q +χ1(b)| ≤ 2(n + 1)q +n +2 , +if χ1 = · · · = χn+1, +and +|Sn(χ, b)| ≤ 2(n + 1)q +n +2 , +if χi ̸= χj for some i ̸= j. +In the case χ1 = · · · = χn+1, the twisted sum En(χ, b) becomes the following untwisted toric +exponential sum +En(χ, b) = +� +y,z,xi∈F∗q +ψ +� +z + y +� +1 − z +� +x1 + · · · + xn + +b +x1 · · · xn +��� +. +Since the origin is a vertex of ∆ and the polynomial inside the additive character has no constant +term, 1 is a trivial eigenvalue of the middle dimensional cohomology. Removing this trivial eigen- +value from the error term, one gets +|En(χ, b) − (−1)n+2| ≤ (2n + 1)q +n +2 , +if χ1 = · · · = χn+1, +and hence the slightly sharper estimate +|Sn(χ, b) + (q − 1)n + (−1)n+1 +q +χ1(b)| ≤ (2n + 1)q +n +2 , +if χ1 = · · · = χn+1. +This proves Theorem 1.2. +3.3. Proof of Theorem 1.3. Similar to formula (3.4), we relate the untwisted inverted Kloosterman +sum Sk,n(b) to toric exponential sum S∗ +k(f). +Sk,n(b) = +� +x1+···+xn+ +b +x1···xn ̸=0 +xi∈F∗ +qk +ψ +� +Trk +� +1 +x1 + · · · + xn + +b +x1···xn +�� + +14 +XIN LIN AND DAQING WAN += +� +z +� +x1+···+xn+ +b +x1···xn +� +=1 +z, xi∈F∗ +qk +ψ (Trk (z)) += 1 +qk +� +z, xi∈F∗ +qk +y∈Fqk +ψ +� +Trk +� +z + y +� +1 − z +� +x1 + · · · + xn + +b +x1 · · · xn +���� += −(qk − 1)n +qk ++ 1 +qk S∗ +k(f). +(3.8) +where f is the Laurent polynomial given by +f(x1, · · · , xn+2) = xn+1 +� +1 − xn+2 +� +x1 + · · · + xn + +b +x1 · · · xn +�� ++ xn+2 +and +S∗ +k(f) = +� +xi∈F∗ +qk +ψ +� +Trk +� +xn+2 + xn+1 +� +1 − xn+2 +� +x1 + · · · + xn + +b +x1 · · · xn +���� +. +The L-functions associated to Sk,n(b) and S∗ +k(f) are defined as +Ln(b, T) = exp +� ∞ +� +k=1 +Sk,n(b)T k +k +� +and +L∗(f, T) = exp +� ∞ +� +k=1 +S∗ +k(f)T k +k +� +. +It follows from formula (3.8) that +Ln(b, T) = exp +� ∞ +� +k=1 +− +� +qk − 1 +�n · T k +qk · k +� +L∗ (f, T/q) += +n +� +i=0 +exp +� +(−1)n−i+1 +�n +i +� ∞ +� +k=1 +� +qi−1T +�k +k +� +L∗ (f, T/q) += L∗ (f, T/q) +n +� +i=0 +� +1 +1 − qi−1T +�(−1)n−i+1(n +i) +. +(3.9) +The main purpose of this subsection is to determine the slopes and weights of Ln(b, T). Based +on formula (3.9), it suffices to consider L∗(f, T) instead. Let ∆ = ∆(f) denote the Newton +polyhedron corresponding to f. Some of the geometric properties about ∆ have been discussed in +subsection 3.2. In Proposition 3.1, we proved that f is non-degenerate if and only if p ∤ (n + 1). In +this case, the L-function L∗(f, T)(−1)n+1 is a polynomial of degree 2n+2. To determine the slopes +of the reciprocal roots of L∗(f, T)(−1)n+1, we shall compute the Hodge polygon and consider when +it coincides with the Newton polygon. +Proposition 3.2. The Laurent polynomial f is ordinary if and only if p ≡ 1 mod (n + 1). +Proof. By facial decomposition theorem, it suffices to consider f δi for i = 1, 2. Let S(δi) be the +solution set of the following linear system +M(δi) + + + + + +r1 +r2 +... +rn+2 + + + + + = u ∈ Zn+2, +where rj ∈ Q ∩ [0, 1). +(3.10) + +ON INVERTED KLOOSTERMAN SUMS OVER FINITE FIELDS +15 +For i = 1 and a given point u = (x1, . . . , xn+2)T , linear system (3.10) equals to + + + + + + + + + + + + + + + + + + + +x1 = r1 − rn+1, +x2 = r2 − rn+1, +· · · +xn = rn − rn+1, +xn+1 = r1 + · · · + rn+2, +xn+2 = r1 + · · · + rn+1, +where rj ∈ Q ∩ [0, 1). +(3.11) +Note that xj ∈ Z, where 1 ≤ j ≤ n + 2. For any r = (r1, . . . , rn+2)T ∈ S(δ1), we have +r1 = · · · = rn = rn+1 ∈ +Z +n + 1 +and +rn+2 = 0. +Let Sp(δi) denote the prime to p part of S(δi). In particular, Sp(δi) = S(δi) if p ∤ det(M(δi)). +Suppose p ∤ (n + 1), the norm function |r| and |{pr}| are given by +|r| = (n + 1)r1 +and +|{pr}| = (n + 1){pr1}. +Then |r| on Sp(δ1) is stable under the p-action if and only if p ≡ 1 mod (n + 1). To see this, it +suffices to consider the unique point r = ( +1 +n+1, · · · , +1 +n+1, 0) with norm 1. This condition holds for +Sp(δ2) through a similar proof. By Proposition 2.19, we obtain Proposition 3.2. +□ +Theorem 3.3. The n + 2 Hodge numbers of ∆ are {1, 2, 2, · · · , 2, 1}. Namely, +H∆(0) = 1, H∆(1) = · · · = H∆(n) = 2, H∆(n + 1) = 1. +Proof. Let ∆i be the polytope generated by the origin and δi. Let u = (x1, . . . , xn+2)T ∈ C(∆i) +be a lattice point with the weight w(u) = k, where 0 ≤ k ≤ n + 2. For i = 1, 2, consider the linear +system (3.10). Since f δi is diagonal, system (3.10) has a unique solution r = (r1, . . . , rn+2)T for a +fixed point u ∈ C(∆i). In this case, the weight is given by +w(u) = r1 + · · · + rn+2 = |r|. +When i = 1, the linear equations (3.11) has exact one solution u = (0, . . . , 0, k, k)T . Since +xn+2 = �n+1 +j=1 rj = k and 0 ≤ rj < 1, we get the restriction 0 ≤ k < n + 1. The Hodge number +H∆1(k) counts the number of lattice points u of weight k/D in a fundamental domain: That is, +H∆1(k) = +� +1, +for 0 ≤ k < n + 1, +0, +for k ≥ n + 1. +The generating function of H∆1(k) is +H1(x) = 1 + x + · · · + xn. +By formula (2.5), we get the generating function of W∆1(k) as follow. +W1(x) = +∞ +� +k=0 +W∆1(k)xk = +H1(x) +(1 − x)n+2 = 1 − xn+1 +(1 − x)n+3 . +Let H2(x) and W2(x) be the generating function of H∆2(k) and W∆2(k), respectively. Similarly, +we can prove H2(x) = H1(x) and W2(x) = W1(x). The polytope ∆1 +� ∆2 is determined by +V1, . . . , Vn+1, whose generating function is given by +W3(x) = +∞ +� +k=0 +W∆1 +� ∆2(k)xk = 1 − xn+1 +(1 − x)n+2 . +By facial decomposition, we have +W∆(k) = W∆1(k) + W∆2(k) − W∆1 +� ∆2(k), +(3.12) + +16 +XIN LIN AND DAQING WAN +which implies +W(x) = +∞ +� +k=0 +W∆(k)xk = W1(x) + W2(x) − W3(x) = 1 + 2x + · · · + 2xn + xn+1 +(1 − x)n+2 +. +This gives the Hodge numbers of ∆ via formula (2.5), that is, +H∆(0) = 1, H∆(1) = · · · = H∆(n) = 2, H∆(n + 1) = 1. +□ +When f is ordinary, the slopes of L∗(f, T)(−1)n+1 can be deduced from Theorem 3.3. +Theorem 3.4. If p ≡ 1 mod (n + 1), the slope sequence of L∗(f, T)(−1)n+1 is given by +{0, 1, 1, 2, 2, . . . , n, n, n + 1}. +Proof. This theorem follows from Lemma 2.5, Proposition 3.2 and Theorem 3.3. +□ +Note that the converse of this theorem is also true, as Proposition 3.2 shows that the condition +p ≡ 1 mod (n + 1) is a necessary and sufficient condition for f to be ordinary. +Now we are ready to consider the weights for the reciprocal roots of L∗(f, T)(−1)n+1. +Theorem 3.5. Suppose p ∤ (n + 1). We have +L∗(f, T)(−1)n+1 = (1 − T)(1 − qT) +2n +� +i=1 +(1 − βiT). +For each 1 ≤ i ≤ 2n, the reciprocal root βi satisfies |βi| = q +n+2 +2 . +Proof. Since the origin is a vertex of ∆, we decompose the cone C(∆) via boundary decomposition +B(∆). Let N(i) be the number of i-dimensional face Σi of C(∆), where 0 ≤ i ≤ dim∆. For +Newton polyhedron ∆ = ∆(f), we have N(0) = 1 and N(1) = n + 3. Note that Σi is an +open cone and Σi ∈ B(∆). Let Σi be the closure of Σi. For simplicity, we denote the Fredholm +determinants as +D(T) = det (I − TA1(f)) , +D◦ +i (T) = det +� +I − TA1(Σi, f Σi) +� +, +Di(T) = det +� +I − TA1(Σi, f Σi) +� +. +The unique 0-dimensional cone Σ0 is the origin and D0(T) = D◦ +0(T) = 1 − T. When i = 1, each +f Σ1 can be normalized to x by variable substitution. That is, +L∗(f Σ1, T) = exp +� ∞ +� +k=1 +−T k +k +� += 1 − T. +By formula (2.7), we have +D1(T) = +∞ +� +i=0 +� +L∗ � +f Σ1, qiT +��(i +i) += (1 − T) (1 − qT) +� +1 − q2T +� +· · · . +Since the only boundary of Σ1 are Σ1 and Σ0, we get D◦ +1(T) after eliminating D◦ +0(T), i.e., +D◦ +1(T) = D1(T) +D◦ +0(T) = (1 − qT) +� +1 − q2T +� ∞ +� +i=3 +� +1 − qiT +� +. +Theorem 2.16 shows that D(T) can be expressed as a product of D◦ +i (T) as follow. +D(T) = +n+2 +� +i=1 +N(i) +� +j=1 +D◦ +i (T) = (1 − T)(1 − qT)n+3 · · · , + +ON INVERTED KLOOSTERMAN SUMS OVER FINITE FIELDS +17 +Note that L∗(f, T)(−1)n+1 is a polynomial of degree 2(n + 1) if f is non-degenerate. Combining +formula (2.6), we obtain +L∗(f, T)(−1)n+1 = +D(T)D(q2T)(n+2 +2 ) · · · +D(qT)n+2D(q3T)(n+2 +3 ) · · · += (1 − T)(1 − qT) +2n +� +i=1 +(1 − βiT), +where |βi| = q +wi +2 ≤ q +n+2 +2 . That is, wi ≤ n + 2. +If βi is a reciprocal root of L∗(f, T)(−1)n+1, the conjugate βi is a reciprocal root of the conjugate +L-function +L∗(f, T) +(−1)n+1 += L∗(−f, T)(−1)n+1. +By Theorem 2.8, the Newton polygon and Hodge polygon coincide at the end points. Applying this +to the product L∗(f, T)(−1)n+1L∗(f, T) +(−1)n+1 +, we deduce that +2(n+1)2 = 2( +n +� +i=1 +2i+n+1) = ordq(1·q2 · +2n +� +i=1 +βiβi) = 2+ +2n +� +i=1 +wi ≤ 2+2n(n+2) = 2(n+1)2. +It follows that the inequality must be an equality, that is, all wi = n + 2. +□ +Formula (3.9) relates L∗(f, T) to Ln(b, T). The valuations for the reciprocal roots and poles of +Ln(b, T) follow from Theorem 3.4 and 3.5. +Theorem 3.6. Suppose p ∤ (n + 1). We have +Ln(b, T)(−1)n+1 = (1 − T)(n+1) +n +� +j=2 +� +1 − qj−1T +�(n +j)(−1)j−1 2n +� +i=1 +(1 − αiT). +For each 1 ≤ i ≤ 2n, the reciprocal root αi satisfies |αi| = q +n +2 . If p ≡ 1 mod (n + 1), the slope +sequence of the αi’s is given by {0, 1, 1, 2, 2, . . . , n − 1, n − 1, n}. +Based on weights of toric L-function, we get the following slightly more precise upper bound for +its associated exponential sum. +Corollary 3.7. If p ∤ (n + 1), we have +|Sk,n(b) + (qk − 1)n − (−1)n(qk + 1) +qk +| ≤ 2nq +nk +2 . +Proof. Theorem 3.5 implies that +|S∗ +k(f) − (−1)n(qk + 1)| ≤ 2nq +(n+2)k +2 +. +Combining formula (3.8), we get the bound for Sk,n(b). +□ +Remark. We finish this paper with two open problems on the estimates of inverted Kloosterman +sums. If n + 1 is divisible by p, the related Laurent polynomial f is degenerate and thus the results +for toric exponential sums are not tenable. In this case, it is an open problem to determine the +optimal square root cancellation for Sn(χ, b) in general. The case n = 1 with p = 2 is already +handled in [Kat95]. The second question concerns the q-adic slope sequence. If p is not equivalent +to 1 modulo n + 1, the Newton polygon corresponding to f is strictly above its Hodge polygon. +Under this assumption, can one still obtain the explicit q-adic slope sequence? +REFERENCES +[Ang96] Jeff Angel, Finite upper half planes over finite fields, Finite Fields Appl. 2 (1996), no. 1, 62–86. MR 1371720 +[AS87a] Alan Adolphson and Steven Sperber, Newton polyhedra and the degree of the L-function associated to an +exponential sum, Invent. Math. 88 (1987), no. 3, 555–569. MR 884800 +[AS87b] +, Twisted Kloosterman sums and p-adic Bessel functions. II. Newton polygons and analytic continua- +tion, Amer. J. Math. 109 (1987), no. 4, 723–764. MR 900037 +[AS89] +, Exponential Sums and Newton Polyhedra: Cohomology and Estimates, Ann. 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Math. 8 (2004), +no. 3, 427–472. MR 2129244 +[YZ22] +Liping Yang and Hao Zhang, Generic Newton polygons for L-functions of (A, B)-exponential sums, Finite +Fields Appl. 78 (2022), Paper No. 101980, 20. MR 4349885 +DEPARTMENT OF MATHEMATICS, SHANGHAI MARITIME UNIVERSITY, SHANGHAI 201306, PR CHINA. +Email address: xlin1126@hotmail.com +DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CALIFORNIA, IRVINE, CA 92697-3875 USA. +Email address: dwan@math.uci.edu + diff --git a/SNE3T4oBgHgl3EQfDglb/content/tmp_files/load_file.txt b/SNE3T4oBgHgl3EQfDglb/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d7bf0de3277db566a7bd5089baa5705d42171736 --- /dev/null +++ b/SNE3T4oBgHgl3EQfDglb/content/tmp_files/load_file.txt @@ -0,0 +1,838 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf,len=837 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='04287v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='NT] 11 Jan 2023 ON INVERTED KLOOSTERMAN SUMS OVER FINITE FIELDS XIN LIN AND DAQING WAN ABSTRACT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' The classical n-variable Kloosterman sums over finite fields are well understood by Deligne’s theorem from complex point of view and by Sperber’s theorem from p-adic point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' In this paper, we study the complex and p-adic estimates of inverted n-variable Kloosterman sums, addressing a question of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Katz (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' We shall give two complex estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' The first one is elementary based on Gauss sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' The second estimate is deeper, depending on the cohomological results of Adolphson-Sperber, Denef-Loeser and Fu for twisted toric exponential sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' This deeper result assumes that the characteristic p does not divide n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Combining with Dwork’s p-adic theory, we also determine the exact p-adic valuations for zeros and poles of the L-function associated to inverted n-variable Kloosterman sums in the case p ≡ 1 mod (n + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' As we shall see, the inverted n-variable Kloosterman sum is more complicated than the classical n-variable Kloosterman sum in all aspects in the sense that our understanding is less complete, partly because the Hodge numbers are now mostly 2 instead of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' INTRODUCTION Let Fq be the finite field of q elements with characteristic p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Let ψ : Fq → C∗ be a nontrivial additive character and let χ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' , χm : F∗ q → C∗ be multiplicative characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' A classical problem in number theory is to give a good estimate for the mixed character sum � xi∈F∗q χ1(f1) · · · χm(fm)ψ(f), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='1) where f, f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' , fm ∈ Fq � x±1 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' , x±1 n � are Laurent polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Reducing m if necessary, we may assume that all the χi’s are non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Using the Gauss sum G(χ) = � x∈F∗q χ(x)ψ(x), one obtains the well known relation � xi,yj∈F∗q χ1(y1) · · · χm(ym)ψ (f + y1f1 + · · · + ymfm) = G(χ1) · · · G(χm) � xi∈F∗q χ1(f1) · · · χm(fm)ψ(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' As the Gauss sums are well-understood, the study of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='1) is reduced to the study of the following type of twisted toric exponential sum � xi∈F∗q χ1(x1) · · · χn(xn)ψ (f(x1, · · · , xn)) , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='2) where some of the χi’s may be trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' This type of twisted toric exponential sum has been studied extensively in the literature, most notably by Adolphson-Sperber [AS87a, AS89, AS90, AS93] via Dwork’s p-adic cohomology, and by Denef-Loeser [DL91] and Fu [Fu09,Fu16] via Grothendieck’s ℓ-adic cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' For the complex estimate, both approaches depend on Deligne’s theorem on the Weil conjectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' A sharp estimate is obtained when f is non-degenerate with respect to its Newton polyhedron ∆(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' For arbitrary f, the sum is still far from well understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' 11T23, 11L05, 11L07, 11S40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Inverted Kloosterman sums, Exponential sums, L-function, Finite field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' 1 2 XIN LIN AND DAQING WAN An important example of toric exponential sums is the classical n-variable Kloosterman sum, where χ1 = · · · = χn = 1 and f(x1, · · · , xn) = x1 + · · · + xn + b x1 · · · xn , b ∈ F∗ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' In this case, the complex weights were determined by Deligne’s well known theorem, and the p- adic slopes were determined by Sperber’s theorem [Spe80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' It should be noted that for twisted n-variable Kloosterman sum (when some of the χi’s are non-trivial), the p-adic slopes are not completely determined in general, except in the case n = 1 for which Adolphson-Sperber [AS87b] obtained the generic p-adic slopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' If we invert the above Laurent polynomial and consider the following rational function f(x1, · · · , xn) = 1 x1 + · · · + xn + b x1···xn , b ∈ F∗ q, which is no longer a Laurent polynomial, we are led to the so-called inverted Kloosterman sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' The study of such sums goes back to N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Katz [Kat95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' More precisely, in this paper, we study the following twisted inverted n-variable Kloosterman sum defined by Sn(χ, b) = � x1···xn+1=b, xi∈F∗q x1+···+xn+1̸=0 χ1(x1) · · · χn+1(xn+1)ψ � 1 x1 + · · · + xn+1 � = � x1+···+xn+ b x1···xn ̸=0 xi∈F∗q χ1(x1) · · · χn(xn)χn+1 � b x1 · · · xn � ψ � 1 x1 + · · · + xn + b x1···xn � , where b ∈ F∗ q and n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' When n = 1, Katz [Kat95] obtained a sharp upper bound for S1(χ, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' This result along with the papers of Angel [Ang96] and Evans [Eva95] proves that finite upper half plane graphs are Ramanujan in characteristic 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' In [Kat95], Katz raised the question: what can be said for Sn(χ, b) when n ≥ 1?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' The aim of this paper is to study these inverted n-variable Kloosterman sums from both complex and p-adic point of views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' As we shall see, this class of sums is very interesting as various new features and additional difficulties arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' The exact sum introduced in [Kat95] is the following related sum, Tn(χ, b) = � x1···xn+1=1 x1+···+xn+1̸=0 χ1(x1) · · · χn+1(xn+1)ψ � b x1 + · · · + xn+1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Upon the change of variables xi → bxi, one sees that Tn(χ, b) = Sn(χ, 1 bn+1 )χ1 · · · χn+1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Thus, the two families of sums Sn(χ, b) and Tn(χ, b) are essentially equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' We work with Sn(χ, b) as it is the closer inverted analogue of the classical Kloosterman sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' For complex estimate of Sn(χ, b), the best one can hope for would be a square root cancellation in the sum Sn(χ, b), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Sn(χ, b) = On(q n 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' As we shall see, this is not true for n > 2 when χ1 = · · · = χn+1, in which case, Sn(χ, b) has the main term −qn−1χ1(b) whose exponent n − 1 is larger than the exponent n/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' We shall give two different estimates for Sn(χ, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' The first estimate is based on an elementary method via Gauss sums which already shows the new feature of a non-trivial main term when χ1 = · · · = χn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' We obtain the following simple estimate for Sn(χ, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' ON INVERTED KLOOSTERMAN SUMS OVER FINITE FIELDS 3 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Notations as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' If χ1 = · · · = χn+1, we have ����Sn(χ, b) + (q − 1)n q χ1(b) ���� ≤ q n+1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' If χi ̸= χj for some i ̸= j, we have |Sn(χ, b)| ≤ q n+1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' For large q, the error term q(n+1)/2 is not the optimal square root cancellation yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' To obtain the deeper square root cancellation with error term On(qn/2), we reduce Sn(χ, b) to a certain twisted toric exponential sum S∗ k(χ, f) which can be handled by the results of Adolphson-Sperber, Denef- Loeser and Fu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' We check that the related Laurent polynomial f is non-degenerate if p does not divide n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' This gives our second estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Notations as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Suppose p ∤ (n + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' If χ1 = · · · = χn+1, we have |Sn(χ, b) + (q − 1)n − (−1)n q χ1(b)| ≤ (2n + 1)q n 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' If χi ̸= χj for some i ̸= j, we have |Sn(χ, b)| ≤ 2(n + 1)q n 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' It is clear that the estimate in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='2 is better than the estimate in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='1 if q > 4(n + 1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' It is expected that Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='2 (with possibly a better constant) remains true when p divides n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' But in this singular case, the above toric sum results do not apply and one would need a different approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' For p-adic slopes, we focus on the simpler untwisted case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' When χ1 = · · · = χn+1, we study the p-adic valuations for the reciprocal roots and poles of the generating L-function of the inverted n-variable Kloosterman sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' We show that the Newton polygon agrees with the Hodge polygon if and only if p ≡ 1 mod n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' As a consequence, this completely determines the p-adic slope sequence when p ≡ 1 mod n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' This is described more precisely below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Our approach is to reduce the generating L-function to a certain untwisted toric L-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' To construct the relationship between the two L-functions, we need to consider the inverted Klooster- man sum defined over every finite extension Fqk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Suppose χ1 = · · · = χn+1, the inverted n-variable Kloosterman sum over Fqk is defined by Sk,n(b) = � x1+···+xn+ b x1···xn ̸=0 xi∈F∗ qk ψ � Trk � 1 x1 + · · · + xn + b x1···xn �� , where b ∈ F∗ q and Trk : Fqk → Fq is the trace map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' The generating L-function of Sk,n(b) is defined by Ln(b, T) = exp � ∞ � k=1 Sk,n(b)T k k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Applying some systematic results available for the related toric L-function, we obtain the complex and p-adic absolute values for all the reciprocal roots and poles of Ln(b, T) under given restrictions on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Suppose p ∤ (n + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' The L-function is a rational function of the following form: Ln(b, T)(−1)n+1 = (1 − T)(n+1) n � j=2 � 1 − qj−1T �(n j)(−1)j−1 2n � i=1 (1 − αiT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' As complex numbers, the reciprocal roots αi satisfy |αi| = q n 2 for all 1 ≤ i ≤ 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' 4 XIN LIN AND DAQING WAN If p ≡ 1 mod (n + 1), viewing the αi’s as p-adic numbers, the slope sequence {vq(αi)}2n i=1 in increasing order is given by {0, 1, 1, 2, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' , n − 1, n − 1, n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Our results show that for the inverted n-variable Kloosterman sum, if p does not divide n + 1, the primitive middle cohomology has dimension 2n, pure of weight n and with Hodge numbers {1, 2, 2, · · · , 2, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' This is in contrast to the classical n-variable Kloosterman sum, where the middle cohomology has dimension n + 1, pure of weight n and with Hodge numbers {1, 1, 1, · · · , 1, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' The larger Hodge numbers suggest that new features and additional difficulties would likely arise in studying inverted n-variable Kloosterman sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' As a corollary of the first part in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='3, we get a slightly better bound for Sk,n(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' If p ∤ (n + 1), for all integers k ≥ 1, we have |Sk,n(b) + (qk − 1)n − (−1)n(qk + 1) qk | ≤ 2nq nk 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' When k = 1, the exponential sum Sk,n(b) reduces to Sn(χ, b) with χ1 = · · · = χn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Explicitly, the estimate in Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='4 is better than the estimate in the first case of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' We remark that the condition p ≡ 1 mod (n + 1) in the second part of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='3 is necessary and sufficient for the same conclusion to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Thus, if p ̸≡ 1 mod (n + 1), the slope sequence will be strictly different, but we do not know the exact slope sequence in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' In section 2, we review some technical methods including Adolphson-Sperber’s theorems and Dwork’s theory on toric exponentials sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' In section 3, we use these methods to prove the main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' We end the introduction by mentioning several recent references [FW21] [Li21] [YZ22] [CL22] [LC22] which applied some of the related toric techniques in treating different classes of non-toric n variable exponential sums arising from analytic number theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' PRELIMINARIES ON TORIC EXPONENTIAL SUMS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Rationality of the toric L-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Let f ∈ Fq � x±1 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' , x±1 n � be a Laurent polynomial and its associated twisted toric exponential sum is defined to be S∗ k(χ, f) = � xi∈F∗ qk χ1(Nk(x1)) · · · χn(Nk(xn))ψ(Trk(f)), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='1) where Trk : Fqk → Fq is the trace map, Nk : Fqk → Fq is the norm map, χ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' , χn : F∗ q → C∗ are multiplicative characters and ψ : Fq → C∗ is a nontrivial additive character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' A classical problem in number theory is to estimate the absolute values of S∗ k(χ, f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' A well known theorem of Dwork-Bombieri-Grothendieck [Dwo60,Bom66,Gro68] says that the generating L-function of S∗ k(χ, f) is a rational function: L∗(χ, f, T) = exp � ∞ � k=1 S∗ k(χ, f)T k k � = �d1 i=1(1 − αiT) �d2 j=1(1 − βjT) , where all the reciprocal zeros and poles are non-zero algebraic integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' In particular, when all of χi are trivial characters, the untwisted toric exponential sum and the associated L-function are denoted by S∗ k(f) and L∗(f, T) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Through logarithmic derivatives, we have S∗ k(χ, f) = d2 � j=1 βk j − d1 � i=1 αk i , k ∈ Z≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='2) Thus, the estimate of S∗ k(χ, f) is reduced to understanding all the absolute values of the reciprocal zeros αi and poles βj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Deligne’s theorem on Riemann hypothesis [Del80] describes the bounds for ON INVERTED KLOOSTERMAN SUMS OVER FINITE FIELDS 5 all the absolute values of αi and βj in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' The complex absolute values of reciprocal zeros and poles satisfy |αi| = qui/2, |βj| = qvj/2, ui ∈ Z ∩ [0, 2n], vj ∈ Z ∩ [0, 2n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' For non-archimedean absolute values, Deligne proved that |αi|ℓ = |βj|ℓ = 1 when ℓ is a prime and ℓ ̸= p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' For p-adic absolute values, one has |αi|p = q−ri, |βj|p = q−sj, ri ∈ Q ∩ [0, n], sj ∈ Q ∩ [0, n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' The integer ui (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' vj) is called the weight of αi (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' βj) and the rational number ri (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' sj) is called the slope of αi (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' βj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' In the past few decades, there has been tremendous interest in determining the weights and slopes of the generating L-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Without any further condition on the Laurent polynomial f, it is even hard to determine the number of reciprocal roots and poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Most of the existing work about the weights and slopes relies on a suitable smoothness condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' For toric exponential sums, this usually means the non-degenerate condition, see below for the precise definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Let (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='3) f(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' xn) = J � j=1 ajxVj be a Laurent polynomial with aj ∈ F∗ q and Vj = (v1j, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' , vnj) ∈ Zn (1 ≤ j ≤ J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' The Newton polyhedron of f, ∆(f), is defined to be the convex closure in Rn generated by the origin and the lattice points Vj (1 ≤ j ≤ J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' For δ ⊂ ∆(f), let the Laurent polynomial f δ = � Vj∈δ ajxVj be the restriction of f to δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='1 (non-degenerate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' A Laurent polynomial f is called non-degenerate if for each closed face δ of ∆(f) of arbitrary dimension which doesn’t contain the origin, the partial derivatives �∂f δ ∂x1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' , ∂f δ ∂xn � have no common zeros with x1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' xn ̸= 0 over the algebraic closure of Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' When f is non-degenerate, Adolphson-Sperber [AS89] proved that the untwisted toric L-function L∗(f, T)(−1)n−1 is a polynomial and improved the bound for weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='2 ( [AS89]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' For any non-degenerate f ∈ Fq[x±1 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' , x±1 n ], the associated L-function L∗(f, T)(−1)n−1 is a polynomial of degree n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Vol(∆(f)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Namely, L∗(f, T)(−1)n−1 = n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Vol(∆(f)) � i=1 (1 − αiT), αi ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' For any multiplicative characters χi (nontrivial or trivial), the degree and weights of the twisted toric L-function are studied and completed by Adolphson-Sperber [AS91, AS93], [DL91]and Fu [Fu09] under the non-degeneracy assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' These results lead to the following bound for the twisted toric exponential sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='3 ( [DL91] [AS93] [Fu09]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Let f ∈ Fq � x±1 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' , x±1 n � be a Laurent polynomial with ∆ = ∆(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' If f is non-degenerate, one has |S∗ k(χ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' , χn, f)| ≤ n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Vol(∆)q nk 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' For the slopes of the L-function, the situation is somewhat simpler in the untwisted case, oth- erwise, even the description of Adolphson-Sperber’s “Hodge lower bound" is a little cumbersome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Thus, the definitions and theories discussed in the following subsections focus on the untwisted L-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' 6 XIN LIN AND DAQING WAN 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Newton polygon and Hodge polygon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' To determine the q-adic slopes of its reciprocal roots, we introduce the q-adic Newton polygon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='4 (Newton polygon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Let L(T) = �n i=0 aiT i ∈ 1+TQp[T], where Qp is the algebraic closure of Qp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' The q-adic Newton polygon of L(T) is defined to be the lower convex closure of the set of points {(k, ordq(ak)) |k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' , n} in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='5 ( [Kob84]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Notations as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Let L(T) = (1−α1T) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' (1−αnT) be the factoriza- tion of L(T) in terms of reciprocal roots αi ∈ Qp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Let λi = ordq αi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' If λ is the slope of the q-adic Newton polygon of L(T) with horizontal length l, then precisely l of the λi are equal to λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' The q-adic Newton polygon of L∗(f, T)(−1)n−1 is denoted as NP(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='5 relates NP(f) to the q-adic valuation of reciprocal roots of toric L-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' The definition of NP(f) relies on the coefficients of L-function, which makes it hard to compute directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' When f is non-degenerate, Adolphson and Sperber proved that L∗(f, T)(−1)n−1 is a polynomial and NP(f) has a topological lower bound called Hodge polygon, which is easier to determine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Thus, we shall compute Hodge polygon and consider when the Newton polygon coincides with this lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Let ∆ be an n-dimensional integral polytope containing the origin in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' For u ∈ Rn, the weight function w(u) represents the smallest non-negative real number c such that u ∈ c∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Denote w(u) = ∞ if such c doesn’t exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Assume δ is a co-dimension 1 face of ∆ not containing the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Let D(δ) be the least common multiple of the denominators of the coefficients in the linear equation defining δ, normalized to have constant term 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' We define the denominator of ∆ to be the least common multiple of all such D(δ) given by: D = D(∆) = lcmδD(δ), where δ runs over all the co-dimension 1 faces of ∆ that don’t contain the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' It’s easy to check w(Zn) ⊆ 1 D(∆)Z≥0 ∪ {+∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' For a non-negative integer k, let (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='4) W∆(k) = # � u ∈ Zn|w(u) = k D � be the number of lattice points in Zn with weight k/D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Its generating function is known to be a rational function of the following form ∞ � k=0 W∆(k)tk/D = �nD k=0 H∆(k)tk/D (1 − t)n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' This leads to Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='6 (Hodge number).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Let ∆ be an n-dimensional integral polytope containing the origin in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' For a non-negative integer k, the k-th Hodge number of ∆ is defined to be H∆(k) = n � i=0 (−1)i �n i � W∆(k − iD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='5) It is known that H∆(k) = 0, if k > nD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Based on the Hodge numbers, we define the Hodge polygon of a given polyhedron ∆ ∈ Rn as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='7 (Hodge polygon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' The Hodge polygon HP(∆) of ∆ is the lower convex polygon in R2 with vertices (0,0) and Qk = � k � m=0 H∆(m), 1 D k � m=0 mH∆(m) � , k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' , nD, ON INVERTED KLOOSTERMAN SUMS OVER FINITE FIELDS 7 where H∆(k) is the k-th Hodge number of ∆, k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' , nD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' That is, HP(∆) is a polygon starting from origin (0,0) with a slope k/D side of horizontal length H∆(k) for k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' , nD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' The vertex Qk is called a break point if H∆(k + 1) ̸= 0 where k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' , nD − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Note that the horizontal length H∆(k) is the number of lattice points of weight k/D in a certain fundamental domain corresponding to a basis of the p-adic cohomology space used to compute the L-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' By a theorem of Adolphson-Sperber, the Hodge polygon is a lower bound of the corresponding Newton polygon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='8 ( [AS89]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' For every prime p and non-degenerate Laurent polynomial f with ∆(f) = ∆ ⊂ Rn, we have NP(f) ≥ HP(∆), where NP(f) is the q-adic Newton polygon of L∗(f, T)(−1)n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Furthermore, the endpoints of NP(f) and NP(∆) coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='9 (ordinary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' A Laurent polynomial f is called ordinary if NP(f) = HP(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' It is clear that the ordinary property of a Laurent polynomial depends on its Newton polyhedron ∆ and on the coefficients of f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Applying the facial decomposition theorem [Wan93], we reduce the ordinary property of f to its smaller pieces which are easier to deal with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='10 (Facial decomposition theorem [Wan93]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Let f be a non-degenerate Laurent poly- nomial over Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Assume ∆ = ∆(f) is n-dimensional and δ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' , δh are all the co-dimension 1 faces of ∆ which don’t contain the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Let f δi denote the restriction of f to δi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Then f is ordinary if and only if f δi is ordinary for 1 ≤ i ≤ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Boundary decomposition theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Before describing the boundary decomposition, we ex- press the L-function in terms of the Fredholm determinant of an infinite Frobenius matrix via Dwork’s trace formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Dwork’s trace formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Let Qp be the field of p-adic numbers and Ω be the completion of Qp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' A fixed primitive p-th root of unity in Ω is denoted as ζp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Let π be the element of Qp(ζp) satisfies ∞ � m=0 πpm pm = 0, π ≡ ζp − 1 mod (ζp − 1)2, and ordp π = 1 p − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Then, π is a uniformizer of Qp(π) and thus Qp(π) = Qp(ζp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Let Ep(t) be the Artin-Hasse exponential series, Ep(t) = exp � ∞ � m=0 tpm pm � = ∞ � m=0 λmtm ∈ Zp[[x]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' In Dwork’s terminology, a splitting function θ(t) is defined to be θ(t) = Ep(πt) = ∞ � m=0 λmπmtm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' A Laurent polynomial f ∈ Fq[x±1 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' , x±1 n ] is written as f = J � j=1 ¯ajxVj, where Vj ∈ Zn and ¯aj ∈ F∗ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Let aj be the Teichmüller lifting of ¯aj in Ω satisfying aq j = aj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Let F(f, x) = J � j=1 θ(ajxVj) = � r∈Zn Fr(f)xr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' 8 XIN LIN AND DAQING WAN The coefficients are given by Fr(f) = � u ( J � j=1 λujauj j )πu1+···+uJ, r ∈ Zn, where the sum is over all the solutions of the following linear system J � j=1 ujVj = r with uj ∈ Z≥0, and λm is m-th coefficient of the Artin-Hasse exponential series Ep(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Assume ∆ = ∆(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Let L(∆) = Zn ∩ C(∆) be the set of lattice points in the closed cone generated by origin and ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' For a given point r ∈ Rn, define the weight function to be w(r) := inf ⃗u \uf8f1 \uf8f2 \uf8f3 J � j=1 uj| J � j=1 ujVj = r, uj ∈ R≥0 \uf8fc \uf8fd \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' The infinite semilinear Frobenius matrix A1(f) is the following matrix whose rows and columns are indexed by the lattice points in L(∆) with respect to the weights: A1(f) = (ar,s(f)) = (Fps−r(f)πw(r)−w(s)), where r, s ∈ L(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' The infinite linear Frobenius matrix Aa(f) is defined to be Aa(f) = A1(f)Aτ 1(f) · · · Aτ a−1 1 (f), where τ is the absolute Frobenius automorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Dwork’s trace formula can be expressed in terms of the matrix Aa(f) as follows, see [Wan04] Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' We have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='6) L∗(f, T)(−1)n−1 = n � i=0 det(I − TqiAa(f))(−1)i(n i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Equivalently, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='7) det(I − TAa(f)) = ∞ � i=0 � L∗(f, qiT)(−1)n−1�(n+i−1 i ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Now it suffices to understand the determinant det(I−TAa(f)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Based on the fact that ordp Fr(f) ≥ w(r) p−1 , we have the following estimate ordp(ar,s(f)) ≥ w(ps − r) + w(r) − w(s) p − 1 ≥ w(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Let ξ be an element in Ω satisfying ξD = πp−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Then A1(f) can be written in a block form, A1(f) = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed A00 ξA01 · · ξiA0i · · A10 ξA11 · · ξiA1i · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Ai0 ξAi1 · · ξiAii · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 , where the block Aii is a p-adic integral W∆(i)×W∆(i) matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' This implies that the q-adic Newton polygon of det(I −TA1(f)) has a natural lower bound which can be identified with the chain level version of the Hodge polygon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' ON INVERTED KLOOSTERMAN SUMS OVER FINITE FIELDS 9 Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Let P(∆) be the polygon in R2 with vertices (0, 0) and Pk = � k � m=0 W∆(m), 1 D k � m=0 mW∆(m) � , k = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' The chain level version of Adolphson-Sperber’s lower bound and the ordinary property are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='13 ( [AS87a]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' The q-adic Newton polygon of det(I − TAa(f)) lies above P(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='14 ( [Wan04]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Notations as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Assume f is non-degenerate with ∆ = ∆(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Then NP(f) = HP(∆) if and only if the q-adic Newton polygon of det(I − TAa(f)) coincides with its lower bound P(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Boundary decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Let f ∈ Fq[x±1 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' , x±1 n ] with ∆ = ∆(f), where ∆ is an n- dimensional integral convex polyhedron in Rn containing the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Let C(∆) be the cone gener- ated by ∆ in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' The boundary decomposition B(∆) = { the interior of a closed face in C(∆) containing the origin} is the unique interior decomposition of C(∆) into a disjoint union of relatively open cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' If the origin is a vertex of ∆, then it is the unique 0-dimensional open cone in B(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Recall that A1(f) = (ar,s(f)) is the infinite semilinear Frobenius matrix whose rows and columns are indexed by the lattice points in L(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' For Σ ∈ B(∆), we define A1(Σ, f) to be the submatrix of A1(f) with r, s ∈ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Let f Σ be the restriction of f to the closure of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Then A1(Σ, f Σ) denotes the submatrix of A1(f Σ) with r, s ∈ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Let B(∆) = {Σ0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' , Σh} such that dim(Σi) ≤ dim(Σi+1), i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' , h − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Define Bij = (ar,s(f)) with r ∈ Σi and s ∈ Σj (0 ≤ i, j ≤ h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' After a permutation of basis vectors, the infinite semilinear Frobenius matrix can be written as (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='8) A1(f) = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed B00 B01 · · B0h B10 B11 · · B1h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Bh0 Bh1 · · Bhh \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 , where Bij = 0 for i > j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Then det(I −TA1(f)) = �h i=0 det(I −TBii) and we have the boundary decomposition theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='16 (Boundary decomposition [Wan93]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Let f ∈ Fq[x±1 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' , x±1 n ] with ∆ = ∆(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Then we have the following factorization det(I − TA1(f)) = � Σ∈B(∆) det � I − TA1(Σ, f Σ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Diagonal local theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' In this subsection, we introduce some non-degenerate and ordinary criteria when the Laurent polynomial is diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' A Laurent polynomial f ∈ Fq[x±1 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' , x±1 n ] is called diagonal if f has exactly n non-constant terms and ∆(f) is an n-dimensional simplex in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Let f be a diagonal Laurent polynomial over Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Write f(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' xn) = n � j=1 ajxVj, where aj ∈ F∗ q and Vj = (v1j, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' , vnj) ∈ Zn for 1 ≤ j ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Let ∆ = ∆(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' The vertex matrix of ∆ is defined to be M(∆) = (V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' , Vn), 10 XIN LIN AND DAQING WAN where the i-th column is the i-th exponent of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Since f is diagonal, M(∆) is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Suppose f ∈ Fq[x±1 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' , x±1 n ] is diagonal with ∆ = ∆(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Then f is non- degenerate if and only if p is relatively prime to det(M(∆)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Let S(∆) be the solution set of the following linear system M(∆) \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed r1 r2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' rn \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 ≡ 0 (mod1), ri ∈ Q ∩ [0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' It’s easy to prove that S(∆) is an abelian group and its order is given by |det M(∆)| = n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Vol(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='9) Let Sp(∆) denote the prime to p part of S(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' It is an abelian subgroup of order equal to the prime to p factor of det M(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' In particular, Sp(∆) = S(∆) if p is relatively prime to det M(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' By the Stickelberger theorem for Gauss sums, we have the following ordinary criterion for a non- degenerate Laurent polynomial [Wan04].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' A diagonal Laurent polynomial f is ordinary at p if and only if the norm function |r| = r1 + · · · + rn on Sp(∆) is stable under the p-action: That is, for each r ∈ Sp(∆), we have |r| = |{pr}|, where {pr} is the class of pr in Sp(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' PROOF OF THE MAIN THEOREMS We prove the main theorems in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Recall that for integer n ≥ 1, the twisted inverted n-variable Kloost- erman sum is defined to be Sn(χ, b) = � x1···xn+1=b x1+···+xn+1̸=0 χ1(x1) · · · χn+1(xn+1)ψ � 1 x1 + · · · + xn+1 � , where b ∈ F∗ q, ψ : Fq → C∗ is a nontrivial additive character and χ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' , χn+1 : F∗ q → C∗ are multiplicative characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Let χ : F∗ q → C∗ denote a multiplicative character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' By the orthogonality of characters, we have Sn(χ, b) = 1 q(q − 1) � λ,xi∈F∗q � u∈Fq ψ (u (x1 + · · · + xn+1 − λ)) χ1(x1) · · · χn+1(xn+1) × ψ �1 λ � � χ χ �x1 · · · χn+1 b � = 1 q(q − 1) � λ∈F∗q � xi∈F∗q χ1(x1) · · · χn+1(xn+1)ψ � 1 λ � � χ χ �x1 · · · χn+1 b � + 1 q(q − 1) � λ∈F∗q � xi∈F∗q � u∈F∗q ψ (u (x1 + · · · + xn+1 − λ)) × χ1(x1) · · · χn+1(xn+1)ψ � 1 λ � � χ χ �x1 · · · χn+1 b � =S1 + S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='1) Then S1 = 1 q(q − 1) � λ∈F∗q ψ �1 λ � � χ χ−1(b) � xi∈F∗q (χχ1) (x1) · · · (χχn+1) (xn+1) ON INVERTED KLOOSTERMAN SUMS OVER FINITE FIELDS 11 = 1 q(q − 1) � λ∈F∗q ψ �1 λ � � χ χ−1(b) n+1 � i=1 \uf8eb \uf8ed � xi∈F∗q (χχi) (xi) \uf8f6 \uf8f8 = \uf8f1 \uf8f2 \uf8f3 −(q − 1)n q χ1(b), if χ1 = · · · = χn+1, 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='2) If χ is trivial, the Gauss sum G(χ) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' If χ is non-trivial, |G(χ)| = √q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Then S2 = 1 q(q − 1) � λ,u∈F∗q � χ χ−1(b) � xi∈F∗q (χχ1) (x1)ψ(ux1) · · · (χχn+1) (xn+1)ψ(uxn+1) × ψ(−uλ)ψ � 1 λ � = 1 q(q − 1) � λ,u∈F∗q � χ χ−1(b)χn+1χ1 · · · χn+1(u)ψ(−uλ)ψ �1 λ � G(χχ1) · · · G(χχn+1) = 1 q(q − 1) � χ χ−1(b) \uf8eb \uf8ed � λ∈F∗q χn+1χ1 · · · χn+1 � − 1 λ � ψ � 1 λ �\uf8f6 \uf8f8 G(χn+1χ1 · · · χn+1) × G(χχ1) · · · G(χχn+1) = 1 q(q − 1) � χ χ−1(b)χn+1χ1 · · · χn+1(−1)G(χn+1χ1 · · · χn+1)G(χn+1χ1 · · · χn+1) × G(χχ1) · · · G(χχn+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='3) Since |G(χ)| ≤ √q, it follows that |S2| ≤ q n+1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='1) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='2), we can deduce the following bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' ����Sn(χ, b) + (q − 1)n q χ1(b) ���� ≤ q n+1 2 , if χ1 = · · · = χn+1, and |Sn(χ, b)| ≤ q n+1 2 , if χi ̸= χj for some i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' This proves Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' The twisted inverted Kloosterman sum Sn(χ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' b) has the expression Sn(χ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' b) = � x1+···+xn+ b x1···xn ̸=0 xi∈F∗q χ1(x1) · · · χn(xn)χn+1 � b x1 · · · xn � × ψ � 1 x1 + · · · + xn + b x1···xn � = � z � x1+···+xn+ b x1···xn � =1 z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' xi∈F∗q χn+1(b) (χ1χn+1)(x1) · · · (χnχn+1)(xn)ψ (z) = 1 q � z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' xi∈F∗q y∈Fq χn+1(b) (χ1χn+1)(x1) · · · (χnχn+1)(xn) × ψ � z + y � 1 − z � x1 + · · · + xn + b x1 · · · xn ��� 12 XIN LIN AND DAQING WAN = χn+1(b) q \uf8eb \uf8ed � z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' xi∈F∗q (χ1χn+1)(x1) · · · (χnχn+1)(xn)ψ (z) + En(χ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' b) \uf8f6 \uf8f8 = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 −(q − 1)n q χ1(b) + 1 qχ1(b)En(χ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' b),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' if χ1 = · · · = χn+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' 1 qχn+1(b)En(χ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' b),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' if χi ̸= χj for some i ̸= j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='4) where En(χ, b) = � y,z,xi∈F∗q (χ1χn+1)(x1) · · · (χnχn+1)(xn) × ψ � z + y � 1 − z � x1 + · · · + xn + b x1 · · · xn ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' In order to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='2, it suffices to estimate En(χ, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Let f ∈ Fq[x±1 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' , x±1 n+2] be the Laurent polynomial defined by f(x1, · · · , xn+2) = xn+1 � 1 − xn+2 � x1 + · · · + xn + b x1 · · · xn �� + xn+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' As defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='1), En(χ, b) is the twisted toric exponential sum associated to f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Let ∆ = ∆(f) denote the Newton polyhedron corresponding to f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Clearly, dim ∆ = n + 2 and ∆ has n + 4 vertices in Rn+2: V0 = (0, · · · , 0)(the origin), V1 = (1, 0, · · · , 0, 1, 1), V2 = (0, 1, · · · , 0, 1, 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' , Vn = (0, 0, · · · , 1, 1, 1), Vn+1 = (−1, · · · , −1, 1, 1), Vn+2 = (0, · · · , 0, 1, 0) and Vn+3 = (0, · · · , 0, 0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Furthermore, ∆ has exactly 2 co-dimension 1 faces not containing the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Explicitly, they are δ1 : xn+1 = 1 and δ2 : xn+2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Vertices V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' , Vn+2 determine the face δ1 and vertices V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' , Vn+1, Vn+3 determine the face δ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Let M(δi) be the vertex matrix of δi, we have M(δ1) = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 1 0 · · 0 −1 0 0 1 · · 0 −1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' 0 0 · · 1 −1 0 1 1 · · 1 1 1 1 1 · · 1 1 0 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 , M(δ2) = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 1 0 · · 0 −1 0 0 1 · · 0 −1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' 0 0 · · 1 −1 0 1 1 · · 1 1 0 1 1 · · 1 1 1 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='5) Explicitly, each f δi is diagonal for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' The restriction of f to δi is defined by f δi = � Vj∈δi ajxVj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' The denominator D = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' f is non-degenerate if and only if p ∤ (n + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Vol(∆) = 2n + 2 (n + 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='. Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' The denominator D = 1 can be deduced immediately from the equation of δi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Since δ1 and δ2 are the co-dimension 1 faces of ∆(f) not containing the origin, it suffices to prove f δ1 and f δ2 are non-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='18, f δi is non-degenerate if and only if p is relatively prime to det(M(δi)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' By formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='5), det(M(δ1)) = −(n + 1) and det(M(δ2)) = n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='6) This proves (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' ON INVERTED KLOOSTERMAN SUMS OVER FINITE FIELDS 13 V0 V1 V2 V3 V4 FIGURE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' ∆ for n = 1 Let ∆i be the polytope generated by δi and the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' The facial decomposition of ∆ implies that Vol(∆) = Vol(∆1) + Vol(∆2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' By formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='9) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='6), we obtain (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' □ Combining Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='3 with Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='1, if p ∤ (n + 1), we have |En(χ, b)| ≤ (n + 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Vol(∆) · q n+2 2 = 2(n + 1)q n+2 2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='7) where p is the characteristic of Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Putting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='4) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='7) together, we then obtain the following bounds when p ∤ (n + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' |Sn(χ, b) + (q − 1)n q χ1(b)| ≤ 2(n + 1)q n 2 , if χ1 = · · · = χn+1, and |Sn(χ, b)| ≤ 2(n + 1)q n 2 , if χi ̸= χj for some i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' In the case χ1 = · · · = χn+1, the twisted sum En(χ, b) becomes the following untwisted toric exponential sum En(χ, b) = � y,z,xi∈F∗q ψ � z + y � 1 − z � x1 + · · · + xn + b x1 · · · xn ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Since the origin is a vertex of ∆ and the polynomial inside the additive character has no constant term, 1 is a trivial eigenvalue of the middle dimensional cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Removing this trivial eigen- value from the error term, one gets |En(χ, b) − (−1)n+2| ≤ (2n + 1)q n 2 , if χ1 = · · · = χn+1, and hence the slightly sharper estimate |Sn(χ, b) + (q − 1)n + (−1)n+1 q χ1(b)| ≤ (2n + 1)q n 2 , if χ1 = · · · = χn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' This proves Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Similar to formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='4), we relate the untwisted inverted Kloosterman sum Sk,n(b) to toric exponential sum S∗ k(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Sk,n(b) = � x1+···+xn+ b x1···xn ̸=0 xi∈F∗ qk ψ � Trk � 1 x1 + · · · + xn + b x1···xn �� 14 XIN LIN AND DAQING WAN = � z � x1+···+xn+ b x1···xn � =1 z, xi∈F∗ qk ψ (Trk (z)) = 1 qk � z, xi∈F∗ qk y∈Fqk ψ � Trk � z + y � 1 − z � x1 + · · · + xn + b x1 · · · xn ���� = −(qk − 1)n qk + 1 qk S∗ k(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='8) where f is the Laurent polynomial given by f(x1, · · · , xn+2) = xn+1 � 1 − xn+2 � x1 + · · · + xn + b x1 · · · xn �� + xn+2 and S∗ k(f) = � xi∈F∗ qk ψ � Trk � xn+2 + xn+1 � 1 − xn+2 � x1 + · · · + xn + b x1 · · · xn ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' The L-functions associated to Sk,n(b) and S∗ k(f) are defined as Ln(b, T) = exp � ∞ � k=1 Sk,n(b)T k k � and L∗(f, T) = exp � ∞ � k=1 S∗ k(f)T k k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' It follows from formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='8) that Ln(b, T) = exp � ∞ � k=1 − � qk − 1 �n · T k qk · k � L∗ (f, T/q) = n � i=0 exp � (−1)n−i+1 �n i � ∞ � k=1 � qi−1T �k k � L∗ (f, T/q) = L∗ (f, T/q) n � i=0 � 1 1 − qi−1T �(−1)n−i+1(n i) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='9) The main purpose of this subsection is to determine the slopes and weights of Ln(b, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Based on formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='9), it suffices to consider L∗(f, T) instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Let ∆ = ∆(f) denote the Newton polyhedron corresponding to f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Some of the geometric properties about ∆ have been discussed in subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' In Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='1, we proved that f is non-degenerate if and only if p ∤ (n + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' In this case, the L-function L∗(f, T)(−1)n+1 is a polynomial of degree 2n+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' To determine the slopes of the reciprocal roots of L∗(f, T)(−1)n+1, we shall compute the Hodge polygon and consider when it coincides with the Newton polygon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' The Laurent polynomial f is ordinary if and only if p ≡ 1 mod (n + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' By facial decomposition theorem, it suffices to consider f δi for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Let S(δi) be the solution set of the following linear system M(δi) \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed r1 r2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' rn+2 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 = u ∈ Zn+2, where rj ∈ Q ∩ [0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='10) ON INVERTED KLOOSTERMAN SUMS OVER FINITE FIELDS 15 For i = 1 and a given point u = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' , xn+2)T , linear system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='10) equals to \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 x1 = r1 − rn+1, x2 = r2 − rn+1, · · xn = rn − rn+1, xn+1 = r1 + · · · + rn+2, xn+2 = r1 + · · · + rn+1, where rj ∈ Q ∩ [0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='11) Note that xj ∈ Z, where 1 ≤ j ≤ n + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' For any r = (r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' , rn+2)T ∈ S(δ1), we have r1 = · · · = rn = rn+1 ∈ Z n + 1 and rn+2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Let Sp(δi) denote the prime to p part of S(δi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' In particular, Sp(δi) = S(δi) if p ∤ det(M(δi)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Suppose p ∤ (n + 1), the norm function |r| and |{pr}| are given by |r| = (n + 1)r1 and |{pr}| = (n + 1){pr1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Then |r| on Sp(δ1) is stable under the p-action if and only if p ≡ 1 mod (n + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' To see this, it suffices to consider the unique point r = ( 1 n+1, · · · , 1 n+1, 0) with norm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' This condition holds for Sp(δ2) through a similar proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='19, we obtain Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' □ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' The n + 2 Hodge numbers of ∆ are {1, 2, 2, · · · , 2, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Namely, H∆(0) = 1, H∆(1) = · · · = H∆(n) = 2, H∆(n + 1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Let ∆i be the polytope generated by the origin and δi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Let u = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' , xn+2)T ∈ C(∆i) be a lattice point with the weight w(u) = k, where 0 ≤ k ≤ n + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' For i = 1, 2, consider the linear system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Since f δi is diagonal, system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='10) has a unique solution r = (r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' , rn+2)T for a fixed point u ∈ C(∆i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' In this case, the weight is given by w(u) = r1 + · · · + rn+2 = |r|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' When i = 1, the linear equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='11) has exact one solution u = (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' , 0, k, k)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Since xn+2 = �n+1 j=1 rj = k and 0 ≤ rj < 1, we get the restriction 0 ≤ k < n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' The Hodge number H∆1(k) counts the number of lattice points u of weight k/D in a fundamental domain: That is, H∆1(k) = � 1, for 0 ≤ k < n + 1, 0, for k ≥ n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' The generating function of H∆1(k) is H1(x) = 1 + x + · · · + xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' By formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='5), we get the generating function of W∆1(k) as follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' W1(x) = ∞ � k=0 W∆1(k)xk = H1(x) (1 − x)n+2 = 1 − xn+1 (1 − x)n+3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Let H2(x) and W2(x) be the generating function of H∆2(k) and W∆2(k), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Similarly, we can prove H2(x) = H1(x) and W2(x) = W1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' The polytope ∆1 � ∆2 is determined by V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' , Vn+1, whose generating function is given by W3(x) = ∞ � k=0 W∆1 � ∆2(k)xk = 1 − xn+1 (1 − x)n+2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' By facial decomposition, we have W∆(k) = W∆1(k) + W∆2(k) − W∆1 � ∆2(k), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='12) 16 XIN LIN AND DAQING WAN which implies W(x) = ∞ � k=0 W∆(k)xk = W1(x) + W2(x) − W3(x) = 1 + 2x + · · · + 2xn + xn+1 (1 − x)n+2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' This gives the Hodge numbers of ∆ via formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='5), that is, H∆(0) = 1, H∆(1) = · · · = H∆(n) = 2, H∆(n + 1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' □ When f is ordinary, the slopes of L∗(f, T)(−1)n+1 can be deduced from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' If p ≡ 1 mod (n + 1), the slope sequence of L∗(f, T)(−1)n+1 is given by {0, 1, 1, 2, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' , n, n, n + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' This theorem follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='5, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='2 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' □ Note that the converse of this theorem is also true, as Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='2 shows that the condition p ≡ 1 mod (n + 1) is a necessary and sufficient condition for f to be ordinary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Now we are ready to consider the weights for the reciprocal roots of L∗(f, T)(−1)n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Suppose p ∤ (n + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' We have L∗(f, T)(−1)n+1 = (1 − T)(1 − qT) 2n � i=1 (1 − βiT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' For each 1 ≤ i ≤ 2n, the reciprocal root βi satisfies |βi| = q n+2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Since the origin is a vertex of ∆, we decompose the cone C(∆) via boundary decomposition B(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Let N(i) be the number of i-dimensional face Σi of C(∆), where 0 ≤ i ≤ dim∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' For Newton polyhedron ∆ = ∆(f), we have N(0) = 1 and N(1) = n + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Note that Σi is an open cone and Σi ∈ B(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Let Σi be the closure of Σi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' For simplicity, we denote the Fredholm determinants as D(T) = det (I − TA1(f)) , D◦ i (T) = det � I − TA1(Σi, f Σi) � , Di(T) = det � I − TA1(Σi, f Σi) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' The unique 0-dimensional cone Σ0 is the origin and D0(T) = D◦ 0(T) = 1 − T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' When i = 1, each f Σ1 can be normalized to x by variable substitution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' That is, L∗(f Σ1, T) = exp � ∞ � k=1 −T k k � = 1 − T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' By formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='7), we have D1(T) = ∞ � i=0 � L∗ � f Σ1, qiT ��(i i) = (1 − T) (1 − qT) � 1 − q2T � · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Since the only boundary of Σ1 are Σ1 and Σ0, we get D◦ 1(T) after eliminating D◦ 0(T), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=', D◦ 1(T) = D1(T) D◦ 0(T) = (1 − qT) � 1 − q2T � ∞ � i=3 � 1 − qiT � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='16 shows that D(T) can be expressed as a product of D◦ i (T) as follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' D(T) = n+2 � i=1 N(i) � j=1 D◦ i (T) = (1 − T)(1 − qT)n+3 · · · , ON INVERTED KLOOSTERMAN SUMS OVER FINITE FIELDS 17 Note that L∗(f, T)(−1)n+1 is a polynomial of degree 2(n + 1) if f is non-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Combining formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='6), we obtain L∗(f, T)(−1)n+1 = D(T)D(q2T)(n+2 2 ) · · · D(qT)n+2D(q3T)(n+2 3 ) · · · = (1 − T)(1 − qT) 2n � i=1 (1 − βiT), where |βi| = q wi 2 ≤ q n+2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' That is, wi ≤ n + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' If βi is a reciprocal root of L∗(f, T)(−1)n+1, the conjugate βi is a reciprocal root of the conjugate L-function L∗(f, T) (−1)n+1 = L∗(−f, T)(−1)n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='8, the Newton polygon and Hodge polygon coincide at the end points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Applying this to the product L∗(f, T)(−1)n+1L∗(f, T) (−1)n+1 , we deduce that 2(n+1)2 = 2( n � i=1 2i+n+1) = ordq(1·q2 · 2n � i=1 βiβi) = 2+ 2n � i=1 wi ≤ 2+2n(n+2) = 2(n+1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' It follows that the inequality must be an equality, that is, all wi = n + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' □ Formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='9) relates L∗(f, T) to Ln(b, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' The valuations for the reciprocal roots and poles of Ln(b, T) follow from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='4 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Suppose p ∤ (n + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' We have Ln(b, T)(−1)n+1 = (1 − T)(n+1) n � j=2 � 1 − qj−1T �(n j)(−1)j−1 2n � i=1 (1 − αiT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' For each 1 ≤ i ≤ 2n, the reciprocal root αi satisfies |αi| = q n 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' If p ≡ 1 mod (n + 1), the slope sequence of the αi’s is given by {0, 1, 1, 2, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' , n − 1, n − 1, n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Based on weights of toric L-function, we get the following slightly more precise upper bound for its associated exponential sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' If p ∤ (n + 1), we have |Sk,n(b) + (qk − 1)n − (−1)n(qk + 1) qk | ≤ 2nq nk 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='5 implies that |S∗ k(f) − (−1)n(qk + 1)| ≤ 2nq (n+2)k 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Combining formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='8), we get the bound for Sk,n(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' □ Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' We finish this paper with two open problems on the estimates of inverted Kloosterman sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' If n + 1 is divisible by p, the related Laurent polynomial f is degenerate and thus the results for toric exponential sums are not tenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' In this case, it is an open problem to determine the optimal square root cancellation for Sn(χ, b) in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' The case n = 1 with p = 2 is already handled in [Kat95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' The second question concerns the q-adic slope sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' If p is not equivalent to 1 modulo n + 1, the Newton polygon corresponding to f is strictly above its Hodge polygon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Under this 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polygons for L-functions of (A, B)-exponential sums, Finite Fields Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' 78 (2022), Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' 101980, 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' MR 4349885 DEPARTMENT OF MATHEMATICS, SHANGHAI MARITIME UNIVERSITY, SHANGHAI 201306, PR CHINA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Email address: xlin1126@hotmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='com DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CALIFORNIA, IRVINE, CA 92697-3875 USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content=' Email address: dwan@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='uci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} +page_content='edu' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE3T4oBgHgl3EQfDglb/content/2301.04287v1.pdf'} diff --git a/XNFJT4oBgHgl3EQf5S05/content/tmp_files/2301.11669v1.pdf.txt b/XNFJT4oBgHgl3EQf5S05/content/tmp_files/2301.11669v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..855b63fae614962ab6af408016b2bf4a1fe047d0 --- /dev/null +++ b/XNFJT4oBgHgl3EQf5S05/content/tmp_files/2301.11669v1.pdf.txt @@ -0,0 +1,3568 @@ +Predator Extinction arose from Chaos of the Prey: the Chaotic +Behavior of a Homomorphic Two-Dimensional Logistic Map in the +Form of Lotka-Volterra Equations +Wei Shan Lee∗, Hou Fai Chan, Ka Ian Im, Kuan Ieong Chan, and U Hin Cheang +Pui Ching Middle School Macau +Macao Special Administrative Region, People’s Republic of China. +Abstract +A two-dimensional homomorphic logistic map that preserves features of Lotka-Volterra Equations +was proposed. +In order to examine the Lotka-Volterra chaos, in addition to ordinary iteration plots +of population, Lyapunov exponents either calculated directly from eigenvalues of Jacobian of the 2D +logistic mapping, or from time-series algorithms of both Rosenstein and Eckmann et al. were calculated, +among which discrepancies were compared. Bifurcation diagrams may be divided into five categories +depending on different topological shapes, among which flip bifurcation and Neimark-Sacker bifurcation +were observed, the latter showing closed orbits around limit circles in the phase portrait and phase space +diagram. Our model restored the 1D logistic map of the prey at the absence of the predator, as well as the +normal competing behavior between two species when the initial population of the two is equal. In spite +of the possibility for two species going into chaos simultaneously, it is also possible that with the same +inter-species parameters as normal but with predator population 10 times more than that of the prey, +under certain growth rate the latter becomes chaotic, and former dramatically reduces to zero, referring +to total annihilation of the predator species. Interpreting humans as the predator and natural resources +as the prey in the ecological system, the aforementioned conclusion may imply that not only excessive +consumption of the natural resources, but its chaotic state triggered by overpopulation of humans may +also backfire in a manner of total extinction on human species. Fortunately, a little chance may exist for +survival of human race, as isolated fixed points in bifurcation diagram of the predator reveals. +Keywords Chaos, Neimark-Sacker Bifurcation, Logistic Map, Lyapunov Exponents, Lotka-Volterra Equa- +tions, Extinction of Species +1 +Introduction +Understanding interactions between human beings and natural resources plays an important role in estab- +lishing sustainable economy and society. Relationships of these two may be studied by the prey and predator +model after we realize that humans beings may be regarded as the predator while natural resources may be +thought of as the prey[1]. Afterwards, researches on the prey-predator models may be implemented to this +field instinctively[2]-[3]. +Generally speaking, there are two main approaches of studies in the literature to this prey and predator +model. The first is to study differential equations, while the other is to study the iterations in difference +equations, whose forms may be inspired by directly applying the forward Euler’s Scheme to acquire coun- +terpart of the former[4]-[10]. The discrete model could be more promising than the continuous one, because +it has more abundant dynamic characteristics in chaotic behaviors[8], whereas it would be more difficult for +solutions to continuous models to reach chaos in low dimensional cases. Taking some examples about the first +approach, studies[11]-[12] have performed on chaos of Lotka-Volterra differential equations with dimensions +higher than three, and researchers[13] claimed that it is impossible to reach chaos for two species in the +form of differential Lotka-Volterra Equations, whose general solutions were obtained in sinusoidal forms by +Evans and Findley[14]. Additionally, based on the Lotka-Volterra model, Dunbar[15] confirmed the existence +∗email: wslee@g.puiching.edu.mo +1 +arXiv:2301.11669v1 [nlin.CD] 27 Jan 2023 + +of traveling wave solutions for two reaction diffusion systems. Besides that, Das and Gupta[16] proposed +solutions to the fractional-order time derivative Lotka-Volterra equations using an analytical approach for +nonlinear problems known as the homotopy perturbation method (HPM). +On the other hand, there are also several studies on the discrete difference equations. +For instance, +Bessoir and Wolf [17] made pioneering contributions to the application of 1D logistic equation on biological +and ecological studies. The same equation was also used to interpret, analyze and predict data according to +the COVID-19 by many researchers[18]. Mareno and English[19] implemented the 1D logistic equation to +the coupled 2D logistic one, and demonstrated that for large growth rate the system underwent a Neimark- +Sacker bifurcation. Li et al.[20] imposed an equal individual effect intensity, corresponding to equal growth +rate in the 1D logistic map, on the two oligopolists in the homomorphic Kopel model and observed three +different kinds of bifurcation. Furthermore, Elhadi and Sprott[21] proposed a two-dimensional mapping, one +of which is the ordinary 1D logistic map while the other consists of a perturbation term of the former and is +also modulated by the first. Shilnikov and Rulkov[22] studied chaos behaviors in two-dimensional difference +equations that reproduced spike-bursting activities in biological neurons, improving further on the previous +research based on the three-dimensional system of ODEs. In spite of applying the forward Euler’s Scheme to +acquire the difference equation, researchers also made use of exponential forms corresponding to solutions on +the differential equations. For example, Ishaque et al.[23] studied a three dimensional predator-prey-parasite +model with an exponential form describing interactions among healthy or infected Tilapia fish as the prey, and +Pelican birds as the predator. Tassaddiq et al.[24] worked on discrete-time exponential difference equation +of Leslie-Gower predator-prey model together with a Holling type III functional response, and indicated +the advantage on this type of discretization method. Previous study[25] suggested a heteromorphic term +describing the decreasing effects on the predator that was only linear to the population of that species, +contrary to the corresponding quadratic term in the prey. Hassell et al[26] applied the predator-prey model +to insect parasitoids and anthropods, and found out that local movements of the two species may cause +extermination of the entire ecological system with chaos, and it is difficult to maintain population stability +for large growth rate of anthropods. Besides, researchers[27] also pointed out that human misbehavior may +be the reason for an ecological system to go into chaotic states. +However, there is no convincing reason for the prey and the predator to have different forms in the +difference equations. +Intuition in mathematical symmetry naturally came to our mind that a successful +predator-prey difference model should resemble the symmetry structure as in Lotka-Volterra differential +equations. Moreover, solutions to Lotka-Volterra Equations in sinusoidal forms cannot explain extinction of +species. +We proposed homomorphic two-dimensional logistic maps that preserve both forms of Lotka-Volterra +Equations and the 1D logistic equation. In our model, we conjectured a quadractic form in both corresponding +terms of the prey and the predator, treating both species on the equal stance. Structures of Bifurcation +diagrams showed that there could be six different categories in our dynamic system. For each categories, we +examined population iterations, phase portraits, phase space diagrams, and topological types of fixed points. +Lyapunov exponents either calcaulted from eigenvalues of Jacobian of the 2D mapping or from time-series +algorithms either Rosenstein[28] or Eckmann et al.[29] were also calculated. Comparisons among those results +were also discussed. +The advantages of our model include the following. +First, we may be able to establish a standard +bifurcation diagram of 1D logistic map about the prey with nonzero initial predator population, growth rates +in both species, and predation parameters. Second, our model may also describe the normal behavior of rise +and fall on the population of the two species when interacting with each other. Third, besides simultaneous +chaos in both species, the main discovery in our research was that the predator may go extinct under the +circumstance of chaos in the prey that the predator overpopulation should be blamed. +2 +Theorems +We first review the one-dimensional logistic equation and the two-dimensional Lotka-Volterra Equations, +comparing the similarity and difference between the two sets of equations, which inspires us on the idea +to establish two-dimensional logistic equations that maintain important features about both of the above +equations. +2 + +2.1 +Review on the 1D logistic equation and Litka-Volterra Equations +To begin with, the one-dimensional logistic equation may be written as +xn+1 = µ0xn(1 − xn), +(1) +where µ0 denotes to the growth rate. +On the other hand, the two-dimensional Lotka-Volterra Equations[30], [31] describe interactions between +the prey and predator in an environment where there is sufficient food supply for the prey, whose only natural +enemy is the predator. The formula may be written as follows: +� +� +� +� +� +� +� +dx +dt = µ0x − µ1xy; +dy +dt = −ν0y + ν1xy, +(2a) +(2b) +where µ0, µ1, ν0, and ν1 are all positive inter-species parameters. x denotes population of the prey while +y, population of the predator, both being positive real numbers. µ0 and ν1 are, respectively, growth rate +of the prey and the predator. While µ1 refers to the parameter for predation to occur upon the prey at +the presence of predator, ν0 refers to all effects that decrease population of the predator, which may include +disease, death, or emigration[32]. With forward Euler’s scheme, one may immediately write the difference +version of Eq.(2) as follows +�xn+1 − xn = µ0xn − µ1xnyn; +yn+1 − yn = −ν0yn + ν1xnyn. +(3a) +(3b) +However, there is an obvious drawback about the above simultaneous equations: it does not preserve the +feature of 1D logistic equation, because the first term on the right hand side is either linear to xn or yn, +whereas the right hand side in Eq.(1) is quadratic to xn. +We now discuss our idea on establishing the two-dimensional map that resembles interaction terms of the +prey and the predator as in Lotka-Volterra. First, we may rewrite the linear term of x on the right hand side +of Eq.(2a) into quadratic, which looks like µ0x(1 − x), together with the homomorphic corresponding term +in Eq.(2b) as ν0y(1 − y). Thus, the modified Lotka-Volterra Equations[33] are +� +� +� +� +� +� +� +dx +dt = µ0x(1 − x) − µ1xy; +dy +dt = −ν0y(1 − y) + ν1xy, +(4a) +(4b) +whose resemblance in the form of difference equations are, therefore, +�xn+1 − xn = µ0xn(1 − xn) − µ1xnyn; +yn+1 − yn = −ν0yn(1 − yn) + ν1xnyn. +(5a) +(5b) +Even though it seems more reasonable to direct resemblance of Lotka-Volterra Equations, Eq.(5) fails to +restore Eq.(1) when parameters other than µ0 are all set to be zero. Fortunately, we may modify this by +dropping xn and yn terms on the left hand side of the above equations +�xn+1 = µ0xn(1 − xn) − µ1xnyn; +yn+1 = −ν0yn(1 − yn) + ν1xnyn, +(6a) +(6b) +which is the desired form. We divided into two cases to study further about properties of Eq.(5) and Eq.(6) +in Sec. 2.3 and Sec. 2.4. But before that, in next subsection we first discuss a very important lemma that +allows us to study stability behaviors of fixed points. +3 + +2.2 +Stability of fixed points +Suppose a mapping of two-dimensional iterations xn+1 and yn+1 are written as xn+1 = f(x, y) and yn+1 = +g(x, y). The Jacobian is, therefore, +� +� +� +� +� +� +� +� +� +� +� +J = ∂(f, g) +∂(x, y) += +� ∂f +∂x +∂f +∂y +∂g +∂x +∂g +∂y +� +, +(7a) +(7b) +whose eigenvalus are ω0 and ω1. It is well known[6], [8] that a fixed point may be divided into the following +four topological types based on their stability behaviors. First, it could be a sink and locally asymptotic +stable if eigenvalues of Eq.(7) satisfy |ω0| < 1 and |ω1| < 1. Second, it could be a source and locally unstable +if eigenvalues satisfy |ω0| > 1 and |ω1| > 1. Third, a fixed point could be a saddle if one of the absolute +values of the eigenvalues is greater than 1 while the other is smaller than 1. At last, a fixed point could be +non-hyperbolic if one of the absolute values of the eigenvalues is equal to 1. The stability of a non-hyperbolic +fixed point is fragile[34], which means that its stability is easily influenced by the small nonlinear terms. +Instead of calculating the range of eigenvalues directly, most of the time it is more convenient to work with +the quadratic formula consisting of eigenvalues ω0 and ω1, namely, Ω(ω) = ω2 −Tr(J)ω +det(J), where Tr(J) +and det(J) are trace and determinant of Jacobian in Eq.(7), respectively, and there could be a correspondence +on the stability behavior around a fixed point between the roots of the quadratic formual, ω0 and ω1, which +are also eigenvalues of Jacobian, through the following Lemma +Lemma 1. Let Ω(ω) = ω2 − Tr(J)ω + det(J), be a quadratic formula where where Tr(J) and det(J) are trace +and determinant of Jacobian in Eq.(7), respectively. Then +1. If Ω(1) > 0, then +(a) |ω0| < 1 and |ω1| < 1 and hence the fixed point is a sink if and only if Ω(−1) > 0 and det(J) < 1; +(b) |ω0| > 1 and |ω1| > 1 and hence the fixed point is a source if and only if Ω(−1) > 0 and det(J) > 1; +(c) One of |ω0| and |ω1| is smaller than 1 while the other greater than 1 and hence the fixed point is +a saddle if and only if Ω(−1) < 0; +(d) Either |ω0| or |ω1| is equal to 1 and hence the fixed point is a non-hyperbolic whenever +i. ω0 = −1 and ω1 ̸= −1 if and only if Ω(−1) = 0 and Tr(J) ̸= 2. +ii. ω0 and ω1 are a pair of complex conjugates and |ω0| = |ω1| = 1 if and only if |Tr(J)| < 2 and +det(J) = 1. +iii. ω0 = ω1 = −1 if and only if Ω(−1) = 0 and Tr(J) = 2. +2. If Ω(1) = 0, then either |ω0| or |ω1| has to be equal to 1. Therefore the fixed point is a non-hyperbolic. +Absolute value of the other root is greater than, equal to, or smaller than 1 if and only if, correspondingly, +absolute value of det(J) is greater than, equal to, or smaller than 1. +3. If Ω(1) < 0, then either |ω0| or |ω1| has to be real and greater than 1. Therefore, the fixed point is a +saddle. Further, +(a) the other root is smaller or equal to −1 if and only if, correspondingly, Ω(−1) < −1 or Ω(−1) = −1. +(b) absolute value of the other root is smaller than 1 if and only if Ω(−1) > 0. +1 makes it easier for us to study analytically the stability of fixed points. +2.3 +Properties of Eq.(5) +Setting up xn+1 = f(x, y) and yn+1 = g(x, y), the two-dimensional logistic equations in Eq.(5) have the +mappings +�f(x, y) = µ0x(1 − x) − µ1xy + x; +g(x, y) = −ν0y(1 − y) + ν1xy + y. +(8a) +(8b) +4 + +Eq.(8) has Jacobian, as indicated in Eq.(7), +J = +�µ0(1 − 2x) − µ1y + 1 +−µ1x +ν1y +ν0(−1 + 2y) + ν1x + 1 +� +, +(9) +with eigenvalues ω0 and ω1 being, respectively, +� +� +� +� +� +� +� +ω0 = −µ0x + ν0y + 1 + 1 +2(xν1 − yµ1) + 1 +2(µ0 − ν0) + ω +2 +ω1 = −µ0x + ν0y + 1 + 1 +2(xν1 − yµ1) + 1 +2(µ0 − ν0) − ω +2 +(10a) +(10b) +where +ω = +� +4(ν0 + µ1 +2 )2y2 + +�� +(4µ0 − 2ν1)µ1 + 8ν0(µ0 + ν1 +2 ) +� +x − 4(ν0 + µ0)(ν0 + µ1 +2 ) +� +y +(11) ++ 4 +� +(µ0 + ν1 +2 )x − µ0 +2 − ν0 +2 +�2� 1 +2 +Further, fixed points, at which pairs of x and y stay still irrespective of time-series iterations[34], are those +pairs of points (x∗, y∗) such that +�x∗ = µ0x∗(1 − x∗) − µ1x∗y∗ + x∗ +y∗ = ν0y∗(1 − y∗) − ν1x∗y∗ + y∗, +(12a) +(12b) +from which four pairs of fixed points (x∗, y∗) may be derived as +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +E0 = (0, 0) +E1 = (0, 1) +E2 = (1, 0) +E3 = +� ν0(µ1 − µ0) +−µ0ν0 + µ1ν1 +, +µ0(ν1 − ν0) +−µ0ν0 + µ1ν1 +� +, +(13a) +(13b) +(13c) +(13d) +provided that the denominator in Eq.(13d) is not zero. On the contrary, however, when µ0ν0 = µ1ν1, Eq.(5) +has fixed points E0, E1,and E2. +Keeping Lamma 1 in Sec. 2.2 in mind, we may be able to examine the topological type of each fixed +point in Eq.(13) as in Theorem 1: +Theorem 1. The topological types of fixed points in Eq.(13) are +1. For E0 = (0, 0), +� +� +� +� +� +� +� +� +� +Ω(1) = −µ0ν0 +Ω(−1) = (2 + µ0)(2 − ν0) +det(J) = (1 + µ0)(1 − ν0) +Tr(J) = 2 + µ0 − ν0. +Because Ω(1) < 0, therefore, E0 is always a saddle. +2. For E1 = (0, 1), +� +� +� +� +� +� +� +� +� +Ω(1) = ν0(µ0 − µ1) +Ω(−1) = (2 + ν0)(2 + µ0 − µ1) +det(J) = (1 + ν0)(1 + µ0 − µ1) +Tr(J) = 2 + µ0 − µ1 + ν0. +In this case, +5 + +(a) E1 cannot be a sink; +(b) if µ0 > µ1, then E1 is a source; +(c) if µ0 < µ1, then E1 is a saddle. +(d) if µ0 = µ1, then E1 is a non-hyperbole; +3. For E2 = (1, 0), +� +� +� +� +� +� +� +� +� +Ω(1) = −µ0(ν1 − ν0) +Ω(−1) = (2 − µ0)(2 + ν1 − ν0) +det(J) = (1 − µ0)(1 + ν1 − ν0) +Tr(J) = 2 − µ0 + (ν1 − ν0). +In this case, +(a) if µ0 < 2 and ν1 < ν0 < ν1 + 1 and ν1 < ν0, or µ0 < 2 and ν0 = ν1 + 1, or µ0 < 2 and +ν1 + 1 < ν0 < ν1 + 2, then E2 is a sink; +(b) if µ0 > 2 and ν0 > ν1 + 2, then E2 is a source; +(c) if µ0 > 2 and ν1 < ν0 < ν1 + 2, or µ0 < 2 and ν0 > ν1 + 2, or ν1 > ν0, then E2 is a saddle; +(d) if ν1 = ν0, then E2 is a non-hyperbole. +4. For E3 = +� +ν0(µ1−µ0) +−µ0ν0+µ1ν1 , +µ0(ν1−ν0) +−µ0ν0+µ1ν1 +� +, +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +Ω(1) = −µ0ν0(ν0 − ν1)(µ0 − µ1) +µ0ν0 − µ1ν1 +Ω(−1) = −ν0(ν0 − ν1 + 2)µ2 +0 + ν0(µ1 + 2)(ν0 − ν1 + 2)µ0 − 4µ1ν1 +µ0ν0 − µ1ν1 +det(J) = −ν0(ν0 − ν1 + 1)µ2 +0 + ν0(µ1 + 1)(ν0 − ν1 + 1)µ0 − µ1ν1 +µ0ν0 − µ1ν1 +Tr(J) = −µ2 +0ν0 + ν0(ν0 + µ1 − ν1 + 2)µ0 − 2µ1ν1 +µ0ν0 − µ1ν1 +. +In this case, +(a) if µ0 < µ1 and µ2 +0ν0 < µ2 +1ν1, or µ0 > µ1 and µ2 +0ν0 > µ2 +1ν1, then it is a saddle; +(b) if µ2 +0/µ2 +1 = ν1/ν0, we have a non-hyperbole in the interior region. +In order to plot bifurcation diagrams, we further assume that µ1 = αµ0, ν0 = βµ0, and ν1 = γµ0, where +α, β, and γ are parameters. In this case the original µ1, ν0, and ν1 vary with µ0. Under this circumstance, +the nontrivial fixed points E3 becomes +E3 = +�β(α − 1) +αγ − β , γ − β +αγ − β +� +, +which is independent of the growth rate parameter µ0. Eigenvalues of Jacobian at E3 in Eq.(9) is +� +� +� +� +� +� +� +� +� +ω0 = +1 +2αγ − 2β +� +Ξ0 + αγ − β +|αγ − β|Ξ1 +� +ω1 = +1 +2αγ − 2β +� +Ξ0 − αγ − β +|αγ − β|Ξ1 +� +, +(18a) +(18b) +where +� +� +� +� +� +� +� +� +� +Ξ0 = 2αγ − β2µ0 − +� +2 + (α − γ − 1)µ0 +� +β +Ξ1 = µ0 +� +β +� +(4βγ − 4γ2 + β)α2 − (2β2 + 2βγ − 4γ2 + 2β)α + β(β − γ + 1)2 +�� 1 +2 +(19a) +(19b) +6 + +2.4 +Properties of Eq.(6) +Similar to Eq.(8), the two-dimensional logistic equations in Eq.(6) have the mappings +�f(x, y) = µ0x(1 − x) − µ1xy; +g(x, y) = −ν0y(1 − y) + ν1xy. +(20a) +(20b) +Eq.(20) has Jacobian that is slightly different from Eq.(9) +J = +�µ0(1 − 2x) − µ1y +−µ1x +ν1y +ν0(−1 + 2y) + ν1x +� +, +(21) +with eigenvalues ω0 and ω1 being, respectively, +� +� +� +� +� +� +� +ω0 = −µ0x + ν0y + 1 +2(xν1 − yµ1) + 1 +2(µ0 − ν0) + ω +2 +ω1 = −µ0x + ν0y + 1 +2(xν1 − yµ1) + 1 +2(µ0 − ν0) − ω +2 +(22a) +(22b) +where ω is the same as in Eq.(11). Fixed points for Eq.(6) are +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +E′ +0 = (0, 0) +E′ +1 = (0, 1 + 1 +ν0 +) +E′ +2 = (1 − 1 +µ0 +, 0) +E′ +3 = +�−µ0ν0 + µ1ν0 + µ1 + ν0 +−µ0ν0 + µ1ν1 +, −µ0ν0 + µ0ν1 − µ0 − ν1 +−µ0ν0 + µ1ν1 +� +(23a) +(23b) +(23c) +(23d) +However, when µ0ν0 = µ1ν1, Eq.(6) has fixed points E′ +0, E′ +1, and E′ +2. +We may also make use of Lamma 1 in Sec. 2.2 to examine the topological type of each fixed point in +Eq.(6) as in Theorem 2: +Theorem 2. The topological types of fixed points in Eq.(23) are +1. For E′ +0 = (0, 0), +� +� +� +� +� +� +� +� +� +Ω(1) = −(µ0 − 1)(ν0 + 1) +Ω(−1) = −(µ0 + 1)(ν0 − 1) +det(J) = −µ0ν0 +Tr(J) = µ0 − ν0. +In this case, +(a) if µ0 < 1 and ν0 < 1, then E′ +0 is a sink. +(b) E′ +0 cannot be a source. +(c) if µ0 > 1, then E′ +0 is a saddle. +(d) if µ0 = 1, then E′ +0 is a non-hyperbole. +2. For E′ +1 = (0, 1 + 1 +ν0 ), +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +Ω(1) = (ν0 + 1)((µ0 − µ1 − 1)ν0 − µ1) +ν0 +Ω(−1) = (ν0 + 3)((µ0 − µ1 + 1)ν0 − µ1) +ν0 +det(J) = (ν0 + 2)((µ0 − µ1)ν0 − µ1) +ν0 +Tr(J) = ν2 +0 + (µ0 − µ1 + 2)ν0 − µ1 +ν0 +. +In this case, +7 + +(a) E′ +1 cannot be a sink. +(b) if µ0 > µ1ν0+µ1+ν0 +ν0 +, then E′ +1 is a source. +(c) if µ0 < µ1ν0+µ1+ν0 +ν0 +, then E′ +1 is a saddle. +(d) if µ0 = µ1ν0+µ1+ν0 +ν0 +, then E′ +1 is a non-hyperbole. +3. For E′ +2 = (1 − +1 +µ0 , 0), +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +Ω(1) = (µ0 − 1)((ν0 − ν1 + 1)µ0 + ν1) +µ0 +Ω(−1) = (µ0 − 3)((ν0 − ν1 − 1)µ0 + ν1) +µ0 +det(J) = (µ0 − 2)((ν0 − ν1)µ0 + ν1) +µ0 +Tr(J) = −µ2 +0 + (ν0 − ν1 − 2)µ0 + ν1 +µ0 +. +In this case, +(a) if 1 < µ0 < 2 and ν1µ0−µ0−ν1 +µ0 +< ν0 < ν1µ0+µ0−ν1 +µ0 +or µ0 = 2 and ν1 +2 − 1 < ν0 < ν1 +2 + 1 with ν1 > 2, +or 2 < µ0 < 3 and ν1µ0−µ0−ν1 +µ0 +< ν0 < ν1µ0+µ0−ν1 +µ0 +, then E′ +2 is a sink; +(b) if 0 < µ0 < 1 and ν0 < ν1µ0−µ0−ν1 +µ0 +, or µ0 > 3 and ν0 > ν1µ0+µ0−ν1 +µ0 +, then E′ +2 is a source; +(c) if 1 < µ0 < 3 and ν0 > ν1µ0+µ0−ν1 +µ0 +, or µ0 > 3 and ν1µ0−µ0−ν1 +µ0 +< ν0 < ν1µ0+µ0−ν1 +µ0 +, or 0 < µ0 < 1 +and ν0 > ν1µ0−µ0−ν1 +µ0 +, or 1 < µ0 and ν0 < ν1µ0−µ0−ν1 +µ0 +, then E′ +2 is a saddle; +(d) if µ1 = 1 or ν0 = ν1µ0−µ0−ν1 +µ0 +, then E′ +2 is a non-hyperbole. +4. For E′ +3 = +� +ν0(µ1−µ0) +−µ0ν0+µ1ν1 , +µ0(ν1−ν0) +−µ0ν0+µ1ν1 +� +, +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +Ω(1) = −µ0ν0(ν0 − ν1)(µ0 − µ1) +µ0ν0 − µ1ν1 +Ω(−1) = −ν0(ν0 − ν1 + 2)µ2 +0 + ν0(µ1 + 2)(ν0 − ν1 + 2)µ0 − 4µ1ν1 +µ0ν0 − µ1ν1 +det(J) = −ν0(ν0 − ν1 + 1)µ2 +0 + ν0(µ1 + 1)(ν0 − ν1 + 1)µ0 − µ1ν1 +µ0ν0 − µ1ν1 +Tr(J) = −µ2 +0ν0 + ν0(ν0 + µ1 − ν1 + 2)µ0 − 2µ1ν1 +µ0ν0 − µ1ν1 +. +In this case, +(a) if µ0 < µ1 and µ2 +0ν0 < µ2 +1ν1, or µ0 > µ1 and µ2 +0ν0 > µ2 +1ν1, then it is a saddle; +(b) if µ2 +0/µ2 +1 = ν1/ν0, we have a non-hyperbole in the interior region. +In terms of α, β, and γ, E′ +3 in Eq.(23d) is E′ +3 = +� +αβµ0−βµ0+α+β +µ0(αγ−β) +, −βµ0+γµ0−γ−1 +µ0(αγ−β) +� +, at which the eigenvalues +of Jacobian in Eq.(21) is +� +� +� +� +� +� +� +� +� +ω0 = +1 +2αγ − 2β +� +Ξ′ +0 + αγ − β +|αγ − β|Ξ′ +1 +� +ω1 = +1 +2αγ − 2β +� +Ξ′ +0 − αγ − β +|αγ − β|Ξ′ +1 +� +, +(28a) +(28b) +8 + +where +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +Ξ′ +0 = (2γ − 1)α − β2µ0 − +� +αµ0 + (−µ0 + 1)γ − µ0 + 4 +� +β +Ξ′ +1 = +� +(µ0 − 1) +� +(−4βµ0 − 4) α2 + 4β (µ0 − 1) α + β2 (µ0 − 1) +� +γ2 ++ +� +4 (βµ0 + 1)2 α2 − 2β (µ0 − 1) (βµ0 + 1) α − 2β2µ0 (µ0 − 1) (β + 1) +� +γ ++ +� +(βµ0 + 1) α − βµ0 (β + 1) +�2 +� 1 +2 +(29a) +(29b) +(29c) +Eq.(28) shows that whenever Ξ′ 2 +1 +is negative, ω0 and ω1 are complex conjugates. Unlike the previous case, +fixed points E′ +1, E′ +2, and E′ +3 now are dependent on the growth rate µ0. +2.5 +Lyapunov exponents +In addition, the chaotic behavior may be better examined by introducing Lyapunov exponents, which are +defined as, base 2 being chosen to conform to Wolf et al[35], +� +� +� +� +� +� +� +λx ≡ log2 |w0| = ln |w0| +ln 2 +λy ≡ log2 |w1| = ln |w1| +ln 2 . +(30a) +(30b) +The positive value of λx or λy, together with the negative value of total sum of the Lyapunov exponent, +either � λx < 0 or � λy < 0, are strong inference of chaos for the prey or the predator[36]. For comparison, +Lyapunov exponents of time series data of prey and predator populations were also calculated by both +algorithms of Rosenstein[28] and Eckmann et al.[29] with the package of NOnLinear measures for Dynamical +Systems (nolds)[37]. For Rosenstein algorithm, embedding dimension for delay embedding was emb dim = 10, +and the step size between time series data points was set to τ = 1 second. While number of data points +(trajectory len) was set to 20 and was used for the distance trajectories between two neighboring points, +the mean period of time series data, obtained from the fast Fourier transform, was used as the minimal +temporal separation(min tsep) between two neighbors. +Search of the suitable lag was terminated when +number of potential neighbors for a vector was found to be smaller than minimal neighbors, which was set +as min neighbors = 20. At last, the RANSAC-fitting was used for the line fitting. As for the algorithm +proposed by Eckmann et al., the matrix dimension was set to 2, and embedding dimension was also set to +10 as in Rosenstein algorithm. Moreover, τ = 1 s, the minimal number of neighbors (min nb) was 4, and +min tsep = 0 were used in the algorithm. +There are at least four disadvantages for the above algorithms, as mentioned by Escot and Galan[36]: lack +of the ability to estimate full Lyapunov spectrum, not resilient to noise in time-series data, poor detection +performance in nonlinearity with an adequate sample size, and no theoretical derivations for the algorithms +about their consistency and asymptotic distributions, making it impossible to statistical inferences respect +to chaos. +3 +Results and Discussions +In the present study, we focused only on drawings of equations in Sec. 2.4. We observed that there could be +five different bifurcation diagrams with various kinds of combinations of parameters. The first category is +Normal, referring to the normal competitive behavior on the increasing and decreasing on numbers about +species between the prey and the predator. The second category is Standard, referring to the standard +bifurcation diagram as shown in the well-known 1D logistic equation in the prey at the absence of the +predator. The third category is named as Paraclete, referring to overlapping structure in the bifurcation +diagram of the prey. +The fourth category is Extinction, connoting to extinction of the predator when +the prey becomes chaotic. The last category is Vorticella Strange, meaning that the bifurcation diagram +resembles the shape of a vorticella but with more complex inner structures before the two species become +chaotic at the same time. Categories and parameters were summarized in Table 1. initX and initY indicate +9 + +initial values of the prey and the predator, respectively. Special attention should be paid to the cases of +Normal and Extinction, where the inter-species parameters are deliberately made the same but the initial +population were different: for Normal, the two species have the same initial population whereas for Extinction, +the predator has 10 times more population than the prey. The discrepancy on initial population in these two +cases shows completely different evolution consequences. +In the following figures, discussions on Lyapounov exponents, Equation, Rosenstein, Eckmann X, and +Eckmann Y in the legend of λx and λy refer to calculations directly from Eq.(30), from time-series algorithm +of Rosenstein, Eckmann et al of the prey, and Eckmann et al of the predator, respectively. Codes, together +with animations on population iterations, phase portraits and phase diagrams under different growth rates, +may be retrieved via Ref([38]). +initX +initY +α +β +γ +Normal +0.200 +0.200 +1.000 +0.001 +0.500 +Standard +0.100 +0.500 +1.000 +0.100 +0.500 +Paraclete +0.010 +0.100 +5.000 +0.010 +0.900 +Extinction +0.010 +0.100 +1.000 +0.001 +0.500 +Vorticella Strange +0.100 +0.500 +0.875 +0.018 +1.000 +Table 1: Parameters used for equations in Sec. 2.4. +3.1 +Normal +Our dynamical system may describe normal competitiveness between two species. Under the circumstance +of equal initial population, Figure 1a shows steadily increasing population of x at the absence of the predator +when µ0 < 3.0. At the appearance of y after µ0 > 3.0, the prey gradually diminishes with more number of +the predator. +Figure 1b ensures that under this scenario there is no chaos, for the Lyapunou exponents calculated by +every algorithm are negative. However, results are different from algorithms. First for λx, there is a trench +around µ0 = 2 by Equation, whereas all other algorithms fail to reproduce. Rosenstein, Eckmann X, and +Eckmann Y only reproduced shallow dip around 1.5 < µ0 < 2.5. In addition, We observe that Rosenstein +and Eckmann X have quite similar results in the whole range of µ0, except that Rosenstein has a slightly +higher value. Furthermore, Eckmann X and Eckmann Y have almost identical values when µ0 > 1.66, but +Eckmann Y digresses a lot from the other three curves at low growth rate below µ0 = 1.66. As for λy, all +algorithms show close spectrum µ0 > 1.692, with larger values for Rosenstein. The four algorithms divide +into two groups of results below µ0 = 1.692, with Rosenstein and Eckmann X showing an increasing tail that +is different from the other two algorithms showing curves of dropping. +3.2 +Standard +Figure 2a shows bifurcation diagram and Lyapunov exponents of Standard. Our model shows that even +with nonzero initial population and nonzero inter-species relationships of α, β, and γ, we may still acquire +flip bifurcation for 1D logistic equation[39] for the prey at the absence of the predator. A flip bifurcation +is a counterpart in discrete dynamic system to describe the concept of periodic doubling in the continuous +dynamic system[40]. +Figure 2b shows the Lyapunov exponents. Rosenstein algorithm did not show results in in this case, +because singular value decomposition did not converge when doing linear least squares, meaning that positive +or negative infinity appeared when we tried to deal with pseudo-inverse matrix. Eckmann Y cannot work, +either, for y = 0 in the whole range. We may see that for overall trend of λx, both algorithms have λx < 0 +for µ0 < 3.0, whereas λx has both positive and negative values for µ0 > 3.5. +It is widely accepted[41] +that values of Lyapunov exponents occur interchangeably between positive and negative infer chaos, which +is consistent with the shaded area in Figure 2a. Another inconsistency occurs with 3.0 < µ0 < 3.5 with +λx > 0 for Equation but λx < 0 for Eckmann X, where x exhibits a flip bifurcation from 2-cycle into 4-cycle. +Nevertheless, this inconsistency may not be a problem for us to distinguish chaos from happening. There is +10 + +(a) +(b) +Figure 1: Competitive behavior and Lyapunov exponents of Normal. +11 + +0.6 +0.5 +0.4 +0.2 +0.1 +0.0 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +Logistic Map +0.25 +0.20 +0.15 - +y +0.10 +0.05 +000 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +μoLyapunov Exponents +0 +入x +-4 +-6 - +Equation +...... +Rosenstein +Eckmann X +Eckmann Y +-8 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +-0 +-2 +-4 - + g- +-8 + +-10 - +Equation +Rosenstein +Eckmann X +-12 +EckmannY +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +μoa break around µ0 = 2 for Eckmann X in both λx and λy, at which Equation shows a deep trench in λx. +Also, near µ0 ≈ 1.245, Equation shows a smaller trench while Eckmann X produces a rising tail. +Figure 3 studies the population in the course of time (iteration) at µ0 equal to 2.700 (1-cycle), 3.000 +(2-cycle where the flip bifurcation occurs) and 3.500 (4-cycle), 3.700 (at which the system goes into chaos), +3.845 (the system going back to more stable 3-cycle), and 3.945 (where the system returns to chaos again) +in the successive order. Zero predator population is obtained throughout course of time. +Figure 4 studies topological types of fixed points. Plots of imaginary or real part of eigenvalues in Eq.(22) +were evaluated at the fixed points E′ +1 in Eq.(23b) (as shown in the upper row), E′ +2 in Eq.(23c) (as shown +in the middle row), and E′ +3 in Eq.(23d) (as shown in the bottom row). Color-bar at the right-hand side +stands for various growth rate µ0, while circles with four different sizes in the legend represent, from the +smallest to the biggest, topological types of sink, source, saddle, and non-hyperbole. We may see that ω0 +and ω1 at the fixed points E′ +1 were pure real numbers with absolute values greater than 1, making the fixed +point a source for all µ0. While ω0 and ω1 at the fixed points E′ +2 were also pure real numbers, the absolute +values vary across 1, making the fixed point E′ +2 topological types of sink, source, and saddle, with possible +non-hyperboles occurring at either low µ0 = 1 or high µ0 = 3.75 if we apply Theorem 2.3(d). +Figure 5 shows absolute values of eigenvalues ω0 and ω1 on fixed points E′ +1, E′ +2 and E′ +3. Topological types +of fixed points may be double-checked more straightforwardly with the figure. The first column shows that +E′ +1 is a source because ω0 and ω1 are always greater than 1. The second column demonstrates that E′ +2 are +a sink when µ0 < 2.141, and when 2.141 < µ0 < 3, E′ +2 is a saddle. Non-hyperboles can also be examined +at µ0 = 1 for E′ +2, at µ0 = 3 for E′ +2 (in both cases ω0 = 1 and ω1 ̸= 1), and at µ0 = 3.75 for E′ +2 and E′ +3 (in +which ω0 ̸= 1 and ω1 = 1). That µ0 = 3.75 is located in chaos region, making the fixed point vulnerable to +nonlinear terms in the dynamic system. At last, when µ0 > 3, E′ +2 is a source. +Figure 6 shows phase portrait and phase space diagram about Standard. µ0 values are represented by the +color bar at the right-hand side. Figure 6a refers to phase portrait, where topological types are also shown +in the legend. An isolated initial coordinate (0.1, 0.50) marked as a source at the upper-left corner. Flip +bifurcation occurs at (0.665, 0, 000). No limit circles were found in the flip bifurcation. The oblique black +straight line, starting from (0.732, 0.000) to (0.689, 0.062), consisting of fixed points E′ +3 that is enclosed by a +thicker yellow cloak indicates that, along the axis, E′ +3 is a saddle, with nonzero y values. This result seems +to contradict to the previous one, saying that predator population is always zero. However, since our initial +population is (x, y)=(0.1, 0.5), it does not lie in the above range. Therefore, the system with the chosen +inter-species constants is not attracted by the saddle points along the oblique black line, confirming that +with none-zero initial population of the predator and non-zero inter-species constants, the predator could +appear, but only for a while. Afterwards, the predator dies out in the course of time, leaving the prey to be +the only surviving species in the paradisaic. This observation may explain why the predator species may not +survive long in some specific ecological system. Further, Figure 6b is phase space diagram for the prey. It +is meaningless to discuss phase space diagram for the predator because there are only two points, (0.0, 0.0) +and (0.5, −0.5), in this case. +12 + +(a) +(b) +Figure 2: Bifurcation diagram and Lyapunov exponents of Standard. +13 + +1.0 +0.8 +0.6 - +X +0.4 +0.2 +0.0 +0.04 +0.02 +0.00 +0.02 +0.04 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +uoLyapunov Exponents +-2 +-6 +Equation +Eckmann X +EckmannY +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +0 +-2 +-4 +-6 +-8 - +Equation +Eckmann X +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +μo(a) +(b) +(c) +(d) +(e) +(f) +Figure 3: Population vs. iteration of Standard. +14 + +1.0 +0.8 +0.6 +μo = 3.945 +0.4 +0.2 +0.0 +0 +25 +50 +75 +100 +125 +150 +175 +200 +0.5 +0.4 +0.3 +n +0.2 +0.1 +0.0 +0 +25 +50 +75 +100 +125 +150 +175 +200 +n1.0 +0.8 +0.6 +μo = 2.700 +0.4 +0.2 +0.0 +0 +25 +50 +75 +100 +125 +150 +175 +200 +0.5 +0.4 +0.3 +0.1 +0.0 +25 +50 +75 +100 +125 +150 +175 +200 +n1.0 +0.8 +0.6 +μo = 3.000 +0.4 +0.2 +0.0 +0 +25 +50 +75 +100 +125 +150 +175 +200 +0.5 +0.4 +0.3 +0.2 +0.1 +0.0 +0 +25 +50 +75 +100 +125 +150 +175 +200 +n1.0 +0.8 +0.6 +0.4 +Ho= 3.500 +0.2 +0.0 +0 +25 +50 +75 +100 +125 +150 +175 +200 +0.5 +0.4 +0.3 +0.2 +0.1 +0.0 +0 +25 +50 +75 +100 +125 +150 +175 +200 +n1.0 +0.8 +0.6 +μo = 3.700 +0.4 +0.2 +0.0 +0 +25 +50 +75 +100 +125 +150 +175 +200 +0.5 +0.4 +0.3 +0.2 +0.1 +0.0 +0 +25 +50 +75 +100 +125 +150 +175 +200 +n1.0 +0.8 +0.6 +0.4 +0.2 +0.0 +0 +25 +50 +75 +100 +125 +150 +175 +200 +0.5 +0.4 +0.3 +0.2 +0.1 +0.0 +25 +50 +75 +100 +125 +150 +175 +200 +nFigure 4: Analysis on eigenvalues for the case of Standard. Eigenvalues described in Eq.(22) for fixed points +E′ +1 (Eq.(23b)), E′ +2 (Eq.(23c)), and E′ +3 (Eq.(23d)) are plotted in the upper row, middle row, and lower row, +respectively. Types of topology are indicated with circles of various sizes. Color bar stands for different µ0. +. +Figure 5: Absolute values of eigenvalues vs. growth rate at fixed points for Standard. Prominent coordinates +that help us understand stability and distinguish the topological type about a fixed point are recorded as +follows. Upper middle panel: (1.244, 0.000), and (3, 75, 1, 00). Upper-right corner panel: (3.75, 1.00). E′ +2 and +E′ +3 are both non-hyperbolic at µ0 = 1, 3, and 3.75. +15 + +0.04 +0.04 +10.4 +0.02 +0.02 +10.2 +Im(wo) +Im(wi) ++ 00°0 +0.00 +0.02 - +0.02 +9.8 +0.04 - +0.04 +9'6 +2.10 +2.15 +2.20 +2.25 +2.30 +2.35 +2.40 +10.4 +10.2 +10.0 +9.8 +9.6 +2.10 +2.15 +2.20 +2.25 +2.30 +2.35 +2.40 +0.04 - +0.04 +1.00 +0.02 +0.02 +0.75 +Im(wo) +Im(wi) +0.00- +0.00 +0.02 - +0.02 +0.25 +0.04 +0.04 +0.00 +2.0 +-1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.000.25 +0.500.75 +1.00 +1.251.50 +1.75 +2.00 +1.8 +0.04 +0.04 +1.6 +0.02 +0.02 +Im(wo) +Im(wi) +0.00 - +0.00 +0.02 - +0.02 +1.2 +0.04 +0.04 +1.0 J +2.8 +2.6 +2.4 +2.2 +2.0 +-1.8 +1.6 +1.0 +1.2 +1.4 +1.6 +1.8 +1.6 +1.8 +2.0 +2.2 +2.4 +2.6 +2.8 +Re(wo) +1°ml +μo +Re(wi) +Type +source +non-hyperbolic +sink +O +saddle1.8 +10.4 +1.0 - +1.6 +0.8 +10.2 +0.6 +1.4 +0.4 +1.2 +9.8 +0.2 +9.6 +1.0 +0.0 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +2.40 +2.00 +2.8 - +1.75 +2.35 +2.6 +1.50 +2.30 +1.25 +2.4 +1.00 +2.2 +0.75 +2.20 +2.0 +0.50 +2.15 +0.25 +1.8 +2.10 - +0.00 +1.6 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +μo. Column for Ei +μo. Column for E2 +μo. Column for E3(a) +(b) +Figure 6: +Analysis on phase +portrait and phase space dia- +gram about Standard. Topolog- +ical types of sink, source, sad- +dle and non-hyperbole are repre- +sented with different sizes from +the smallest to the largest as +shown in the legends. +Oblique +black line in Figure 6a indicates +saddle fixed points, demonstrat- +ing that the predator cannot sur- +vive long in the ecological sys- +tem. Figure 6b shows the phase +space of x. Phase diagram of the +predator is not shown here, be- +cause it only contains two points +(0.0, 0.0), and (0.5, −0.5), which +makes the figure boring. +16 + +Type +0.5 +non-hyperbolic +sink +saddle +O +source +0.4 +y +0.2 +0.1 +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +μo +X0.6 +0.4 +0.2 - +0.0 +0.4 +0.6 +-0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +μo3.3 +Paraclete +Figure 7a represents two sets of overlapping bifurcation diagrams: one is the Standard, the other with +vorticella-shaped, which starts to appear after µ0 = 2.29. +Between 2.29 < µ0 < 2.45, shaded regions +appearing vertically with gaps are not chaos but transient states of population. At µ0 = 3.256, the vorticella- +shaped has bifurcation that start to be chaotic, at which we would explain later that it should be classified +as Neimark-Sacker bifurcation, and mingles together with the flip bifurcation of Standard after µ0 = 3.46, at +which the four-cycles occurs. +Figure 7b shows the Lyapunov exponents for Paraclete. For the same reason in Standard, Rosenstein +algorithm did not show results in in this case, either. We may see that for overall trend of λx, algorithms +of both Equation and Eckmann X have λx < 0 for µ0 < 3.0, whereas λx has both positive and negative +values for µ0 > 3.0, inferring chaos. Eckmann Y only shows a small portion, not spectrum, because y = 0 for +µ0 < 2.29 leads to failure on producing full spectrum of both λx and λy. Therefore, where we may see only +some segment of λx after µ0 > 3.0 that almost overlaps with the curve of Eckmann X. Inconsistency also +occurs at low growth rate in λy spectrum, for Equation shows stern-drooping tails while Eckmann X shows +a raising-up one. +Figure 8 studies the population in the course of time (iteration) at µ0 equal to 2.790 (stable population), +3.025 (2-cycle), 3.255 (at which Neimark-sacker bifurcation is on the way), 3.460 (4-cycle), 3.845 (3-cycle in +Standard), and 3.945 (chaos region) in the successive order. +Figure 9 studies topological types of fixed points. Plots of imaginary or real part of eigenvalues in Eq.(22) +were evaluated at the fixed points E′ +1 in Eq.(23b) (as shown in the upper row), E′ +2 in Eq.(23c) (as shown in +the middle row), and E′ +3 in Eq.(23d) (as shown in the bottom row). Color-bar at the right-hand side stands +for various growth rate µ0, while circles with four different sizes in the legend represent, from the smallest +to the biggest, topological types of sink, source, saddle, and non-hyperbole. We may see that ω0 and ω1 at +the fixed points E′ +1 were pure real numbers with absolute values greater than 1, making the fixed point a +source for all µ0. While ω0 and ω1 at the fixed points E′ +2 were also pure real numbers, the absolute values +vary across 1, making the fixed point E′ +2 topological types of sink, source, and saddle, with only one possible +non-hyperbole occurring at |ω0| = 1. +Figure 10 shows absolute values of eigenvalues ω0 and ω1 on fixed points E′ +1, E′ +2 and E′ +3. Topological +types of fixed points may be double-checked more straightforwardly with the figure. The first column shows +that E′ +1 is a source because ω0 and ω1 are always greater than 1. The second column demonstrates that E′ +2 +are a sink when µ0 < 2.141, and when 2.141 < µ0 < 3, E′ +2 is a saddle. At last, when µ0 > 3, E′ +2 is a source. +As we look closely into the third column, it shows that E′ +3 is non-hyperbolic at µ0 ≈ 2.137 for ω0 ̸= 1 +and ω1 = 1, which is vulnerable to nonlinear terms in the dynamic system. It explains why the transient +state under that growth rate is not shown in Figure 7a. Also, bending points along curves plotted in the +third-column figures demonstrate that transient states occur within 2.258 < µ0 < 2.475. More interestingly, +the third column manifests that E′ +3 represents Neimark-Sacker bifurcation at µ0 = 3.25636 because of the fol- +lowing facts: first, ω0 and ω1 are complex conjugates with modulus 1, and second, as µ0 varies across 3.25636 +from smaller to larger value, topological type of E′ +3 changes from a sink (stable) to a source (unstable)[19]. +Figure 11 shows phase portrait and phase space diagram about Paraclete. µ0 values are represented by +the color bar at the right-hand side. Figure 11a refers to phase portrait, showing Neimark-Sacker bifurcation +established at (0.352, 0.068) with µ0 ≈ 3.256 at the center of limit circles. +Black straight lines indicate +oblique axis consisting of E′ +3, including sink and source, and horizontal axis composed of E′ +2, including +source and saddle, under different µ0. Topological types are also shown in the legend. Dots with larger µ0 +representing chaos spread outside around the limit circles. Figure 11b is phase space diagram for the prey +centered at (0.352, 0) and Figure 11c refers to phase space diagram for the predator centered at (0, 0.068). +Not surprisingly, the center of limit circles in Figure 11b has the same x value as that of Figure 11a. Similarly, +same y value for the center of limit circles in Figure 11c and in Figure 11a. +17 + +(a) +(b) +Figure 7: Bifurcation diagram and Lyapunov exponents of Paraclete. +18 + +1.0 +0.8 +0.6 +X +0.4 - +0.2 +0.0 +0.150 +0.125 +0.100 +>0.075 +0.050 +0.025 +0.000 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +uo0 +-2 - +-4 +6- +Equation +Eckmann X +EckmannY +-8 +2 +-2 +-4 - +-6于 +Equation +Eckmann X +-8 +Eckmann Y +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +μo(a) +(b) +(c) +(d) +(e) +(f) +Figure 8: Population vs. iteration of Paraclete. +19 + +1.0 +0.8 +0.6 +0.4 +0.2 +μo = 2.790 +0.0 +0 +25 +50 +75 +100 +125 +150 +175 +200 +0.150 +0.125 +0.100 +0.075 +0.050 +0.025 +0.000 - +0.025 +0.050 +25 +50 +75 +100 +125 +150 +175 +200 +n1.0 +0.8 +0.6 +0.4 +0.2 +μo = 3.025 +0.0 +0 +25 +50 +75 +100 +125 +150 +175 +200 +0.150 +0.125 +0.100 +0.075 +0.050 +0.025 +0.000 +0.025 +-0.050 +0 +25 +50 +75 +100 +125 +150 +175 +200 +n1.0 +0.8 +0.6 +X +0.4 +0.2 +μo = 3,255 +0.0 +0 +25 +50 +100 +125 +150 +175 +200 +0.150 +0.125 +0.100 +0.075 +0.050 +0.025 +0.000 +0.025 +0.050 +0 +25 +50 +75 +100 +125 +150 +175 +200 +n1.0 +0.8 +0.6 +ux +0.4 +0.2 +μo = 3.460 +0.0 +0 +25 +50 +75 +100 +125 +150 +175 +200 +0.150 +0.125 +0.100 +0.075 +0.050 +0.025 +0.000 +-0.025 +-0.050 +0 +25 +50 +75 +100 +125 +150 +175 +200 +n1.0 +0.8 +0.6 +0.4 +0.2 +以o -:3.845. +0.0 +0 +25 +50 +75 +100 +125 +150 +175 +200 +0.150 +0.125 +0.100 +0.075 +0.050 +0.025 +0.000 +-0.025 +-0.050 +0 +25 +50 +75 +100 +125 +150 +175 +200 +n1.0 +0.8 +0.6 +n +X +0.4 +0.2 +3.945 +0.0 +0 +25 +50 +75 +100 +125 +150 +175 +200 +0.150 +0.125 +0.100 +0.075 +0.050 +0.025 +0.000 +-0.025 +-0.050 +0 +25 +50 +75 +100 +125 +150 +175 +200 +nFigure 9: Analysis on eigenvalues for the case of Paraclete. Legends have the same meaning described in +Figure 4. +Figure 10: Absolute values of eigenvalues vs. growth rate at fixed points for Paraclete. Important coordinates: +Upper middle panel (2.14, 1.00). Upper-right corner panel (2.137, 1.000), (2.457, 0.440), and (3.256, 1.000). +Lower-right corner panel (2.258, 0.000), (2.475, 0.427), and (3.256, 1.000). +20 + +0.04 +0.04 +515.0 +0.02 +0.02 +512.5 +Im(wo) +Im(wi) +0.00 +0.00 +0.02 - +0.02 +507.5 +0.04 - +0.04 +505.0 +2.010 +2.015 +2.020 +2.025 +2.030 +2.035 +2.040 +-516 +-514 +-512 +510 +508 +-506 +504 +2.010 +2.015 +2.020 +2.025 +2.030 +2.035 +2.040 +3 +0.04 - +0.04 +2.5 +0.02 +2.0 - +0.02 +Im(wo) +Im(wi) ++00'0 +0.00 +0.02 - +1.0 +0.02 +0.5 +0.04 +0.04 +EEEELEEF +0.0 +2.0 +1.5 +-1.0 +0.5 +0.0 +0.5 +1.0 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +0.000.250.500.751.00 +1.251.501.752.00 +2 +1.25 于 +0.00 +1.50 +1.00 +0.25 +0.50 +1.25 +Im(wi) +0.50 +0.75 +1.00 +0.75 +0.25 +1.25 +0.50 +0.00 +0.8 +0.6 +0.4 +0.2 +0.0 +0.2 +0.4 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +Re(wi) +[wol +μo +Re(wo) +Type +source +non-hyperbolic +saddle +sink516 +2.5 +1.6 +514 +1.4 +2.0 +512 +1.2 +1.5 +1.0 +1.0 +508 +0.8 +506 +0.5 +0.6 +504 +0.0 +0.4 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +2.040 +2.00 +1.2 +1.75 +2.035 +1.50 +1.0 +2.030 +1.25 +0.8 - +1.00 +0.6 +0.75 +2.020 +0.4 +0.50 +2.015 +0.2 +0.25 +2.010 +0.00 +0.0 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +μo. Column for Ei +μo. Column for E2 +μo. Column for E3(a) +(b) +(c) +Figure 11: +Analysis on phase +portrait +and +phase +space +diagram +about +Paraclete. +(11a)Phase +portrait +shows +Neimark-Sacker bifurcation es- +tablished at (0.352, 0.068) with +µ0 ≈ 3.256 located at the center +of limit circles. +Oblique axis +and horizontal axis consisting +of E′ +3 +and E′ +2 +with different +µ0, respectively. +We may see +from the legend that topological +types of E′ +3 +are mostly sink +and source, while those of E′ +2 +are mostly source and saddle. +Phase portrait and phase space +diagram for the prey have same +x in (11b) for the center limit +circle. Similarly, phase portrait +and phase space diagram for the +predator have same y in (11c) +for the center limit circle. +21 + +Type +non-hyperbolic +sink +saddle +0.14 +source +0.12 +0.10 +0.08- +0.06- +0.04 - +0.02 +0.00 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +μo0.6 +0.4 +0.2 - +0.0 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +μo0.075 +0.050 +0.025 +0.000 - +0.025 +0.050 +0.075 +0.100 +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0.12 +0.14 +V +μo3.4 +Extinction +Figure 12a shows the bifurcation diagram of Extinction. There is a flip bifurcation for x around µ0 ≈ 2.99; +however, the 2-cycle collides at µ0 ≈ 3.165 and returns back to 1-cycle. Shaded regions between 3.165 < µ0 < +3.434 for x and 3.10 < µ0 < 3.343 for y do not refer to chaos but belong to transient states. Afterwards, the +bifurcation diagram goes back to Normal for both x and y, and represent pleasant conditions of predictable +values before x goes into 3-cycle at µ0 ≈ 3.828 as in Standard, where y drops to zero cliff-fallingly at +µ0 ≈ 3.824, manifesting to extinction of the predator for the prey is around the 3-cycle state. Fortunately, +there could be still few little chances for the predator to survive at (µ0, y) = (3.845, 0.219), (3.887, 0.226), +and (3.897, 0.228), where we may see three isolated fixed points appear with a vertical tail of transient states. +When the prey becomes fully chaotic, the predator population reduces back to zero dramatically again, and +never has any further opportunity to rise back. This astonishing phenomenon could be the most profound +finding in the study, which states that the prey in chaos generated by overpopulation of the predator would +erase the entire predator species. +Figure 12b shows Lyapunov exponents for Extinction that is similar to those in Figure 1b, except for the +regions at 3.0 < µ0 < 3.18 and µ0 > 3.86, the former showing a bum by Equation, which is also the same +region for the prey to be in 2-cycle, and the latter presenting chaos for x and extinction for y. Discrepancy +appears for Eckmann Y that is different from the other when µ0 < 1.662, where it shows a decreasing tail +while the other algorithms show an increasing trend. Unlike Figure 1b, whose results show that Eckmann +X is always in between Rosenstein and Eckmann Y in the range of 1.5 < µ0 < 3.0, Figure 12b shows more +intertwines at µ0 = 1.886 for λx, and at µ0 = 1.717 and µ0 = 1.891 for λy. Similar to Figure 1b, the four +algorithms also divides into two groups of curves below µ0 = 1.717, with Rosenstein and Eckmann X going +upward, together with Equation and Eckmann Y going downward. +Figure 13 shows population iteration of Extinction. As we can see, flip bifurcation starts at µ0 = 3.000 as +in Figure(13a), while in Figure(13b) the two fixed points collide, and after transient states (n > 200), they +have tendency to merging into a single fixed point, as we explained earlier in Figure 12a on the characteristics +about the shaded region between 3.165 < µ0 < 3.434 for x. We further demonstrate that at µ0 = 3.500 in +Figure(13c), after transient(n > 175), bifurcation collapses to one. Furthermore, 3-cycle is opened in x at +µ0 = 3.84, as indicated in Figure(13d), with extinction of y at the exactly the same moment. However, a +sunlight of survival for y is shed on the window at µ0 = 3.845, as shown in Figure(13e), where the two species +may still exist under predictable population. Finally, Figure 13f portends the predator extinction under +chaos of the prey. +Figure 14 studied topological types of fixed points. +Plots of imaginary or real part of eigenvalues in +Eq.(22) were evaluated also at the three fixed points E′ +1, E′ +2, and E′ +3 as in Figure 4. We may see that ω0 +at the fixed points E′ +1 is pure real numbers with absolute values greater than 1, whereas ω1 keeps the value +−1000 for all µ0, making the fixed point a source (unstable) for all µ0. While ω0 < 1 and ω1 > 1 at the +fixed points E′ +2, the fixed point E′ +2 has a topological type of saddle, with possible non-hyperboles occurring +at |ω0| = 1 and |ω1| = 0. +Stability of fixed points may also be examined in Figure 20. Figures at the first column shows that that +E′ +1 is a source, for both eigenvalues have absolute values greater than 1. In the second column, we see that +E′ +2 changes its stability from a sink to source when µ0 varies at 3, at which flip bifurcation occurs; meanwhile, +E′ +3 changes from source to sink. +Figure 16 shows phase portrait and phase diagrams for Extinction. No limit circles are found in this +particular case. +22 + +(a) +(b) +Figure 12: Bifurcation diagram and Lyapunov exponents of Extinction. +23 + +Logistic Map +1.0 +0.8 - +0.6 +0.4 +0.2 +0.0 +0.20 - +0.15 - +0.10 - +0.05 - +0.00- +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +μoLyapunovExponents +0 +-2 +-6- +Equation +Rosenstein +Eckmann X +Eckmann Y +8 +0- +-2- +-4 +-6 +-8 + +Equation +-10 - +Rosenstein +Eckmann X +EckmannY +-12 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +μo(a) +(b) +(c) +(d) +(e) +(f) +Figure 13: Population vs. iteration of Extinction. +24 + +1.0 - +0.8 - +0.6 +n +0.4 +μo = 3.000 +0.2 +0.0 +0 +25 +50 +75 +100 +125 +150 +175 +200 +0.30 +0.25 - +0.20 +0.15 +0.10 - +0.05 +0.00 +0.05 +0 +25 +50 +100 +125 +150 +175 +200 +n1.0 - +0.8 - +0.6 +u +0.4 +μo = 3.375 +0.2 +0.0 +0 +25 +50 +75 +100 +125 +150 +175 +200 +0.30 +0.25 - +0.20 +0.15 +0.10 - +0.05 +0.00 +0.05 +0 +25 +50 +75 +100 +125 +150 +175 +200 +n1.0 - +0.8 - +0.6 +n +0.4 +μo = 3.500 +0.2 +0.0 +0 +25 +50 +75 +100 +125 +150 +175 +200 +0.30 +0.25 - +0.20 +0.15 +0.10 - +0.05 +0.00 +0.05 +0 +25 +50 +75 +100 +125 +150 +175 +200 +n1.0 +0.8 - +0.6 +0.4 +μo = 3.840 +0.2 +0.0 +0 +25 +50 +75 +100 +125 +150 +175 +200 +0.30 +0.25 - +0.20 - +0.15 +0.10 - +0.05 +0.00 - +0.05 +0 +25 +50 +100 +125 +150 +175 +200 +n1.0 - +0.8 - +0.6 +0.4 +μo = 3.845 +0.2 +0.0 +0 +25 +50 +75 +100 +125 +150 +175 +200 +0.30 +0.25 - +0.20 +0.15 +0.10 - +0.05 +0.00 +0.05 +0 +25 +50 +75 +100 +125 +150 +175 +200 +n1.0 - +0.8 - +0.6 +C +0.4 +μo = 3.945 +0.2 +0.0 +0 +25 +50 +75 +100 +125 +150 +175 +200 +0.30 +0.25 - +0.20 +0.15 +0.10 - +0.05 +0.00 - +0.05 +0 +25 +50 +75 +100 +125 +150 +175 +200 +nFigure 14: Analysis on eigenvalues for the case of Extinction. Legends have the same meaning described in +Figure 4. +Figure 15: Absolute values of eigenvalues vs. growth rate at fixed points for Extinction. From the figures at +the first column, it is clearly shown that E′ +1 is a source. Also, at µ0 = 3 where flip bifurcation occurs, E′ +2 +changes its stability from a sink to source, whereas E′ +3 from source to sink. +25 + +0.04 +0.04 +1040- +0.02 +0.02 +1020 +Im(wo) +Im(wi) +0.00 +0.00 +0.02 - +0.02 +980 - +0.04 +0.04 +096 +2.00102.00152.0020 +2.00252.00302.00352.0040 +1040 +1020 +-1000 +980 +-960 +2.0010 2.0015 2.0020 2.00252.0030 2.00352.0040 +3 +1.5 +0.04 +0.04 +0.02 +0.02 +Im(wo) +Im(wi) +1.0 +Iml +0.00- +0.00 +0.02 - +0.02 +0.5 +0.04 +0.04 ++ 00 +2.0 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +0.000.250.500.751.00 +1.251.501.75 +2.00 +0.04 +0.04 +1.5 +0.02 +0.02 +Im(wo) +Im(wi) +00°0 +0.00 +0.02 - +0.02 +0.5 +0.04 +0.04 +1.8-1.6-1.4 +-1.2 +-1.0 +0.80.60.40.2 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +1.8 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +1.8 +Re(wo) +Re(wi) +[wol +μo +Type +source +non-hyperbolic +sink +O +saddle1.8 +1.4 +1040 +1.6 +1.2 +1.4 +1020 +1.0 +1.2 +0.8 +1.0 +0.6 +0.8 +980 +0.6 +0.4 +0.4 +0.2 +960 +0.2 +0.0 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +1.8 +2.0040 +2.00 +1.6 +1.75 +2.0035 +1.4 J +1.50 +2.0030 +1.2 +1.25 +1.0 +1.00 +0.8 +0.75 +2.0020 +0.6 +0.50 +2.0015 +0.4 +0.25 +0.2 +2.0010 +0.00 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +μo. Column for Ei +μo. Column for E2 +μo. Column for Es(a) +(b) +(c) +Figure 16: +Analysis on phase +portrait and phase space dia- +gram about Extinction. No limit +circles were found in this case. +26 + +Type +0.25 +non-hyperbolic +sink +O +saddle +source +0.15 - +y +0.10 - +0.05 +0.00 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +μo0.6 +0.4 +0.2 - +0.0 +0.4 +0.6 +-0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +μo0.04 +0.02 +0.00 +0.02 +-0.04 - +-0.06 +-0.08 +0.10 +0.00 +0.05 +0.10 +0.15 +0.20 +μo +y3.5 +Vorticella Strange +The last bifurcation type in our study is Vorticella Strange, which means that bifurcation diagram looks +like a vorticella, only with more complicated internal structures. +Figure 17a shows bifurcation diagram. +When µ0 < 2.0, x grows steadily without y. After presence of the predator, population of the prey starts to +decrease. Both species show predictable population before µ0 = 3.025, at which we classify a Neimark-Sacker +bifurcation as in Paraclete. Transient states occur between 3.025 < µ0 < 3.200, as indicated in the shaded +region. Later, a 6-cycle appears at µ0 = 3.24, which is also confirmed in Figure 18b. The 6-cycle becomes +3-cycle at µ0 = 3.40, as shown in Figure 18c, followed by chaos at µ0 = 3.485, which is also demonstrated +in Figure 18d. The system goes back to 6-cycle around µ0 = 3.540, as we may also confirm in Figure 18e. +Afterwards, the system goes back to chaos, as shown in Figure 18f. +Figure 17b shows Lyapunov spectrum of Voticella Strange. All four algorithms barely show positive spectra +for both x and y before µ0 = 3.5. On the contrary, afterµ0 = 3.5, four algorithms show positive Lyapunov +exponents, where the system falls into chaos. For λx, Eckmann Y shows a decreasing tail below µ0 < 1.5, +inconsistent from the other algorithms. Besides Equation showing two valleys between 1.500 < µ0 < 2.304 +and 3.224 < µ0 < 3.463, the other three algorithms only provide flat spectra in the two regions. +Figure 19 shows analysis on eigenvalues of Vorticella Strange. At the first row, we see that both eigenvalues +have zero imaginary parts, with absolute real parts of both greater than 1 (see also first column in Figure +20). Thus, E′ +1 is a source. Similar analysis may also be done on E′ +2, as shown in the second row in Figure +19 as well as the second column in Figure 20. At µ0 = 2.0, it turns from a sink to a saddle, and it maintains +as a saddle between 2.0 < µ0 < 3.0, after which it turns to a source. Finally, the third column in Figure +20 demonstrates that Neimark-Sacker bifurcation occurs at µ0 ≈ 3.025, with coordinates (0.341, 0.377), as +shown in Figure 21a. +Similar bifurcation diagram was found by Hu et al.[10], and was identified as the Hopf bifurcation. +However, the criteria for Hopf bifurcation in the two-dimensional system includes that the two eigenvalues +are purely conjugate imaginary pair with zero real part[42]. Since Figure 19 shows that ω1 has a non-zero +real part, our Vorticella Strange cannot be a Hopf bifurcation. +For the sake of integrity, phase portrait and phase space diagram of Vorticell Strange are also shown in +Figure 21. Similar detailed meaning is discussed in the Section of Paraclete. +4 +Conclusions +We successfully built up a discrete-time model of two-dimensional mapping that resembles features of dif- +ferential Lotka-Volterra Equations. We studied the topological types of fixed points, and found out that +fascinating feather-like structures were formed around limit circles pierced through by the axis of fixed points +in the phase portraits and phase space diagrams under various growth rates with Neimark-Sacker bifurcation. +We further divided our dynamical systems into five categories by different shapes of bifurcation diagrams: +Normal, Standard, Paraclete, Extinction, and Vorticella Strange. In every case, we studied the stability and +topological types of the fixed points by criteria discussed in Lamma 1. In addition to plot the population +vs. iteration, we also calculated Lyapunov exponents both by eigenvalues of Jacobian of mapping functions, +and by Rosenstein algorithm and Eckmann et al. algorithm. Discrepancies clearly showed that Lyapunov +exponents calculated by time-series algorithms may be unreliable within the range of chaos and at low growth +rates, as well as when the Lyapunov exponents change dramatically. Furthermore, our model not only re- +gained the 1D logistic mapping of the prey under zero predator yet with non-zero initial predator population +and with non-zero inter-species constants, but it also showed the normal competitiveness of the prey and +the predator without chaos. The quintessence in the current research was that, besides the possibility for +the prey and the predator to become chaotic altogether, it is also probable for the predator to go extinct at +the chaotic state of the prey. In other words, human overpopulation would cause chaos in natural resources, +ultimately in return erase the entire human race. Luckily, even under this difficult circumstance slim chance +is still left upon us to continue our race under some specific growth rate, as we may see some isolated fixed +points remain in the predator bifurcation diagram. +Our model may inspire conjecture on other relationships between two physical quantities, because mathe- +matically what we demonstrated was that one quantity may dramatically reduce to zero at the state of chaos +27 + +(a) +(b) +Figure 17: Bifurcation diagram and Lyapunov exponents of Vorticella Strange. +28 + +Logistic Map +1.0 +0.8 +0.6 +0.4 +0.2 - +0.0 +1.0 +0.8 - +0.6 +0.4 +0.2 - +0'0 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +μoLyapunov Exponents +0- +-2 +Equation +6 +Rosenstein +Eckmann X +Eckmann Y +2 +F9- +Equation +8 +Rosenstein +Eckmann X +EckmannY +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +uo(a) +(b) fig: +(c) +(d) +(e) +(f) +Figure 18: Population vs. iteration of Vorticella Strange. +29 + +1.0 - +μo = 3,025 +0.8 +0.6 +0.4 +0.2 +0.0 +50 +75 +100 +125 +150 +175 +200 +1.0 +0.8 +0.6 +0.4 +.. +0.2 +0.0 +0 +25 +50 +75 +100 +125 +150 +175 +200 +n1.0 - +μo = 3.240 +0.8 +0.6 +0.4 +0.2 +0.0 +25 +50 +75 +100 +125 +150 +175 +200 +1.0 +0.8 +0.6 +0.4 +0.2 +0.0 +0 +25 +50 +75 +100 +125 +150 +175 +200 +n1.0 - +μo = 3.400 +0.8 +0.6 +0.4 +0.2 +0.0 +25 +50 +75 +100 +125 +150 +175 +200 +1.0 +0.8 +0.6 +0.4 +0.2 +0.0 +0 +25 +50 +75 +100 +125 +150 +175 +200 +n1.0 +μo = 3.485 +0.8 +0.6 +0.4 +0.2 +0.0 +0 +25 +50 +75 +100 +125 +150 +175 +200 +1.0 +0.8 +0.6 +0.4 +0.2 +0.0 +0 +25 +50 +75 +100 +125 +150 +175 +200 +n1.0 +μo = 3.540 +0.8 +0.6 +0.4 +0.2 +0.0 +0 +25 +50 +75 +100 +125 +150 +175 +200 +1.0 +0.8 +0.6 +0.4 +0.2 +0.0 +0 +25 +50 +75 +100 +125 +150 +175 +200 +n1.0 +μo = 3.700 +0.8 +0.6 +0.4 +0.2 +0.0 +0 +25 +50 +75 +100 +125 +150 +175 +200 +1.0 +0.8 +0.6 +0.4 +0.2 +0.0 +25 +50 +75 +100 +125 +150 +175 +200 +nFigure 19: Analysis on eigenvalues for the case of Vorticella Strange. +Legends have the same meaning +described in Figure 4. +. +Figure 20: Absolute values of eigenvalues vs. growth rate at fixed points for Vorticella Strange. Impor- +tant coordinates include: Upper middle panel (3.025, 1.935). Upper-right corner panel (2.299, 0.491), and +(3.035, 1.000). Lower-left corner panel (2.069, 0.000), and (2.370, 0.490). +30 + +48.5 +0.04 +0.04 +48.4 +0.02 +Im(wi) +0.02 +Im(wo) +0.00 - +0.00 + +0.02 +0.02 +48.2 +0.04 +0.04 +2.02 +2.03 +2.04 +2.05 +2.06 +2.07 +-48.50-48.45-48.40-48.35-48.30-48.2548.20-48.15 +2.02 +2.03 +2.04 +2.05 +2.06 +2.07 +3 +0.04 +0.04 +0.02 +0.02 +2 +Im(wo) +Im(wi) +ml +00°0 ++00'0 +0.02 +0.02 +0.04 +0.04 +0+ +2.0 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +0.000.250.50°0.75 +1.00 +1.25 +1.501.75 +1.25 +0.00 +1.50 +1.00 +0.25 +Im(wi) +0.50 +1.25 +0.50 +0.75 +0.25 +1.00 +0.75 +0.00 +1.25 +0.50- +0.6 +0.4 +0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +Re(wo) +Re(wi) +[wol +μo +Type +source +non-hyperbolic +saddle +sink48.50 +1.6 +48.45 +2.5 +1.4 +48.40 +2.0 +48.35 +1.2 +1.5 +1.0 +48.25 +1.0 +0.8 +48.20 +0.5 +0.6 +48.15 +0.0 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +2.07 +1.4 +1.75 +1.2 +2.06 +1.50 +1.0 +1.25 手 +2.05 +0.8 +[wol +1.00 +2.04 +0.6 +0.75 +0.50 +0.4 +2.03 +0.25 手 +0.2 +2.02 +0.00 +0.0 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +μo. Column for Ei +μo. Column for E2 +μo. Column for E3(a) +(b) +(c) +Figure 21: +Analysis on phase +portrait and phase space dia- +gram about Vorticella Strange. +(21a)Phase +portrait +shows +Neimark-Sacker bifurcation es- +tablished at (0.341, 0.377) with +µ0 ≈ 3.025. See also caption in +Figure 11. +31 + +Type +1.0 +non-hyperbolic +sink +O +saddle +source +0.8 - +0.6 +y +0.4 +0.2 +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +μo +X0.6 +0.4 - +0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +μo0.4 +0.2 +0.0 - +0.2 +0.4 - +0.6 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +μo +yof the other. Therefore, it could be highly possible that, for example, the superconducting state, which refers +to the zero resistance, may be achieved with chaos of some physical quantity that has relationship between +resistance in the form of simultaneous difference equations. +5 +Acknowledgment +We thank Pui Ching Middle School in Macau PRC for the kindness to support this research project. +References +[1] +Peter Roopnarine, Ecology and the Tragedy of the Commons, Sustainability 2013, 5(2), 749-773. +[2] +Ankit Kumar et al., A Computer-Based Simulation Showing Balance of the Population of Predator and +Prey and the Effects of Human Intervention, 2021 IOP Conf. Ser.: Mater. Sci. Eng. 1031 012049 +[3] +Cheng Sok Kin et al., Predicting Earth’s Carrying Capacity of Human Population as the Predator +and the Natural Resources as the Prey in the Modified Lotka-Volterra Equations with Time-dependent +Parameters, arXiv:1904.05002v2 [q-bio.PE]. 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Volterra, ”Variazioni e fluttuazioni del numero d’individui in specie animali conviventi”. Mem. Acad. +Lincei Roma. 2: 31–113, 1926. +[32] Lotka-Volterra Equation Wikipedia, https://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra e +quations +[33] Immanuel M. Bonze, Lotka-Volterra Equation and Replicator Dynamics: A Two-Dimensional Classifi- +cation, Biol. Cybern 48, 201-221, 1983. Eq 4 in our work has the same form as Eq 2 in this reference. +[34] Steve H. Strogatz, Nonlinear Dynamics and Chaos, p.156, Addison Wesley, 2nd Ed. +[35] Alan Wolf, Jack B. Swift, Harry L. Swinney and John A. Vastano, Determining Lyapunov Exponents +from a Time Series, Physica 16D, 285-317, 1985. +[36] Lorenzo Escot and Julio E. Sandubete Galan, ”A brief methodological note on chaos theory and its +recent applications based on new computer resources”, Revista: ENERGEIA (ISSN: 1666-5732) Vol. +VII, N´um. 1, 2020, pp 53-64. +33 + +[37] Sch¨olzel, Christopher. (2019, June 16). Nonlinear measures for dynamical systems (Version 0.5.2). Zen- +odo. http://doi.org/10.5281/zenodo.3814723 +[38] Codes and animations may be retrieved via https://github.com/weishanlee/LotkaVolterraChaos +[39] Stephen T. Thornton and Jerry B. Marrion, Classical Dynamics of Particles and Systems, 5th Ed., CH4, +ISBN-10:0534408966 +[40] IGI Global https://www.igi-global.com/dictionary/flip-bifurcation/11262 +[41] Herbert Goldstein, Charles Poole, John Safko, Classical Mechanics, 3rd Ed, CH11, ISBN-10:0321188977. +[42] J. Hale, and H Ko¸cak, Dynamics and Bifurcations. CH 11, Vol. 3, Berlin: Springer-Verlag (1991). +ISBN-13: 9783540971412. +34 + diff --git a/XNFJT4oBgHgl3EQf5S05/content/tmp_files/load_file.txt b/XNFJT4oBgHgl3EQf5S05/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1d97a656342b87ba8f28cc593749ceec21e5eac2 --- /dev/null +++ b/XNFJT4oBgHgl3EQf5S05/content/tmp_files/load_file.txt @@ -0,0 +1,2233 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf,len=2232 +page_content='Predator Extinction arose from Chaos of the Prey: the Chaotic Behavior of a Homomorphic Two-Dimensional Logistic Map in the Form of Lotka-Volterra Equations Wei Shan Lee∗, Hou Fai Chan, Ka Ian Im, Kuan Ieong Chan, and U Hin Cheang Pui Ching Middle School Macau Macao Special Administrative Region, People’s Republic of China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Abstract A two-dimensional homomorphic logistic map that preserves features of Lotka-Volterra Equations was proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' In order to examine the Lotka-Volterra chaos, in addition to ordinary iteration plots of population, Lyapunov exponents either calculated directly from eigenvalues of Jacobian of the 2D logistic mapping, or from time-series algorithms of both Rosenstein and Eckmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' were calculated, among which discrepancies were compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Bifurcation diagrams may be divided into five categories depending on different topological shapes, among which flip bifurcation and Neimark-Sacker bifurcation were observed, the latter showing closed orbits around limit circles in the phase portrait and phase space diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Our model restored the 1D logistic map of the prey at the absence of the predator, as well as the normal competing behavior between two species when the initial population of the two is equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' In spite of the possibility for two species going into chaos simultaneously, it is also possible that with the same inter-species parameters as normal but with predator population 10 times more than that of the prey, under certain growth rate the latter becomes chaotic, and former dramatically reduces to zero, referring to total annihilation of the predator species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Interpreting humans as the predator and natural resources as the prey in the ecological system, the aforementioned conclusion may imply that not only excessive consumption of the natural resources, but its chaotic state triggered by overpopulation of humans may also backfire in a manner of total extinction on human species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Fortunately, a little chance may exist for survival of human race, as isolated fixed points in bifurcation diagram of the predator reveals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Keywords Chaos, Neimark-Sacker Bifurcation, Logistic Map, Lyapunov Exponents, Lotka-Volterra Equa- tions, Extinction of Species 1 Introduction Understanding interactions between human beings and natural resources plays an important role in estab- lishing sustainable economy and society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Relationships of these two may be studied by the prey and predator model after we realize that humans beings may be regarded as the predator while natural resources may be thought of as the prey[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Afterwards, researches on the prey-predator models may be implemented to this field instinctively[2]-[3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Generally speaking, there are two main approaches of studies in the literature to this prey and predator model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' The first is to study differential equations, while the other is to study the iterations in difference equations, whose forms may be inspired by directly applying the forward Euler’s Scheme to acquire coun- terpart of the former[4]-[10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' The discrete model could be more promising than the continuous one, because it has more abundant dynamic characteristics in chaotic behaviors[8], whereas it would be more difficult for solutions to continuous models to reach chaos in low dimensional cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Taking some examples about the first approach, studies[11]-[12] have performed on chaos of Lotka-Volterra differential equations with dimensions higher than three, and researchers[13] claimed that it is impossible to reach chaos for two species in the form of differential Lotka-Volterra Equations, whose general solutions were obtained in sinusoidal forms by Evans and Findley[14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Additionally, based on the Lotka-Volterra model, Dunbar[15] confirmed the existence ∗email: wslee@g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='puiching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='mo 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='11669v1 [nlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='CD] 27 Jan 2023 of traveling wave solutions for two reaction diffusion systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Besides that, Das and Gupta[16] proposed solutions to the fractional-order time derivative Lotka-Volterra equations using an analytical approach for nonlinear problems known as the homotopy perturbation method (HPM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' On the other hand, there are also several studies on the discrete difference equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' For instance, Bessoir and Wolf [17] made pioneering contributions to the application of 1D logistic equation on biological and ecological studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' The same equation was also used to interpret, analyze and predict data according to the COVID-19 by many researchers[18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Mareno and English[19] implemented the 1D logistic equation to the coupled 2D logistic one, and demonstrated that for large growth rate the system underwent a Neimark- Sacker bifurcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' [20] imposed an equal individual effect intensity, corresponding to equal growth rate in the 1D logistic map, on the two oligopolists in the homomorphic Kopel model and observed three different kinds of bifurcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Furthermore, Elhadi and Sprott[21] proposed a two-dimensional mapping, one of which is the ordinary 1D logistic map while the other consists of a perturbation term of the former and is also modulated by the first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Shilnikov and Rulkov[22] studied chaos behaviors in two-dimensional difference equations that reproduced spike-bursting activities in biological neurons, improving further on the previous research based on the three-dimensional system of ODEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' In spite of applying the forward Euler’s Scheme to acquire the difference equation, researchers also made use of exponential forms corresponding to solutions on the differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' For example, Ishaque et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' [23] studied a three dimensional predator-prey-parasite model with an exponential form describing interactions among healthy or infected Tilapia fish as the prey, and Pelican birds as the predator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Tassaddiq et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' [24] worked on discrete-time exponential difference equation of Leslie-Gower predator-prey model together with a Holling type III functional response, and indicated the advantage on this type of discretization method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Previous study[25] suggested a heteromorphic term describing the decreasing effects on the predator that was only linear to the population of that species, contrary to the corresponding quadratic term in the prey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Hassell et al[26] applied the predator-prey model to insect parasitoids and anthropods, and found out that local movements of the two species may cause extermination of the entire ecological system with chaos, and it is difficult to maintain population stability for large growth rate of anthropods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Besides, researchers[27] also pointed out that human misbehavior may be the reason for an ecological system to go into chaotic states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' However, there is no convincing reason for the prey and the predator to have different forms in the difference equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Intuition in mathematical symmetry naturally came to our mind that a successful predator-prey difference model should resemble the symmetry structure as in Lotka-Volterra differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Moreover, solutions to Lotka-Volterra Equations in sinusoidal forms cannot explain extinction of species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' We proposed homomorphic two-dimensional logistic maps that preserve both forms of Lotka-Volterra Equations and the 1D logistic equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' In our model, we conjectured a quadractic form in both corresponding terms of the prey and the predator, treating both species on the equal stance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Structures of Bifurcation diagrams showed that there could be six different categories in our dynamic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' For each categories, we examined population iterations, phase portraits, phase space diagrams, and topological types of fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Lyapunov exponents either calcaulted from eigenvalues of Jacobian of the 2D mapping or from time-series algorithms either Rosenstein[28] or Eckmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' [29] were also calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Comparisons among those results were also discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' The advantages of our model include the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' First, we may be able to establish a standard bifurcation diagram of 1D logistic map about the prey with nonzero initial predator population, growth rates in both species, and predation parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Second, our model may also describe the normal behavior of rise and fall on the population of the two species when interacting with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Third, besides simultaneous chaos in both species, the main discovery in our research was that the predator may go extinct under the circumstance of chaos in the prey that the predator overpopulation should be blamed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' 2 Theorems We first review the one-dimensional logistic equation and the two-dimensional Lotka-Volterra Equations, comparing the similarity and difference between the two sets of equations, which inspires us on the idea to establish two-dimensional logistic equations that maintain important features about both of the above equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='1 Review on the 1D logistic equation and Litka-Volterra Equations To begin with, the one-dimensional logistic equation may be written as xn+1 = µ0xn(1 − xn), (1) where µ0 denotes to the growth rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' On the other hand, the two-dimensional Lotka-Volterra Equations[30], [31] describe interactions between the prey and predator in an environment where there is sufficient food supply for the prey, whose only natural enemy is the predator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' The formula may be written as follows: � � � � � � � dx dt = µ0x − µ1xy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' dy dt = −ν0y + ν1xy, (2a) (2b) where µ0, µ1, ν0, and ν1 are all positive inter-species parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' x denotes population of the prey while y, population of the predator, both being positive real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' µ0 and ν1 are, respectively, growth rate of the prey and the predator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' While µ1 refers to the parameter for predation to occur upon the prey at the presence of predator, ν0 refers to all effects that decrease population of the predator, which may include disease, death, or emigration[32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' With forward Euler’s scheme, one may immediately write the difference version of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (2) as follows �xn+1 − xn = µ0xn − µ1xnyn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' yn+1 − yn = −ν0yn + ν1xnyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (3a) (3b) However, there is an obvious drawback about the above simultaneous equations: it does not preserve the feature of 1D logistic equation, because the first term on the right hand side is either linear to xn or yn, whereas the right hand side in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (1) is quadratic to xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' We now discuss our idea on establishing the two-dimensional map that resembles interaction terms of the prey and the predator as in Lotka-Volterra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' First, we may rewrite the linear term of x on the right hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (2a) into quadratic, which looks like µ0x(1 − x), together with the homomorphic corresponding term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (2b) as ν0y(1 − y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Thus, the modified Lotka-Volterra Equations[33] are � � � � � � � dx dt = µ0x(1 − x) − µ1xy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' dy dt = −ν0y(1 − y) + ν1xy, (4a) (4b) whose resemblance in the form of difference equations are, therefore, �xn+1 − xn = µ0xn(1 − xn) − µ1xnyn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' yn+1 − yn = −ν0yn(1 − yn) + ν1xnyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (5a) (5b) Even though it seems more reasonable to direct resemblance of Lotka-Volterra Equations, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (5) fails to restore Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (1) when parameters other than µ0 are all set to be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Fortunately, we may modify this by dropping xn and yn terms on the left hand side of the above equations �xn+1 = µ0xn(1 − xn) − µ1xnyn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' yn+1 = −ν0yn(1 − yn) + ν1xnyn, (6a) (6b) which is the desired form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' We divided into two cases to study further about properties of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (5) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (6) in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='3 and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' But before that, in next subsection we first discuss a very important lemma that allows us to study stability behaviors of fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='2 Stability of fixed points Suppose a mapping of two-dimensional iterations xn+1 and yn+1 are written as xn+1 = f(x, y) and yn+1 = g(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' The Jacobian is, therefore, � � � � � � � � � � � J = ∂(f, g) ∂(x, y) = � ∂f ∂x ∂f ∂y ∂g ∂x ∂g ∂y � , (7a) (7b) whose eigenvalus are ω0 and ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' It is well known[6], [8] that a fixed point may be divided into the following four topological types based on their stability behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' First, it could be a sink and locally asymptotic stable if eigenvalues of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (7) satisfy |ω0| < 1 and |ω1| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Second, it could be a source and locally unstable if eigenvalues satisfy |ω0| > 1 and |ω1| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Third, a fixed point could be a saddle if one of the absolute values of the eigenvalues is greater than 1 while the other is smaller than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' At last, a fixed point could be non-hyperbolic if one of the absolute values of the eigenvalues is equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' The stability of a non-hyperbolic fixed point is fragile[34], which means that its stability is easily influenced by the small nonlinear terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Instead of calculating the range of eigenvalues directly, most of the time it is more convenient to work with the quadratic formula consisting of eigenvalues ω0 and ω1, namely, Ω(ω) = ω2 −Tr(J)ω +det(J), where Tr(J) and det(J) are trace and determinant of Jacobian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (7), respectively, and there could be a correspondence on the stability behavior around a fixed point between the roots of the quadratic formual, ω0 and ω1, which are also eigenvalues of Jacobian, through the following Lemma Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Let Ω(ω) = ω2 − Tr(J)ω + det(J), be a quadratic formula where where Tr(J) and det(J) are trace and determinant of Jacobian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (7), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Then 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' If Ω(1) > 0, then (a) |ω0| < 1 and |ω1| < 1 and hence the fixed point is a sink if and only if Ω(−1) > 0 and det(J) < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (b) |ω0| > 1 and |ω1| > 1 and hence the fixed point is a source if and only if Ω(−1) > 0 and det(J) > 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (c) One of |ω0| and |ω1| is smaller than 1 while the other greater than 1 and hence the fixed point is a saddle if and only if Ω(−1) < 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (d) Either |ω0| or |ω1| is equal to 1 and hence the fixed point is a non-hyperbolic whenever i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' ω0 = −1 and ω1 ̸= −1 if and only if Ω(−1) = 0 and Tr(J) ̸= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' ω0 and ω1 are a pair of complex conjugates and |ω0| = |ω1| = 1 if and only if |Tr(J)| < 2 and det(J) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' ω0 = ω1 = −1 if and only if Ω(−1) = 0 and Tr(J) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' If Ω(1) = 0, then either |ω0| or |ω1| has to be equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Therefore the fixed point is a non-hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Absolute value of the other root is greater than, equal to, or smaller than 1 if and only if, correspondingly, absolute value of det(J) is greater than, equal to, or smaller than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' If Ω(1) < 0, then either |ω0| or |ω1| has to be real and greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Therefore, the fixed point is a saddle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Further, (a) the other root is smaller or equal to −1 if and only if, correspondingly, Ω(−1) < −1 or Ω(−1) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (b) absolute value of the other root is smaller than 1 if and only if Ω(−1) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' 1 makes it easier for us to study analytically the stability of fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='3 Properties of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (5) Setting up xn+1 = f(x, y) and yn+1 = g(x, y), the two-dimensional logistic equations in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (5) have the mappings �f(x, y) = µ0x(1 − x) − µ1xy + x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' g(x, y) = −ν0y(1 − y) + ν1xy + y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (8a) (8b) 4 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (8) has Jacobian, as indicated in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (7),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' J = �µ0(1 − 2x) − µ1y + 1 −µ1x ν1y ν0(−1 + 2y) + ν1x + 1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (9) with eigenvalues ω0 and ω1 being,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' � � � � � � � ω0 = −µ0x + ν0y + 1 + 1 2(xν1 − yµ1) + 1 2(µ0 − ν0) + ω 2 ω1 = −µ0x + ν0y + 1 + 1 2(xν1 − yµ1) + 1 2(µ0 − ν0) − ω 2 (10a) (10b) where ω = � 4(ν0 + µ1 2 )2y2 + �� (4µ0 − 2ν1)µ1 + 8ν0(µ0 + ν1 2 ) � x − 4(ν0 + µ0)(ν0 + µ1 2 ) � y (11) + 4 � (µ0 + ν1 2 )x − µ0 2 − ν0 2 �2� 1 2 Further,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' fixed points,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' at which pairs of x and y stay still irrespective of time-series iterations[34],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' are those pairs of points (x∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' y∗) such that �x∗ = µ0x∗(1 − x∗) − µ1x∗y∗ + x∗ y∗ = ν0y∗(1 − y∗) − ν1x∗y∗ + y∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (12a) (12b) from which four pairs of fixed points (x∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' y∗) may be derived as � � � � � � � � � � � � � � � E0 = (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' 0) E1 = (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' 1) E2 = (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' 0) E3 = � ν0(µ1 − µ0) −µ0ν0 + µ1ν1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' µ0(ν1 − ν0) −µ0ν0 + µ1ν1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (13a) (13b) (13c) (13d) provided that the denominator in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (13d) is not zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' On the contrary, however, when µ0ν0 = µ1ν1, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (5) has fixed points E0, E1,and E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Keeping Lamma 1 in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='2 in mind, we may be able to examine the topological type of each fixed point in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (13) as in Theorem 1: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' The topological types of fixed points in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (13) are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' For E0 = (0, 0), � � � � � � � � � Ω(1) = −µ0ν0 Ω(−1) = (2 + µ0)(2 − ν0) det(J) = (1 + µ0)(1 − ν0) Tr(J) = 2 + µ0 − ν0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Because Ω(1) < 0, therefore, E0 is always a saddle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' For E1 = (0, 1), � � � � � � � � � Ω(1) = ν0(µ0 − µ1) Ω(−1) = (2 + ν0)(2 + µ0 − µ1) det(J) = (1 + ν0)(1 + µ0 − µ1) Tr(J) = 2 + µ0 − µ1 + ν0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' In this case, 5 (a) E1 cannot be a sink;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (b) if µ0 > µ1, then E1 is a source;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (c) if µ0 < µ1, then E1 is a saddle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (d) if µ0 = µ1, then E1 is a non-hyperbole;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' For E2 = (1, 0), � � � � � � � � � Ω(1) = −µ0(ν1 − ν0) Ω(−1) = (2 − µ0)(2 + ν1 − ν0) det(J) = (1 − µ0)(1 + ν1 − ν0) Tr(J) = 2 − µ0 + (ν1 − ν0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' In this case, (a) if µ0 < 2 and ν1 < ν0 < ν1 + 1 and ν1 < ν0, or µ0 < 2 and ν0 = ν1 + 1, or µ0 < 2 and ν1 + 1 < ν0 < ν1 + 2, then E2 is a sink;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (b) if µ0 > 2 and ν0 > ν1 + 2, then E2 is a source;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (c) if µ0 > 2 and ν1 < ν0 < ν1 + 2, or µ0 < 2 and ν0 > ν1 + 2, or ν1 > ν0, then E2 is a saddle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (d) if ν1 = ν0, then E2 is a non-hyperbole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' For E3 = � ν0(µ1−µ0) −µ0ν0+µ1ν1 , µ0(ν1−ν0) −µ0ν0+µ1ν1 � , � � � � � � � � � � � � � � � � � � � � � � � � � � � Ω(1) = −µ0ν0(ν0 − ν1)(µ0 − µ1) µ0ν0 − µ1ν1 Ω(−1) = −ν0(ν0 − ν1 + 2)µ2 0 + ν0(µ1 + 2)(ν0 − ν1 + 2)µ0 − 4µ1ν1 µ0ν0 − µ1ν1 det(J) = −ν0(ν0 − ν1 + 1)µ2 0 + ν0(µ1 + 1)(ν0 − ν1 + 1)µ0 − µ1ν1 µ0ν0 − µ1ν1 Tr(J) = −µ2 0ν0 + ν0(ν0 + µ1 − ν1 + 2)µ0 − 2µ1ν1 µ0ν0 − µ1ν1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' In this case, (a) if µ0 < µ1 and µ2 0ν0 < µ2 1ν1, or µ0 > µ1 and µ2 0ν0 > µ2 1ν1, then it is a saddle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (b) if µ2 0/µ2 1 = ν1/ν0, we have a non-hyperbole in the interior region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' In order to plot bifurcation diagrams, we further assume that µ1 = αµ0, ν0 = βµ0, and ν1 = γµ0, where α, β, and γ are parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' In this case the original µ1, ν0, and ν1 vary with µ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Under this circumstance, the nontrivial fixed points E3 becomes E3 = �β(α − 1) αγ − β , γ − β αγ − β � , which is independent of the growth rate parameter µ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Eigenvalues of Jacobian at E3 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (9) is � � � � � � � � � ω0 = 1 2αγ − 2β � Ξ0 + αγ − β |αγ − β|Ξ1 � ω1 = 1 2αγ − 2β � Ξ0 − αγ − β |αγ − β|Ξ1 � , (18a) (18b) where � � � � � � � � � Ξ0 = 2αγ − β2µ0 − � 2 + (α − γ − 1)µ0 � β Ξ1 = µ0 � β � (4βγ − 4γ2 + β)α2 − (2β2 + 2βγ − 4γ2 + 2β)α + β(β − γ + 1)2 �� 1 2 (19a) (19b) 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='4 Properties of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (6) Similar to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (8), the two-dimensional logistic equations in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (6) have the mappings �f(x, y) = µ0x(1 − x) − µ1xy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' g(x, y) = −ν0y(1 − y) + ν1xy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (20a) (20b) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (20) has Jacobian that is slightly different from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (9) J = �µ0(1 − 2x) − µ1y −µ1x ν1y ν0(−1 + 2y) + ν1x � , (21) with eigenvalues ω0 and ω1 being, respectively, � � � � � � � ω0 = −µ0x + ν0y + 1 2(xν1 − yµ1) + 1 2(µ0 − ν0) + ω 2 ω1 = −µ0x + ν0y + 1 2(xν1 − yµ1) + 1 2(µ0 − ν0) − ω 2 (22a) (22b) where ω is the same as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='(11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Fixed points for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (6) are � � � � � � � � � � � � � � � � � � � � � � � E′ 0 = (0, 0) E′ 1 = (0, 1 + 1 ν0 ) E′ 2 = (1 − 1 µ0 , 0) E′ 3 = �−µ0ν0 + µ1ν0 + µ1 + ν0 −µ0ν0 + µ1ν1 , −µ0ν0 + µ0ν1 − µ0 − ν1 −µ0ν0 + µ1ν1 � (23a) (23b) (23c) (23d) However, when µ0ν0 = µ1ν1, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (6) has fixed points E′ 0, E′ 1, and E′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' We may also make use of Lamma 1 in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='2 to examine the topological type of each fixed point in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (6) as in Theorem 2: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' The topological types of fixed points in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (23) are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' For E′ 0 = (0, 0), � � � � � � � � � Ω(1) = −(µ0 − 1)(ν0 + 1) Ω(−1) = −(µ0 + 1)(ν0 − 1) det(J) = −µ0ν0 Tr(J) = µ0 − ν0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' In this case, (a) if µ0 < 1 and ν0 < 1, then E′ 0 is a sink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (b) E′ 0 cannot be a source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (c) if µ0 > 1, then E′ 0 is a saddle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (d) if µ0 = 1, then E′ 0 is a non-hyperbole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' For E′ 1 = (0, 1 + 1 ν0 ), � � � � � � � � � � � � � � � � � � � � � � � � � � � Ω(1) = (ν0 + 1)((µ0 − µ1 − 1)ν0 − µ1) ν0 Ω(−1) = (ν0 + 3)((µ0 − µ1 + 1)ν0 − µ1) ν0 det(J) = (ν0 + 2)((µ0 − µ1)ν0 − µ1) ν0 Tr(J) = ν2 0 + (µ0 − µ1 + 2)ν0 − µ1 ν0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' In this case, 7 (a) E′ 1 cannot be a sink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (b) if µ0 > µ1ν0+µ1+ν0 ν0 , then E′ 1 is a source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (c) if µ0 < µ1ν0+µ1+ν0 ν0 , then E′ 1 is a saddle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (d) if µ0 = µ1ν0+µ1+ν0 ν0 , then E′ 1 is a non-hyperbole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' For E′ 2 = (1 − 1 µ0 , 0), � � � � � � � � � � � � � � � � � � � � � � � � � � � Ω(1) = (µ0 − 1)((ν0 − ν1 + 1)µ0 + ν1) µ0 Ω(−1) = (µ0 − 3)((ν0 − ν1 − 1)µ0 + ν1) µ0 det(J) = (µ0 − 2)((ν0 − ν1)µ0 + ν1) µ0 Tr(J) = −µ2 0 + (ν0 − ν1 − 2)µ0 + ν1 µ0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' In this case, (a) if 1 < µ0 < 2 and ν1µ0−µ0−ν1 µ0 < ν0 < ν1µ0+µ0−ν1 µ0 or µ0 = 2 and ν1 2 − 1 < ν0 < ν1 2 + 1 with ν1 > 2, or 2 < µ0 < 3 and ν1µ0−µ0−ν1 µ0 < ν0 < ν1µ0+µ0−ν1 µ0 , then E′ 2 is a sink;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (b) if 0 < µ0 < 1 and ν0 < ν1µ0−µ0−ν1 µ0 , or µ0 > 3 and ν0 > ν1µ0+µ0−ν1 µ0 , then E′ 2 is a source;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (c) if 1 < µ0 < 3 and ν0 > ν1µ0+µ0−ν1 µ0 , or µ0 > 3 and ν1µ0−µ0−ν1 µ0 < ν0 < ν1µ0+µ0−ν1 µ0 , or 0 < µ0 < 1 and ν0 > ν1µ0−µ0−ν1 µ0 , or 1 < µ0 and ν0 < ν1µ0−µ0−ν1 µ0 , then E′ 2 is a saddle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (d) if µ1 = 1 or ν0 = ν1µ0−µ0−ν1 µ0 , then E′ 2 is a non-hyperbole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' For E′ 3 = � ν0(µ1−µ0) −µ0ν0+µ1ν1 , µ0(ν1−ν0) −µ0ν0+µ1ν1 � , � � � � � � � � � � � � � � � � � � � � � � � � � � � Ω(1) = −µ0ν0(ν0 − ν1)(µ0 − µ1) µ0ν0 − µ1ν1 Ω(−1) = −ν0(ν0 − ν1 + 2)µ2 0 + ν0(µ1 + 2)(ν0 − ν1 + 2)µ0 − 4µ1ν1 µ0ν0 − µ1ν1 det(J) = −ν0(ν0 − ν1 + 1)µ2 0 + ν0(µ1 + 1)(ν0 − ν1 + 1)µ0 − µ1ν1 µ0ν0 − µ1ν1 Tr(J) = −µ2 0ν0 + ν0(ν0 + µ1 − ν1 + 2)µ0 − 2µ1ν1 µ0ν0 − µ1ν1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' In this case, (a) if µ0 < µ1 and µ2 0ν0 < µ2 1ν1, or µ0 > µ1 and µ2 0ν0 > µ2 1ν1, then it is a saddle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (b) if µ2 0/µ2 1 = ν1/ν0, we have a non-hyperbole in the interior region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' In terms of α, β, and γ, E′ 3 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (23d) is E′ 3 = � αβµ0−βµ0+α+β µ0(αγ−β) , −βµ0+γµ0−γ−1 µ0(αγ−β) � , at which the eigenvalues of Jacobian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (21) is � � � � � � � � � ω0 = 1 2αγ − 2β � Ξ′ 0 + αγ − β |αγ − β|Ξ′ 1 � ω1 = 1 2αγ − 2β � Ξ′ 0 − αγ − β |αγ − β|Ξ′ 1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (28a) (28b) 8 where � � � � � � � � � � � � � � � � � � � � � � � � � � � Ξ′ 0 = (2γ − 1)α − β2µ0 − � αµ0 + (−µ0 + 1)γ − µ0 + 4 � β Ξ′ 1 = � (µ0 − 1) � (−4βµ0 − 4) α2 + 4β (µ0 − 1) α + β2 (µ0 − 1) � γ2 + � 4 (βµ0 + 1)2 α2 − 2β (µ0 − 1) (βµ0 + 1) α − 2β2µ0 (µ0 − 1) (β + 1) � γ + � (βµ0 + 1) α − βµ0 (β + 1) �2 � 1 2 (29a) (29b) (29c) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (28) shows that whenever Ξ′ 2 1 is negative, ω0 and ω1 are complex conjugates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Unlike the previous case, fixed points E′ 1, E′ 2, and E′ 3 now are dependent on the growth rate µ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 Lyapunov exponents In addition, the chaotic behavior may be better examined by introducing Lyapunov exponents, which are defined as, base 2 being chosen to conform to Wolf et al[35], � � � � � � � λx ≡ log2 |w0| = ln |w0| ln 2 λy ≡ log2 |w1| = ln |w1| ln 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (30a) (30b) The positive value of λx or λy, together with the negative value of total sum of the Lyapunov exponent, either � λx < 0 or � λy < 0, are strong inference of chaos for the prey or the predator[36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' For comparison, Lyapunov exponents of time series data of prey and predator populations were also calculated by both algorithms of Rosenstein[28] and Eckmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' [29] with the package of NOnLinear measures for Dynamical Systems (nolds)[37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' For Rosenstein algorithm, embedding dimension for delay embedding was emb dim = 10, and the step size between time series data points was set to τ = 1 second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' While number of data points (trajectory len) was set to 20 and was used for the distance trajectories between two neighboring points, the mean period of time series data, obtained from the fast Fourier transform, was used as the minimal temporal separation(min tsep) between two neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Search of the suitable lag was terminated when number of potential neighbors for a vector was found to be smaller than minimal neighbors, which was set as min neighbors = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' At last, the RANSAC-fitting was used for the line fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' As for the algorithm proposed by Eckmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=', the matrix dimension was set to 2, and embedding dimension was also set to 10 as in Rosenstein algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Moreover, τ = 1 s, the minimal number of neighbors (min nb) was 4, and min tsep = 0 were used in the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' There are at least four disadvantages for the above algorithms, as mentioned by Escot and Galan[36]: lack of the ability to estimate full Lyapunov spectrum, not resilient to noise in time-series data, poor detection performance in nonlinearity with an adequate sample size, and no theoretical derivations for the algorithms about their consistency and asymptotic distributions, making it impossible to statistical inferences respect to chaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' 3 Results and Discussions In the present study, we focused only on drawings of equations in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' We observed that there could be five different bifurcation diagrams with various kinds of combinations of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' The first category is Normal, referring to the normal competitive behavior on the increasing and decreasing on numbers about species between the prey and the predator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' The second category is Standard, referring to the standard bifurcation diagram as shown in the well-known 1D logistic equation in the prey at the absence of the predator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' The third category is named as Paraclete, referring to overlapping structure in the bifurcation diagram of the prey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' The fourth category is Extinction, connoting to extinction of the predator when the prey becomes chaotic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' The last category is Vorticella Strange, meaning that the bifurcation diagram resembles the shape of a vorticella but with more complex inner structures before the two species become chaotic at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Categories and parameters were summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' initX and initY indicate 9 initial values of the prey and the predator, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Special attention should be paid to the cases of Normal and Extinction, where the inter-species parameters are deliberately made the same but the initial population were different: for Normal, the two species have the same initial population whereas for Extinction, the predator has 10 times more population than the prey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' The discrepancy on initial population in these two cases shows completely different evolution consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' In the following figures, discussions on Lyapounov exponents, Equation, Rosenstein, Eckmann X, and Eckmann Y in the legend of λx and λy refer to calculations directly from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (30), from time-series algorithm of Rosenstein, Eckmann et al of the prey, and Eckmann et al of the predator, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Codes, together with animations on population iterations, phase portraits and phase diagrams under different growth rates, may be retrieved via Ref([38]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' initX initY α β γ Normal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='200 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='500 Standard 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='500 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='500 Paraclete 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='100 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='900 Extinction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='500 Vorticella Strange 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='875 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='018 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='000 Table 1: Parameters used for equations in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='1 Normal Our dynamical system may describe normal competitiveness between two species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Under the circumstance of equal initial population, Figure 1a shows steadily increasing population of x at the absence of the predator when µ0 < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' At the appearance of y after µ0 > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0, the prey gradually diminishes with more number of the predator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Figure 1b ensures that under this scenario there is no chaos, for the Lyapunou exponents calculated by every algorithm are negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' However, results are different from algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' First for λx, there is a trench around µ0 = 2 by Equation, whereas all other algorithms fail to reproduce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Rosenstein, Eckmann X, and Eckmann Y only reproduced shallow dip around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 < µ0 < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' In addition, We observe that Rosenstein and Eckmann X have quite similar results in the whole range of µ0, except that Rosenstein has a slightly higher value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Furthermore, Eckmann X and Eckmann Y have almost identical values when µ0 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='66, but Eckmann Y digresses a lot from the other three curves at low growth rate below µ0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' As for λy, all algorithms show close spectrum µ0 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='692, with larger values for Rosenstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' The four algorithms divide into two groups of results below µ0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='692, with Rosenstein and Eckmann X showing an increasing tail that is different from the other two algorithms showing curves of dropping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='2 Standard Figure 2a shows bifurcation diagram and Lyapunov exponents of Standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Our model shows that even with nonzero initial population and nonzero inter-species relationships of α, β, and γ, we may still acquire flip bifurcation for 1D logistic equation[39] for the prey at the absence of the predator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' A flip bifurcation is a counterpart in discrete dynamic system to describe the concept of periodic doubling in the continuous dynamic system[40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Figure 2b shows the Lyapunov exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Rosenstein algorithm did not show results in in this case, because singular value decomposition did not converge when doing linear least squares, meaning that positive or negative infinity appeared when we tried to deal with pseudo-inverse matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Eckmann Y cannot work, either, for y = 0 in the whole range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' We may see that for overall trend of λx, both algorithms have λx < 0 for µ0 < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0, whereas λx has both positive and negative values for µ0 > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' It is widely accepted[41] that values of Lyapunov exponents occur interchangeably between positive and negative infer chaos, which is consistent with the shaded area in Figure 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Another inconsistency occurs with 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 < µ0 < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 with λx > 0 for Equation but λx < 0 for Eckmann X, where x exhibits a flip bifurcation from 2-cycle into 4-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Nevertheless, this inconsistency may not be a problem for us to distinguish chaos from happening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' There is 10 (a) (b) Figure 1: Competitive behavior and Lyapunov exponents of Normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 Logistic Map 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='15 - y 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='05 000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 μoLyapunov Exponents 0 入x 4 6 - Equation .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='. Rosenstein Eckmann X Eckmann Y 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0 2 4 - g- 8 + 10 - Equation Rosenstein Eckmann X 12 EckmannY 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 μoa break around µ0 = 2 for Eckmann X in both λx and λy, at which Equation shows a deep trench in λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Also, near µ0 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='245, Equation shows a smaller trench while Eckmann X produces a rising tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Figure 3 studies the population in the course of time (iteration) at µ0 equal to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='700 (1-cycle), 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='000 (2-cycle where the flip bifurcation occurs) and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='500 (4-cycle), 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='700 (at which the system goes into chaos), 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='845 (the system going back to more stable 3-cycle), and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='945 (where the system returns to chaos again) in the successive order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Zero predator population is obtained throughout course of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Figure 4 studies topological types of fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Plots of imaginary or real part of eigenvalues in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (22) were evaluated at the fixed points E′ 1 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (23b) (as shown in the upper row), E′ 2 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (23c) (as shown in the middle row), and E′ 3 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (23d) (as shown in the bottom row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Color-bar at the right-hand side stands for various growth rate µ0, while circles with four different sizes in the legend represent, from the smallest to the biggest, topological types of sink, source, saddle, and non-hyperbole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' We may see that ω0 and ω1 at the fixed points E′ 1 were pure real numbers with absolute values greater than 1, making the fixed point a source for all µ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' While ω0 and ω1 at the fixed points E′ 2 were also pure real numbers, the absolute values vary across 1, making the fixed point E′ 2 topological types of sink, source, and saddle, with possible non-hyperboles occurring at either low µ0 = 1 or high µ0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='75 if we apply Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='3(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Figure 5 shows absolute values of eigenvalues ω0 and ω1 on fixed points E′ 1, E′ 2 and E′ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Topological types of fixed points may be double-checked more straightforwardly with the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' The first column shows that E′ 1 is a source because ω0 and ω1 are always greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' The second column demonstrates that E′ 2 are a sink when µ0 < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='141, and when 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='141 < µ0 < 3, E′ 2 is a saddle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Non-hyperboles can also be examined at µ0 = 1 for E′ 2, at µ0 = 3 for E′ 2 (in both cases ω0 = 1 and ω1 ̸= 1), and at µ0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='75 for E′ 2 and E′ 3 (in which ω0 ̸= 1 and ω1 = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' That µ0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='75 is located in chaos region, making the fixed point vulnerable to nonlinear terms in the dynamic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' At last, when µ0 > 3, E′ 2 is a source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Figure 6 shows phase portrait and phase space diagram about Standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' µ0 values are represented by the color bar at the right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Figure 6a refers to phase portrait, where topological types are also shown in the legend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' An isolated initial coordinate (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='50) marked as a source at the upper-left corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Flip bifurcation occurs at (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='665, 0, 000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' No limit circles were found in the flip bifurcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' The oblique black straight line, starting from (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='732, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='000) to (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='689, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='062), consisting of fixed points E′ 3 that is enclosed by a thicker yellow cloak indicates that, along the axis, E′ 3 is a saddle, with nonzero y values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' This result seems to contradict to the previous one, saying that predator population is always zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' However, since our initial population is (x, y)=(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5), it does not lie in the above range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Therefore, the system with the chosen inter-species constants is not attracted by the saddle points along the oblique black line, confirming that with none-zero initial population of the predator and non-zero inter-species constants, the predator could appear, but only for a while.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Afterwards, the predator dies out in the course of time, leaving the prey to be the only surviving species in the paradisaic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' This observation may explain why the predator species may not survive long in some specific ecological system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Further, Figure 6b is phase space diagram for the prey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' It is meaningless to discuss phase space diagram for the predator because there are only two points, (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0) and (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5), in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' 12 (a) (b) Figure 2: Bifurcation diagram and Lyapunov exponents of Standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' 13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='6 - X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 uoLyapunov Exponents 2 6 Equation Eckmann X EckmannY 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0 2 4 6 8 - Equation Eckmann X 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 μo(a) (b) (c) (d) (e) (f) Figure 3: Population vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' iteration of Standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' 14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='6 μo = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='945 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0 25 50 75 100 125 150 175 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='3 n 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0 25 50 75 100 125 150 175 200 n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='6 μo = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='700 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0 25 50 75 100 125 150 175 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 25 50 75 100 125 150 175 200 n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='6 μo = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0 25 50 75 100 125 150 175 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0 25 50 75 100 125 150 175 200 n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='4 Ho= 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0 25 50 75 100 125 150 175 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0 25 50 75 100 125 150 175 200 n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='6 μo = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='700 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0 25 50 75 100 125 150 175 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0 25 50 75 100 125 150 175 200 n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0 25 50 75 100 125 150 175 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 25 50 75 100 125 150 175 200 nFigure 4: Analysis on eigenvalues for the case of Standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Eigenvalues described in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (22) for fixed points E′ 1 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (23b)), E′ 2 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (23c)), and E′ 3 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (23d)) are plotted in the upper row, middle row, and lower row, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Types of topology are indicated with circles of various sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Color bar stands for different µ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Figure 5: Absolute values of eigenvalues vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' growth rate at fixed points for Standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Prominent coordinates that help us understand stability and distinguish the topological type about a fixed point are recorded as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Upper middle panel: (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='244, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='000), and (3, 75, 1, 00).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Upper-right corner panel: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='75, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='00).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Column for Ei μo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Column for E2 μo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Column for E3(a) (b) Figure 6: Analysis on phase portrait and phase space dia- gram about Standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Topolog- ical types of sink, source, sad- dle and non-hyperbole are repre- sented with different sizes from the smallest to the largest as shown in the legends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Oblique black line in Figure 6a indicates saddle fixed points, demonstrat- ing that the predator cannot sur- vive long in the ecological sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Figure 6b shows the phase space of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Phase diagram of the predator is not shown here, be- cause it only contains two points (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0), and (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5), which makes the figure boring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' 16 Type 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 non-hyperbolic sink saddle O source 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='4 y 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 μo X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='2 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 μo3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='3 Paraclete Figure 7a represents two sets of overlapping bifurcation diagrams: one is the Standard, the other with vorticella-shaped, which starts to appear after µ0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Between 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='29 < µ0 < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='45, shaded regions appearing vertically with gaps are not chaos but transient states of population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' At µ0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='256, the vorticella- shaped has bifurcation that start to be chaotic, at which we would explain later that it should be classified as Neimark-Sacker bifurcation, and mingles together with the flip bifurcation of Standard after µ0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='46, at which the four-cycles occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Figure 7b shows the Lyapunov exponents for Paraclete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' For the same reason in Standard, Rosenstein algorithm did not show results in in this case, either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' We may see that for overall trend of λx, algorithms of both Equation and Eckmann X have λx < 0 for µ0 < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0, whereas λx has both positive and negative values for µ0 > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0, inferring chaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Eckmann Y only shows a small portion, not spectrum, because y = 0 for µ0 < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='29 leads to failure on producing full spectrum of both λx and λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Therefore, where we may see only some segment of λx after µ0 > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 that almost overlaps with the curve of Eckmann X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Inconsistency also occurs at low growth rate in λy spectrum, for Equation shows stern-drooping tails while Eckmann X shows a raising-up one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Figure 8 studies the population in the course of time (iteration) at µ0 equal to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='790 (stable population), 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='025 (2-cycle), 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='255 (at which Neimark-sacker bifurcation is on the way), 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='460 (4-cycle), 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='845 (3-cycle in Standard), and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='945 (chaos region) in the successive order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Figure 9 studies topological types of fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Plots of imaginary or real part of eigenvalues in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (22) were evaluated at the fixed points E′ 1 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (23b) (as shown in the upper row), E′ 2 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (23c) (as shown in the middle row), and E′ 3 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (23d) (as shown in the bottom row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Color-bar at the right-hand side stands for various growth rate µ0, while circles with four different sizes in the legend represent, from the smallest to the biggest, topological types of sink, source, saddle, and non-hyperbole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' We may see that ω0 and ω1 at the fixed points E′ 1 were pure real numbers with absolute values greater than 1, making the fixed point a source for all µ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' While ω0 and ω1 at the fixed points E′ 2 were also pure real numbers, the absolute values vary across 1, making the fixed point E′ 2 topological types of sink, source, and saddle, with only one possible non-hyperbole occurring at |ω0| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Figure 10 shows absolute values of eigenvalues ω0 and ω1 on fixed points E′ 1, E′ 2 and E′ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Topological types of fixed points may be double-checked more straightforwardly with the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' The first column shows that E′ 1 is a source because ω0 and ω1 are always greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' The second column demonstrates that E′ 2 are a sink when µ0 < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='141, and when 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='141 < µ0 < 3, E′ 2 is a saddle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' At last, when µ0 > 3, E′ 2 is a source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' As we look closely into the third column, it shows that E′ 3 is non-hyperbolic at µ0 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='137 for ω0 ̸= 1 and ω1 = 1, which is vulnerable to nonlinear terms in the dynamic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' It explains why the transient state under that growth rate is not shown in Figure 7a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Also, bending points along curves plotted in the third-column figures demonstrate that transient states occur within 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='258 < µ0 < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='475.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' More interestingly, the third column manifests that E′ 3 represents Neimark-Sacker bifurcation at µ0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='25636 because of the fol- lowing facts: first, ω0 and ω1 are complex conjugates with modulus 1, and second, as µ0 varies across 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='25636 from smaller to larger value, topological type of E′ 3 changes from a sink (stable) to a source (unstable)[19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Figure 11 shows phase portrait and phase space diagram about Paraclete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' µ0 values are represented by the color bar at the right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Figure 11a refers to phase portrait, showing Neimark-Sacker bifurcation established at (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='352, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='068) with µ0 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='256 at the center of limit circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Black straight lines indicate oblique axis consisting of E′ 3, including sink and source, and horizontal axis composed of E′ 2, including source and saddle, under different µ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Topological types are also shown in the legend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Dots with larger µ0 representing chaos spread outside around the limit circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Figure 11b is phase space diagram for the prey centered at (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='352, 0) and Figure 11c refers to phase space diagram for the predator centered at (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='068).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Not surprisingly, the center of limit circles in Figure 11b has the same x value as that of Figure 11a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Similarly, same y value for the center of limit circles in Figure 11c and in Figure 11a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' 17 (a) (b) Figure 7: Bifurcation diagram and Lyapunov exponents of Paraclete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' 18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='6 X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='4 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='100 >0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 uo0 2 - 4 6- Equation Eckmann X EckmannY 8 2 2 4 - 6于 Equation Eckmann X 8 Eckmann Y 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 μo(a) (b) (c) (d) (e) (f) Figure 8: Population vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' iteration of Paraclete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' 19 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='2 μo = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='790 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0 25 50 75 100 125 150 175 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='000 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='050 25 50 75 100 125 150 175 200 n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='2 μo = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0 25 50 75 100 125 150 175 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='050 0 25 50 75 100 125 150 175 200 n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='6 X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='2 μo = 3,255 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0 25 50 100 125 150 175 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='050 0 25 50 75 100 125 150 175 200 n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='6 ux 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='2 μo = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='460 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0 25 50 75 100 125 150 175 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='050 0 25 50 75 100 125 150 175 200 n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='2 以o -:3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='845.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0 25 50 75 100 125 150 175 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='050 0 25 50 75 100 125 150 175 200 n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='6 n X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='945 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0 25 50 75 100 125 150 175 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='050 0 25 50 75 100 125 150 175 200 nFigure 9: Analysis on eigenvalues for the case of Paraclete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Legends have the same meaning described in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Figure 10: Absolute values of eigenvalues vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' growth rate at fixed points for Paraclete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Important coordinates: Upper middle panel (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='14, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='00).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Upper-right corner panel (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='137, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='000), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='457, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='440), and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='256, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Lower-right corner panel (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='258, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='000), (2.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 μo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Column for Ei μo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Column for E2 μo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Column for E3(a) (b) (c) Figure 11: Analysis on phase portrait and phase space diagram about Paraclete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (11a)Phase portrait shows Neimark-Sacker bifurcation es- tablished at (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='352, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='068) with µ0 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='256 located at the center of limit circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Oblique axis and horizontal axis consisting of E′ 3 and E′ 2 with different µ0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' We may see from the legend that topological types of E′ 3 are mostly sink and source, while those of E′ 2 are mostly source and saddle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Phase portrait and phase space diagram for the prey have same x in (11b) for the center limit circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Similarly, phase portrait and phase space diagram for the predator have same y in (11c) for the center limit circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' 21 Type non-hyperbolic sink saddle 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='14 source 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='08- 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='14 V μo3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='4 Extinction Figure 12a shows the bifurcation diagram of Extinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' There is a flip bifurcation for x around µ0 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='99;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' however, the 2-cycle collides at µ0 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='165 and returns back to 1-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Shaded regions between 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='165 < µ0 < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='434 for x and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='10 < µ0 < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='343 for y do not refer to chaos but belong to transient states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Afterwards, the bifurcation diagram goes back to Normal for both x and y, and represent pleasant conditions of predictable values before x goes into 3-cycle at µ0 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='828 as in Standard, where y drops to zero cliff-fallingly at µ0 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='824, manifesting to extinction of the predator for the prey is around the 3-cycle state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Fortunately, there could be still few little chances for the predator to survive at (µ0, y) = (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='845, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='219), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='887, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='226), and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='897, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='228), where we may see three isolated fixed points appear with a vertical tail of transient states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' When the prey becomes fully chaotic, the predator population reduces back to zero dramatically again, and never has any further opportunity to rise back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' This astonishing phenomenon could be the most profound finding in the study, which states that the prey in chaos generated by overpopulation of the predator would erase the entire predator species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Figure 12b shows Lyapunov exponents for Extinction that is similar to those in Figure 1b, except for the regions at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 < µ0 < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='18 and µ0 > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='86, the former showing a bum by Equation, which is also the same region for the prey to be in 2-cycle, and the latter presenting chaos for x and extinction for y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Discrepancy appears for Eckmann Y that is different from the other when µ0 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='662, where it shows a decreasing tail while the other algorithms show an increasing trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Unlike Figure 1b, whose results show that Eckmann X is always in between Rosenstein and Eckmann Y in the range of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 < µ0 < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0, Figure 12b shows more intertwines at µ0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='886 for λx, and at µ0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='717 and µ0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='891 for λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Similar to Figure 1b, the four algorithms also divides into two groups of curves below µ0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='717, with Rosenstein and Eckmann X going upward, together with Equation and Eckmann Y going downward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Figure 13 shows population iteration of Extinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' As we can see, flip bifurcation starts at µ0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='000 as in Figure(13a), while in Figure(13b) the two fixed points collide, and after transient states (n > 200), they have tendency to merging into a single fixed point, as we explained earlier in Figure 12a on the characteristics about the shaded region between 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='165 < µ0 < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='434 for x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' We further demonstrate that at µ0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='500 in Figure(13c), after transient(n > 175), bifurcation collapses to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Furthermore, 3-cycle is opened in x at µ0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='84, as indicated in Figure(13d), with extinction of y at the exactly the same moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' However, a sunlight of survival for y is shed on the window at µ0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='845, as shown in Figure(13e), where the two species may still exist under predictable population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Finally, Figure 13f portends the predator extinction under chaos of the prey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Figure 14 studied topological types of fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Plots of imaginary or real part of eigenvalues in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' (22) were evaluated also at the three fixed points E′ 1, E′ 2, and E′ 3 as in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' We may see that ω0 at the fixed points E′ 1 is pure real numbers with absolute values greater than 1, whereas ω1 keeps the value −1000 for all µ0, making the fixed point a source (unstable) for all µ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' While ω0 < 1 and ω1 > 1 at the fixed points E′ 2, the fixed point E′ 2 has a topological type of saddle, with possible non-hyperboles occurring at |ω0| = 1 and |ω1| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Stability of fixed points may also be examined in Figure 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Figures at the first column shows that that E′ 1 is a source, for both eigenvalues have absolute values greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' In the second column, we see that E′ 2 changes its stability from a sink to source when µ0 varies at 3, at which flip bifurcation occurs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' meanwhile, E′ 3 changes from source to sink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Figure 16 shows phase portrait and phase diagrams for Extinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' No limit circles are found in this particular case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' 22 (a) (b) Figure 12: Bifurcation diagram and Lyapunov exponents of Extinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' 23 Logistic Map 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='8 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='20 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='15 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='10 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='05 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='00- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 μoLyapunovExponents 0 2 6- Equation Rosenstein Eckmann X Eckmann Y 8 0- 2- 4 6 8 + Equation 10 - Rosenstein Eckmann X EckmannY 12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 μo(a) (b) (c) (d) (e) (f) Figure 13: Population vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' iteration of Extinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' 24 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='8 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='6 n 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='4 μo = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0 25 50 75 100 125 150 175 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='25 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='10 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='05 0 25 50 100 125 150 175 200 n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='8 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='6 u 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='4 μo = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='375 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0 25 50 75 100 125 150 175 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='25 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='10 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='05 0 25 50 75 100 125 150 175 200 n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='8 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='6 n 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='4 μo = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0 25 50 75 100 125 150 175 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='25 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='10 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='05 0 25 50 75 100 125 150 175 200 n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='8 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='4 μo = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='840 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0 25 50 75 100 125 150 175 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='25 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='20 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='10 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='00 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='05 0 25 50 100 125 150 175 200 n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='8 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='4 μo = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='845 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0 25 50 75 100 125 150 175 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='25 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='10 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='05 0 25 50 75 100 125 150 175 200 n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='8 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='6 C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='4 μo = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='945 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0 25 50 75 100 125 150 175 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='25 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='10 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='00 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='05 0 25 50 75 100 125 150 175 200 nFigure 14: Analysis on eigenvalues for the case of Extinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Legends have the same meaning described in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Figure 15: Absolute values of eigenvalues vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' growth rate at fixed points for Extinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' From the figures at the first column, it is clearly shown that E′ 1 is a source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Also, at µ0 = 3 where flip bifurcation occurs, E′ 2 changes its stability from a sink to source, whereas E′ 3 from source to sink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='04 1040- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='02 1020 Im(wo) Im(wi) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='02 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='02 980 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='04 096 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='00102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='00152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0020 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='00252.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='00302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='00352.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0040 1040 1020 1000 980 960 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0010 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0015 2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 μo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Column for Ei μo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Column for E2 μo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Column for Es(a) (b) (c) Figure 16: Analysis on phase portrait and phase space dia- gram about Extinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' No limit circles were found in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' 26 Type 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='25 non-hyperbolic sink O saddle source 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='15 - y 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='10 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 μo0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='2 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 μo0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='04 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='20 μo y3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 Vorticella Strange The last bifurcation type in our study is Vorticella Strange, which means that bifurcation diagram looks like a vorticella, only with more complicated internal structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Figure 17a shows bifurcation diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' When µ0 < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0, x grows steadily without y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' After presence of the predator, population of the prey starts to decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Both species show predictable population before µ0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='025, at which we classify a Neimark-Sacker bifurcation as in Paraclete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Transient states occur between 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='025 < µ0 < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='200, as indicated in the shaded region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Later, a 6-cycle appears at µ0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='24, which is also confirmed in Figure 18b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' The 6-cycle becomes 3-cycle at µ0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='40, as shown in Figure 18c, followed by chaos at µ0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='485, which is also demonstrated in Figure 18d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' The system goes back to 6-cycle around µ0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='540, as we may also confirm in Figure 18e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Afterwards, the system goes back to chaos, as shown in Figure 18f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Figure 17b shows Lyapunov spectrum of Voticella Strange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' All four algorithms barely show positive spectra for both x and y before µ0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' On the contrary, afterµ0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5, four algorithms show positive Lyapunov exponents, where the system falls into chaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' For λx, Eckmann Y shows a decreasing tail below µ0 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5, inconsistent from the other algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Besides Equation showing two valleys between 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='500 < µ0 < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='304 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='224 < µ0 < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='463, the other three algorithms only provide flat spectra in the two regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Figure 19 shows analysis on eigenvalues of Vorticella Strange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' At the first row, we see that both eigenvalues have zero imaginary parts, with absolute real parts of both greater than 1 (see also first column in Figure 20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Thus, E′ 1 is a source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Similar analysis may also be done on E′ 2, as shown in the second row in Figure 19 as well as the second column in Figure 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' At µ0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0, it turns from a sink to a saddle, and it maintains as a saddle between 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 < µ0 < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0, after which it turns to a source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Finally, the third column in Figure 20 demonstrates that Neimark-Sacker bifurcation occurs at µ0 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='025, with coordinates (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='341, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='377), as shown in Figure 21a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Similar bifurcation diagram was found by Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' [10], and was identified as the Hopf bifurcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' However, the criteria for Hopf bifurcation in the two-dimensional system includes that the two eigenvalues are purely conjugate imaginary pair with zero real part[42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Since Figure 19 shows that ω1 has a non-zero real part, our Vorticella Strange cannot be a Hopf bifurcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' For the sake of integrity, phase portrait and phase space diagram of Vorticell Strange are also shown in Figure 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Similar detailed meaning is discussed in the Section of Paraclete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' 4 Conclusions We successfully built up a discrete-time model of two-dimensional mapping that resembles features of dif- ferential Lotka-Volterra Equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' We studied the topological types of fixed points, and found out that fascinating feather-like structures were formed around limit circles pierced through by the axis of fixed points in the phase portraits and phase space diagrams under various growth rates with Neimark-Sacker bifurcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' We further divided our dynamical systems into five categories by different shapes of bifurcation diagrams: Normal, Standard, Paraclete, Extinction, and Vorticella Strange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' In every case, we studied the stability and topological types of the fixed points by criteria discussed in Lamma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' In addition to plot the population vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' iteration, we also calculated Lyapunov exponents both by eigenvalues of Jacobian of mapping functions, and by Rosenstein algorithm and Eckmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Discrepancies clearly showed that Lyapunov exponents calculated by time-series algorithms may be unreliable within the range of chaos and at low growth rates, as well as when the Lyapunov exponents change dramatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Furthermore, our model not only re- gained the 1D logistic mapping of the prey under zero predator yet with non-zero initial predator population and with non-zero inter-species constants, but it also showed the normal competitiveness of the prey and the predator without chaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' The quintessence in the current research was that, besides the possibility for the prey and the predator to become chaotic altogether, it is also probable for the predator to go extinct at the chaotic state of the prey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' In other words, human overpopulation would cause chaos in natural resources, ultimately in return erase the entire human race.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Luckily, even under this difficult circumstance slim chance is still left upon us to continue our race under some specific growth rate, as we may see some isolated fixed points remain in the predator bifurcation diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Our model may inspire conjecture on other relationships between two physical quantities, because mathe- matically what we demonstrated was that one quantity may dramatically reduce to zero at the state of chaos 27 (a) (b) Figure 17: Bifurcation diagram and Lyapunov exponents of Vorticella Strange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' 28 Logistic Map 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='2 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='8 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content="2 - 0'0 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 μoLyapunov Exponents 0- 2 Equation 6 Rosenstein Eckmann X Eckmann Y 2 F9- Equation 8 Rosenstein Eckmann X EckmannY 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 uo(a) (b) fig: (c) (d) (e) (f) Figure 18: Population vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' iteration of Vorticella Strange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' 29 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 - μo = 3,025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 50 75 100 125 150 175 200 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='. 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0 25 50 75 100 125 150 175 200 n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 - μo = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='240 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 25 50 75 100 125 150 175 200 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0 25 50 75 100 125 150 175 200 n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 - μo = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 25 50 75 100 125 150 175 200 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0 25 50 75 100 125 150 175 200 n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 μo = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='485 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0 25 50 75 100 125 150 175 200 1.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 0 25 50 75 100 125 150 175 200 n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 μo = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='700 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+page_content='341, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='377) with µ0 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' See also caption in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' 31 Type 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 non-hyperbolic sink O saddle source 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='8 - 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='0 μo yof the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Therefore, it could be highly possible that, for example, the superconducting state, which refers to the zero resistance, may be achieved with chaos of some physical quantity that has relationship between resistance in the form of simultaneous difference equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' 5 Acknowledgment We thank Pui Ching Middle School in Macau PRC for the kindness to support this research project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' References [1] Peter Roopnarine, Ecology and the Tragedy of the Commons, 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+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='3814723 [38] Codes and animations may be retrieved via https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content='com/weishanlee/LotkaVolterraChaos [39] Stephen T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Thornton and Jerry B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' Marrion, Classical Dynamics of Particles and Systems, 5th Ed.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' CH 11, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' 3, Berlin: Springer-Verlag (1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' ISBN-13: 9783540971412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} +page_content=' 34' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFJT4oBgHgl3EQf5S05/content/2301.11669v1.pdf'} diff --git a/ZNFPT4oBgHgl3EQfuTUu/content/tmp_files/2301.13155v1.pdf.txt b/ZNFPT4oBgHgl3EQfuTUu/content/tmp_files/2301.13155v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..518da5e4a2b8e9900215c2149d53bd51609cea77 --- /dev/null +++ b/ZNFPT4oBgHgl3EQfuTUu/content/tmp_files/2301.13155v1.pdf.txt @@ -0,0 +1,1417 @@ +Published as a conference paper at ICLR 2023 +ADVANCING RADIOGRAPH REPRESENTATION LEARN- +ING WITH MASKED RECORD MODELING +Hong-Yu Zhou1,2∗ +Chenyu Lian1∗ +Liansheng Wang1 +Yizhou Yu2 +1School of Informatics, Xiamen University +2Department of Computer Science, The University of Hong Kong +whuzhouhongyu@gmail.com, dopaminel@foxmail.com, +lswang@xmu.edu.cn, yizhouy@acm.org +ABSTRACT +Modern studies in radiograph representation learning (R2L) rely on either self- +supervision to encode invariant semantics or associated radiology reports to in- +corporate medical expertise, while the complementarity between them is barely +noticed. To explore this, we formulate the self- and report-completion as two com- +plementary objectives and present a unified framework based on masked record +modeling (MRM). In practice, MRM reconstructs masked image patches and +masked report tokens following a multi-task scheme to learn knowledge-enhanced +semantic representations. With MRM pre-training, we obtain pre-trained mod- +els that can be well transferred to various radiography tasks. Specifically, we +find that MRM offers superior performance in label-efficient fine-tuning. For in- +stance, MRM achieves 88.5% mean AUC on CheXpert using 1% labeled data, +outperforming previous R2L methods with 100% labels. On NIH ChestX-ray, +MRM outperforms the best performing counterpart by about 3% under small la- +beling ratios. Besides, MRM surpasses self- and report-supervised pre-training in +identifying the pneumonia type and the pneumothorax area, sometimes by large +margins. Code and models are available at https://github.com/RL4M/ +MRM-pytorch. +1 +INTRODUCTION +Findings: +[MASK] cardiac, mediastinal and +hilar [MASK] are normal. +[MASK] [MASK] [MASK] normal. +lungs are [MASK] . +… +… +Inputs +Report +Radiograph +Paired +Representations +Findings: +the cardiac, mediastinal and +hilar contours are normal. +pulmonary vasculature is normal. +lungs are clear . +Outputs +Figure 1: Illustration. +MRM learns trans- +ferable radiograph representations via recon- +structing masked records, i.e., masked radio- +graph patches and masked reports tokens. +Radiograph representation learning (R2L) has been +among the core problems of medical image analysis. +Previously, downstream radiograph analysis tasks +counts on pre-trained models on ImageNet (Deng +et al., 2009) or large X-ray datasets (Wang et al., +2017; Irvin et al., 2019; Johnson et al., 2019; Bus- +tos et al., 2020) to alleviate the shortage of expert +labeling. The emergence of self-supervised repre- +sentation learning (Doersch et al., 2015; Agrawal +et al., 2015; Wang & Gupta, 2015) provides a choice +to conduct pre-training with negligible human inter- +vention by exploiting self-supervision. However, the +self-supervised paradigm ignores the introduction of +medical expertise (e.g., anatomy), reducing its trans- +ferability to downstream tasks with limited label in- +formation. +On the other hand, free-text radiology reports written +by experienced radiologists often contain rich do- +main knowledge. To leverage this, researchers developed automated rule-based labelers (Wang +et al., 2017; Irvin et al., 2019) to extract structured labels from unstructured texts. Nevertheless, +∗Work done while visiting Xiamen University. First two authors contributed equally. +1 +arXiv:2301.13155v1 [cs.CV] 30 Jan 2023 + +Published as a conference paper at ICLR 2023 +these labelers have several practical limitations. First, some procedures of the label extraction work- +flow, such as rulemaking and natural language processing, still require the intensive involvement of +experts and engineers. Besides, the developed labelers can hardly adapt to new scenarios due to the +fixed rules and lexicons. +Against this background, report-supervised R2L was proposed (Zhang et al., 2020) to acquire super- +vision from radiology reports. In practice, this paradigm leverages words and sentences in free-text +reports as supervision to guide deep neural networks to learn radiograph representations, outper- +forming the archetypical label- and self-supervised pre-training by observable margins in various +downstream tasks (Zhang et al., 2020; Zhou et al., 2022). The report-supervised R2L highlights +the importance of the incorporation of domain knowledge. This differs from the self-supervised +paradigm, which focuses on learning invariant semantic representations. Nonetheless, current stud- +ies view the self- and report-supervised R2L as separate, discrete choices, preventing their combina- +tions. +Driven by this analysis, we present a unified framework based on masked record modeling (MRM), +where the self- and report-completion tasks are modeled as two complementary objectives. Specif- +ically, masked image reconstruction integrates semantics into pre-trained models, while masked +report restoration facilitates the incorporation of medical expertise. +As a result, MRM learns +knowledge-enhanced semantic representations that generalize well. In practice, MRM masks ran- +dom patches and tokens from the input radiograph and associated radiology report with high mask- +ing ratios. Following a multi-task scheme, MRM asks the radiography pre-trained model to learn +visual representations that can not only reconstruct the missing patches but also restore the missing +tokens from the non-masked token embeddings along with mask tokens. +With MRM pre-training, we can train radiography models on MIMIC-CXR (Johnson et al., 2019) +with improved generalization performance. With a pre-trained ViT-B/16 model, we achieve 88.5% +mean AUC when fine-tuned on CheXpert (Irvin et al., 2019) with only 1% labels. This outperforms +all previous counterparts with 100% labeled data. On NIH ChestX-ray (Wang & Gupta, 2015), +MRM surpasses the report-supervised paradigm by about 3% when the labeling ratios1 are 1% and +10%. On pneumonia identification tasks, MRM outperforms self- and report-supervised baselines, +sometimes by substantial margins. These observations help verify the effectiveness of MRM in +learning more transferable radiograph representations. +2 +RELATED WORK +2.1 +REPORT-SUPERVISED RADIOGRAPH REPRESENTATION LEARNING +Recently, report-supervised learning (Zhang et al., 2020; Liao et al., 2021; Huang et al., 2021; Zhou +et al., 2022; Boecking et al., 2022) emerges as a new R2L paradigm that automatically acquires +supervision from free-text radiology reports. Zhang et al. (2020) proposed ConVIRT to contrast the +radiograph features with latent embeddings of sentences in radiology reports. Liao et al. (2021) and +Huang et al. (2021) explored the alignment between local patches and words in the report. Zhou et al. +(2022) presented a Transformer-based R2L framework that conducts autoregressive report modeling +and study-report matching. Report-supervised R2L takes the advantage of label-supervised learning, +which is the incorporation of domain knowledge. Compared to the self-supervised paradigm, report- +supervised R2L lays no emphasis on learning semantically invariant representations. To address +the discrepancy between them, we formalize self- and report-completion as two complementary +objectives, based on which we propose to encode both semantics and medical expertise into latent +representations following a multi-task scheme. +2.2 +VISUAL REPRESENTATION LEARNING VIA IMAGE-LANGUAGE PRE-TRAINING +Learning visual representations from image-language pairs has achieved tremendous success in nat- +ural image tasks (Sariyildiz et al., 2020; Desai & Johnson, 2021; Radford et al., 2021; Mu et al., +2021; Li et al., 2021; Geng et al., 2022; Wang et al., 2022; Chen et al., 2022; Singh et al., 2022; Yu +et al., 2022; Dou et al., 2022; Arici et al., 2021; Kwon et al., 2022). Similar to ConVIRT (Zhang +1The labeling ratio X% means that X% of the training set from a fully annotated downstream dataset are +used for supervised fine-tuning. +2 + +Published as a conference paper at ICLR 2023 +Tokenizer +…… There is no focal +consolidation, effusion, +or pneumothorax. ……. +No free air below the +right hemidiaphragm is +seen. …… +Masked language modeling (MLM) loss +GAP +Duplication +Low-resolution input +…… There [MASK] no +[MASK] consolidation, +[MASK] , or [MASK]. ……. +No [MASK] air [MASK] +the [MASK] [MASK] is +seen. …… +…… There is no focal +consolidation, effusion, +or pneumothorax. ……. +No free air below the +right hemidiaphragm is +seen. …… +Paired +Rand. mask +(ratio = 50%) +Rand. mask +(ratio = 75%) +Image +encoder ++ +Report +decoder +Image +decoder +Sub-sampling +Masked image modeling (MIM) loss +High-resolution output +… +… +Non-masked report token embed. +Non-masked image patch embed. +Non-masked hybrid token embed. +Mask token embed. +Input report +Masked report generation +Hybrid representations +Masked report restoration +Input radiograph +Masked LR image generation +Radiograph representations +Masked HR image restoration +Figure 2: OVERVIEW. During the pre-training stage, MRM requires the image encoder to provide +radiograph representations to simultaneously support the restoration of masked radiograph patches +and masked associated radiology report tokens. The masked language and image modeling losses +are only calculated on image and report tokens highlighted in pink . Embed., LR, and HR stand +for embeddings, low-resolution, and high-resolution, respectively. +et al., 2020), image-language contrast has been widely adopted to conduct pre-training (Radford +et al., 2021; Mu et al., 2021; Li et al., 2021; Yu et al., 2022; Dou et al., 2022; Gao et al., 2022). +Nowadays, efforts have been made to train a unified encoder for vision and language data (Geng +et al., 2022; Wang et al., 2022; Chen et al., 2022; Geng et al., 2022). Akin to our approach, SLIP (Mu +et al., 2021) combines SimCLR (Chen et al., 2020) and CLIP (Radford et al., 2021) to train a vi- +sion encoder using image-language pairs. However, SLIP only slightly outperforms SimCLR in +fine-tuning, while requiring large batch sizes and tens of millions of image-language pairs for pre- +training. In contrast, our MRM surpasses various self-supervised methodologies by large margins +and can be pre-trained using only hundreds of thousands of radiograph-report pairs, enabling effec- +tive medical visual representation learning with limited annotations and computing resources. +3 +MASKED RECORD MODELING +We propose MRM (i.e., Masked Record Modeling) to learn radiograph representations using record- +level supervision. As the name implies, MRM acquires supervision signals from both radiographs +and associated radiology reports. The motivation behind is to learn knowledge-enhanced semantic +latent representations by reconstructing masked radiograph patches and masked radiology report +tokens in medical records. +Fig. 2 presents an overview of MRM. We first apply random masking to each low-resolution radio- +graph and its associated radiology report (with different high masking ratios). Then, we forward +the obtained non-masked image patches to the image encoder to acquire non-masked image patch +embeddings. These embeddings serve two purposes: (i) assist non-masked report tokens to restore +the masked report tokens; (ii) restore the high-resolution masked radiograph patches. To achieve +the first goal, we add the globally averaged radiograph representation to each non-masked report +token embedding and pass the resulting hybrid representations to the report decoder for masked re- +port restoration. As for the second purpose, we conduct a novel patch restoration task to explicitly +encode more local details into radiograph representations by reconstructing high-resolution patches +from low-resolution inputs. +3.1 +REPORT COMPREHENSION +Masked report generation. In our scenario, each radiology report is associated with a radiograph. +To convert the free-text report into tokens, we use WordPiece (Wu et al., 2016) as the default to- +kenizer, whose vocabulary has approximately 30k tokens. After tokenization, we randomly mask +3 + +2CPublished as a conference paper at ICLR 2023 +a number of report tokens with [MASK]. Compared to BERT (Devlin et al., 2018) that randomly +masks 15% tokens, we use a 50% probability of masking each token in the report. The insight +behind the use of a higher masking ratio is that we want the model to lean more upon the image +embeddings to finish the report-completion task. +Hybrid representations for storing multi-modal information. We then transform non-masked +report tokens into token embeddings using a simple lookup table2, which stores randomly initialized +embeddings of a fixed dictionary and size. In practice, we retrieve embeddings using indices. Then, +the global embedding of the associated radiograph is added to each non-masked token embedding. +The resulting non-masked hybrid embeddings are supposed to include the multi-modal information +from the radiograph and associated radiology report, which ought to be helpful for restoring the +masked tokens. +Masked report restoration. To reconstruct the masked tokens, we forward latent embeddings of +both hybrid tokens and mask tokens to the report decoder (a light-weight transformer model), where +fixed (i.e., unlearnable) positional embeddings are added to encode the position information. We +train the report decoder using the masked language modeling objective. +3.2 +RADIOGRAPH UNDERSTANDING +Masked image generation with low resolution. We propose to learn radiograph representations +by reconstructing high-resolution radiograph patches from low-resolution inputs. The motivation +behind is to encode more local information into latent embeddings via super-resolution imaging. +As shown in Fig. 2, we sub-sample each high-resolution radiograph by a factor of two to generate +a low-resolution input. Following He et al. (2022), we split low-resolution radiograph into non- +overlapping image patches, where 75% patches are randomly masked. +Radiograph representations. We add fixed unlearnable positional embeddings to linearly trans- +formed non-masked image patches. +Next, we forward the resulting patch embeddings to the +transformer-based image encoder, which produces non-masked image patch embeddings. Then, the +global average pooling (GAP) is applied to all non-masked embeddings, whereby a global feature +is obtained. Here, we hypothesize that the image-level information brought by the global feature +is helpful to the restoration of masked report tokens. Based on this hypothesis, we duplicate and +add the global feature to each non-masked report token embedding, producing the hybrid token +embeddings that encode the multi-modal information. +Masked image restoration with high resolution. Non-masked image and mask token represen- +tations with added positional embeddings are passed to the image decoder for the restoration of +masked radiograph patches. Specifically, the image decoder is required to restore a high-resolution +(2× the input resolution) patch from each input token via a shared fully-connected (FC) layer (across +all tokens). In practice, the proposed restoration procedure explicitly requires the learned image rep- +resentations to include more local details that often matter a lot in medical diagnosis. +3.3 +MULTI-TASK MODELING +Suppose each input radiograph consists of two set IM and IN . The masked set IM={x1, . . . , xh} +(ground truths) contains h high-resolution image patches that serve as reconstruction targets. The +non-masked set IN ={s1, . . . , sk} comprises k low-resolution patches that are treated as model in- +puts. Likewise, we denote the associated radiology report using the masked set RM={u1, . . . , up} +(ground truths) and the non-masked set RN ={v1, . . . , vq} (inputs). Here, x, s, u, and v stand for +the masked image patch, non-masked image patch, masked report token, and non-masked report +token, respectively. For model parameters, we use ΘE, ΘD, and ΘR to denote the parameters of the +image encoder, image decoder, and report decoder, respectively. +For the restoration of masked report tokens, we forward hybrid representations to the report decoder +and minimize the negative log-likelihood function. During the training stage, the objective function +2https://pytorch.org/docs/stable/generated/torch.nn.Embedding.html. +4 + +Published as a conference paper at ICLR 2023 +LR (i.e., the MLM loss in Fig. 2) of the above optimization procedure can be summarized as follows: +LR(RM, RN , IN ) = − +p +� +i=1 +log P (ui | v1:q, s1:k; ΘE, ΘR) , +(1) +where P stands for the conditional probability. We ignore the mask tokens for simplicity. +Similarly, we can formalize the objective function of the high-resolution masked radiograph restora- +tion (cf. the MIM loss in Fig. 2) as follows: +LI(IM, IN ) = MSE (fΘD(fΘE(s1:k)), x1:h) . +(2) +In practice, we adopt the mean squared error (MSE) to measure the differences between the predicted +and ground-truth image patches with high resolution, where all pixel values are normalized to [0, 1]. +The total multi-task training objective function (to be minimized) of the multi-modal restoration is +as follows: +L(RM, RN , IM, IN ) = LR(RM, RN , IN ) + λLI(IM, IN ) +(3) +where λ is a hyper-parameter that controls the relative impacts of two objective functions. After +pre-training, we can transfer the weight parameters of the image encoder (i.e., ΘE) to various down- +stream tasks for fine-tuning. +4 +EXPERIMENTS +In this section, we mainly compare MRM against report- and self-supervised R2L methodologies on +5 well-established public datasets. Average results are reported over three training runs. +4.1 +MIMIC-CXR FOR PRE-TRAINING +We conduct pre-training on MIMIC-CXR (Johnson et al., 2019), one of the largest X-ray datasets, +that contains more than 370,000 radiograph images from over 220,000 patient studies. Each radio- +graph is paired with one associated radiology report. +4.2 +DATASETS FOR FINE-TUNING +We validate the transferability of learned radiograph representations on X-ray based classification +and segmentation tasks via end-to-end fine-tuning. Specifically, we evaluate the pre-trained model +on 4 X-ray datasets in the classification tasks, which are NIH ChestX-ray (Wang et al., 2017), +CheXpert (Irvin et al., 2019), RSNA Pneumonia (Shih et al., 2019), and COVID-19 Image Data +Collection (Cohen et al., 2020). For the segmentation task, we fine-tune the pre-trained model on +SIIM-ACR Pneumothorax Segmentation.3 +CheXpert introduces a multi-label classification problem on chest X-rays. We follow the official +guideline (Irvin et al., 2019) and report the model performance on 5 selected pathologies, i.e., at- +electasis, cardiomegaly, consolidation, edema, and pleural effusion. Considering the official test +set of CheXpert is not available to the public, we follow ConVIRT (Zhang et al., 2020) to regard +the official validation set as the test set. Meanwhile, we randomly sample 5,000 images from the +official training set to build the validation set. The training/validation/test split each constitutes +218,414/5,000/234 images of the whole dataset. +RSNA Pneumonia defines a binary classification problem, where each chest radiograph is cat- +egorized as either pneumonia or normal. +We adopt the official data split, where the train- +ing/validation/test set comprises 25,184/1,500/3,000 images, respectively. +NIH ChestX-ray consists of about 112,120 frontal-view chest radiograph images, where a multi- +label classification problem on 14 chest pathologies is introduced. The training/validation/test split +each constitutes 70%/10%/20% of the whole dataset. +COVID-19 Image Data Collection is a relatively small dataset, which involves 900 chest radio- +graphs. We follow Zhou et al. (2022) to conduct fine-tuning on this small-scale dataset to investigate +3https://www.kaggle.com/c/siim-acr-pneumothorax-segmentation. +5 + +Published as a conference paper at ICLR 2023 +Methods +Input +Size +Pre-train. +Data +CheXpert +RSNA Pneumonia +SIIM +1% +10% +100% +1% +10% +100% +10% +100% +Our MRM +224 +MI-CXR +88.5± 0.7 +88.5± 0.6 +88.7± 0.3 +91.3± 0.6 +92.7± 0.4 +93.3± 0.4 +73.2± 0.5 +91.4± 0.3 +CNN-based +ConVIRT +224 +CheXpert +85.9 +86.8 +87.3 +77.4 +80.1 +81.3 +43.2 +59.9 +GLoRIA +224 +CheXpert +86.6 +87.8 +88.1 +86.1 +88.0 +88.6 +46.9 +63.4 +ConVIRT +224 +MI-CXR +87.0 +88.1 +88.1 +88.8 +91.5 +92.7 +- +- +MedKLIP† +224 +MI-CXR +- +- +- +87.3 +88.0 +89.3 +72.1 +79.4 +BioViL +480 +PubMed + +MI-III/CXR +- +- +- +88.1 +88.4 +89.1 +- +- +Transformer-based +GLoRIA∗ +224 +MI-CXR +86.5± 0.8 +87.5± 0.6 +87.8± 0.5 +89.7± 0.8 +91.2± 0.5 +92.1± 0.3 +71.8± 0.7 +90.9± 0.4 +REFERS +224 +MI-CXR +87.2± 0.8 +88.1± 0.5 +88.2± 0.3 +89.4± 0.7 +91.6± 0.7 +92.7± 0.4 +72.1± 0.5 +89.7± 0.2 +M3AE +224 +MI-CXR +86.2± 0.6 +87.3± 0.6 +87.9± 0.4 +89.0± 0.5 +90.8± 0.6 +92.3± 0.3 +72.0± 0.7 +90.4± 0.3 +MGCA† +224 +MI-CXR +88.8 +89.1 +89.7 +89.1 +89.9 +90.8 +59.3 +64.2 +Table 1: COMPARISONS ON CHEXPERT, RSNA PNEUMONIA, AND SIIM. We report AUC scores +of different labeling ratios when fine-tuning on CheXpert and RSNA Pneumonia. In compari- +son, dice scores are presented on SIIM. The best results are bolded. MI- stands for the MIMIC +dataset series. Note that ResNet-50 and ViT-B/16 are treated as the default backbones for CNN- +and Transformer-based methods, respectively. * denotes our implementation of GLoRIA using ViT- +B/16. Approaches with † leverage disease-level annotations for pre-training. Specifically, numbers +of MGCA on CheXpert and RSNA Pneumonia are linear classification results. +Labeling Ratios +Methods +Average +Atelectasis +Cardiomegaly +Consolidation +Edema +Effusion +Emphysema +Fibrosis +Hernia +Infiltration +Mass +Nodule +Pleural Thickening +Pneumonia +Pneumothorax +1% +Our MRM +79.4± 0.8 +78.8 +90.3 +80.0 +86.5 +86.9 +82.0 +71.9 +90.0 +67.2 +82.3 +69.6 +72.3 +69.6 +84.0 +MedKLIP +77.2 +- +- +- +- +- +- +- +- +- +- +- +- +- +- +REFERS +76.7 +77.5 +85.6 +78.6 +84.9 +85.4 +79.5 +72.3 +77.1 +67.5 +76.2 +66.5 +71.6 +69.3 +81.7 +Model Genesis +70.3 +72.1 +67.1 +75.8 +76.1 +80.6 +72.6 +64.8 +73.5 +65.7 +65.2 +62.2 +67.6 +64.8 +76.2 +C2L +71.1 +75.1 +67.1 +77.6 +75.1 +83.4 +71.5 +66.8 +70.0 +63.8 +70.1 +66.2 +68.1 +65.7 +74.4 +Context Restoration +67.8 +69.1 +64.4 +73.2 +73.8 +78.1 +70.0 +62.1 +70.2 +65.2 +62.4 +59.1 +65.0 +62.2 +73.8 +TransVW +71.3 +74.5 +68.9 +76.7 +79.8 +81.1 +67.9 +68.7 +68.2 +66.8 +66.5 +66.2 +68.5 +68.8 +75.0 +ImageNet Pre-training +69.8 +73.3 +69.6 +76.0 +81.7 +80.5 +67.1 +64.9 +64.8 +65.8 +67.0 +62.3 +65.7 +65.0 +74.0 +10% +Our MRM +84.0± 0.5 +82.3 +90.9 +81.1 +89.0 +88.8 +92.2 +84.8 +94.0 +70.1 +86.6 +75.1 +78.6 +74.3 +88.4 +MedKLIP +78.9 +- +- +- +- +- +- +- +- +- +- +- +- +- +- +REFERS +80.9 +80.1 +89.8 +79.5 +87.8 +87.5 +88.2 +77.2 +86.1 +69.6 +82.0 +72.8 +74.2 +72.2 +85.6 +Model Genesis +76.0 +77.2 +72.8 +77.5 +85.7 +85.2 +81.0 +75.3 +78.0 +68.4 +73.1 +69.5 +72.2 +67.7 +80.4 +C2L +76.6 +78.0 +75.5 +77.5 +84.1 +85.7 +81.2 +73.7 +79.5 +67.4 +77.5 +71.7 +72.0 +67.3 +81.9 +Context Restoration +73.8 +75.5 +70.6 +77.1 +84.5 +84.2 +79.4 +73.1 +67.5 +68.1 +70.9 +66.9 +71.7 +65.2 +79.1 +TransVW +74.4 +76.5 +70.8 +77.6 +83.0 +84.8 +79.7 +69.9 +74.7 +68.5 +72.1 +68.3 +72.4 +63.2 +79.6 +ImageNet Pre-training +74.4 +74.2 +79.8 +75.9 +85.7 +83.2 +80.4 +72.1 +74.0 +64.1 +71.7 +65.6 +69.6 +66.2 +79.7 +100% +Our MRM +85.9± 0.3 +84.2 +93.0 +82.2 +91.0 +89.6 +94.3 +86.7 +94.4 +71.8 +88.2 +78.5 +81.4 +77.3 +90.2 +MedKLIP +83.2 +- +- +- +- +- +- +- +- +- +- +- +- +- +- +REFERS +84.7 +83.0 +92.3 +82.1 +90.2 +88.7 +91.4 +83.9 +93.3 +74.1 +85.5 +76.7 +78.5 +77.0 +89.1 +Model Genesis +81.0 +78.8 +84.5 +79.2 +87.8 +86.6 +89.7 +81.0 +85.2 +71.1 +81.9 +73.2 +75.8 +73.0 +85.6 +C2L +82.2 +81.1 +90.2 +81.0 +88.1 +88.0 +88.3 +80.8 +86.8 +72.0 +82.7 +74.1 +76.2 +75.3 +85.9 +Context Restoration +78.7 +75.8 +82.9 +76.4 +86.6 +84.8 +88.2 +78.6 +83.0 +70.0 +79.6 +69.5 +73.2 +69.4 +84.0 +TransVW +81.7 +79.8 +85.0 +80.0 +88.2 +87.1 +90.1 +81.8 +85.9 +72.3 +82.6 +74.4 +76.6 +74.0 +86.1 +ImageNet Pre-training +80.0 +78.3 +89.3 +77.6 +87.9 +85.9 +87.4 +78.5 +88.8 +65.9 +79.9 +70.7 +74.5 +71.0 +84.7 +Table 2: COMPARISONS ON NIH CHESTX-RAY. Besides self-supervised and transfer learning +baselines, we also present the performance of REFERS and MedKLIP (with competitive perfor- +mance on CheXpert, RSNA Pneumonia, and SIIM) as references. AUC scores are displayed. The +highest AUC scores in each labeling ratio are bolded. +the effectiveness of various pre-training methodologies when the amount of annotations is limited. +There are two tasks included. The first task requires the model to distinguish COVID-19 cases +from non-COVID-19 pneumonia cases, where the training/validation/test set comprises 356/54/99 +radiographs, respectively. The second task is to distinguish viral pneumonia cases from bacterial +pneumonia ones, where the training/validation/test set contains 297/43/86 cases, respectively. +SIIM-ACR Pneumothorax Segmentation (SIIM) aims to facilitate the development of segmen- +tation models to identify pneumothorax disease in chest radiographs. SIIM contains over 120,000 +frontal-view chest X-rays with precise manual segmentation of pneumothorax. We follow Huang +et al. (2021) to construct the training/validation/test split, where each constitutes 70%/15%/15% of +the whole dataset. +6 + +Published as a conference paper at ICLR 2023 +4.3 +BASELINES +4.3.1 +REPORT-SUPERVISED METHODOLOGIES +We first compare MRM against a range of pre-training approaches, which use radiology reports +as supervision to learn radiograph representations. There are 4 report-supervised approaches in- +volved in the baseline comparisons, which are ConVIRT (Zhang et al., 2020), GLoRIA (Huang +et al., 2021), BioViL (Boecking et al., 2022), and REFERS (Zhou et al., 2022). Specifically, Con- +VIRT (Zhang et al., 2020) proposed to learn medical visual representations by contrasting paired +radiographs and sentences from radiology reports. GLoRIA (Huang et al., 2021) improved ConVIRT +by contrasting radiograph sub-regions and words in the reports. BioViL (Boecking et al., 2022) and +REFERS (Zhou et al., 2022) incorporated masked language modeling loss into contrastive learn- +ing. Moreover, REFERS introduced a multi-view fusion attention to better align the representations +of each radiograph and its associated report. In addition, MGCA (Wang et al., 2023) and Med- +KLIP (Wu et al., 2023) were included as two recent baselines4. Apart from above baselines, we also +include M3AE (Geng et al., 2022), a recent masked multi-modal pre-training method aside from the +application to medical data, for comparison. +In experiments, we fine-tune pre-trained models of MRM and other report-supervised methods on +CheXpert (classification), RSNA Pneumonia (classification), and SIIM (segmentation). +4.3.2 +SELF-SUPERVISED AND TRANSFER LEARNING METHODS +Besides report-supervised approaches, we also include self-supervised and transfer learning ap- +proaches in our comparisons. Specifically, Context Restoration (Chen et al., 2019), Model Gen- +esis (Zhou et al., 2021), and TransVW (Haghighi et al., 2021) are based on predictive SSL, while +C2L (Zhou et al., 2020) was developed on top of contrastive learning. In addition, MRM is also +compared to ImageNet pre-training (Wang et al., 2017). In practice, we conduct the comparisons +with self-supervised and transfer learning approaches on NIH ChestX-ray and COVID-19 Image +Data Collection. +Methods +COVID-19 vs. Others +Viral vs. Bacterial +Our MRM +85.8± 0.4 +91.5± 0.3 +REFERS +82.1 +80.4 +Model Genesis +76.0 +71.8 +C2L +77.8 +73.0 +Context Restoration +74.6 +69.8 +TransVW +76.1 +71.5 +ImageNet Pre-training +74.1 +70.3 +Table 3: COMPARISONS ON COVID- +19 IMAGE DATA COLLECTION. +The +best results are bolded. +Methods +1% +10% +100% +Our MRM +79.4 +84.0 +85.9 +- Masked modeling & Super-resolution restoration +69.9 +75.2 +80.3 +- Masked report modeling (LR) +74.7 +81.3 +85.1 +- Masked radiograph modeling (LI) +76.7 +82.2 +84.7 +- Super-resolution restoration +78.8 +83.7 +85.7 ++ Hybrid features for image restoration +78.9 +83.6 +85.7 +Table 4: ABLATIONS ON NIH CHESTX-RAY. +AUC scores of three different labeling ratios are +reported. The best results are bolded. +4.4 +RESULTS +4.4.1 +COMPARISONS WITH REPORT-SUPERVISED BASELINES +In Table 1, we present the comparative results with report-supervised methodologies on CheXpert, +RSNA Pneumonia, and SIIM (segmentation). Specifically, we investigate the performance when +fine-tuning with limited and full supervision. We provide reconstruction examples and segmentation +results in the appendix. +From Table 1, we observe no obvious performance gap between CNN- and Transformer-based +report-supervised pre-training methods. For instance, after implementing GLoRIA on MIMIC-CXR +with ViT-B/16, we observe performance drops and improvements on CheXpert and RSNA Pneumo- +nia, respectively. This contrast demonstrates that replacing CNN with Transformer may not bring +performance gains to report-supervised pre-training. Among all baselines, REFERS is the best per- +forming approach. +Nonetheless, MRM consistently outperforms various baselines on all three datasets under different +labeling ratios. Specifically, MRM maintains more advantages over previous pre-training method- +4MGCA (Wang et al., 2023) and MedKLIP (Wu et al., 2023) were released after the submission deadline of +ICLR 2023. We added results in the camera ready for better comparisons. +7 + +Published as a conference paper at ICLR 2023 +Fusion +1% +10% +100% +GAP +79.4 +84.0 +85.9 +GMP +77.2 +83.8 +86.0 +(a) Multi-modal fusion strategies +PI +1% +10% +100% +75% +79.4 +84.0 +85.9 +50% +78.8 +84.4 +86.1 +0% +75.4 +82.0 +84.4 +(b) Masking ratios for radiographs +λ +1% +10% +100% +1 +79.4 +84.0 +85.9 +2 +78.6 +83.5 +85.4 +0.5 +79.0 +83.7 +85.9 +(c) Loss controlling factor λ +Table 5: Ablation studies on choices of multi-modal fusion and hyper-parameters. GAP and GMP +stand for global average and maximum pooling, respectively. Experiments are performed on NIH +ChestX-ray under various labeling ratios. The best results are bolded. +ologies when fine-tuning with limited annotations, which is quite meaningful for medical image +analysis as large amounts of specialist annotations (from radiologists or clinicians) are usually hard +to access. It is worth noting that MRM achieves 88.5% when using only 1% labeled data on CheX- +pert, better than previous counterparts with 100% annotations. +We see that M3AE generally performs worse than our MRM in all labeling ratios, especially under +extremely limited data. The underperformance of M3AE may be attributed to the fact that it requires +a large amount of multi-modal data to learn transferable joint representations for images and texts. +4.4.2 +COMPARISONS WITH SELF-SUPERVISED AND TRANSFER LEARNING BASELINES +Table 2 presents the average and per-class classification results on NIH ChestX-ray. Compared to +self-supervised learning baselines, MRM achieves large improvements in almost every chest pathol- +ogy. On average, MRM outperforms C2L (Zhou et al., 2020) and TransVW (Haghighi et al., 2021), +the best two self-supervised pre-training methodologies, by about 8% when the amount of available +labeled data is extremely limited (i.e., the labeling ratio is 1%). Similarly, we also observe remark- +able improvements when comparing MRM to ImageNet Pre-training (Wang et al., 2017). These +phenomena demonstrate that the radiograph representations learned by MRM are more transferable +than those from previous self-supervised and transfer learning methods. +Compared to REFERS (i.e., the best performing report-supervised baseline in Table 1), MRM still +provides notable improvements in most chest pathologies. Specifically, MRM is more advantageous +when the amount of labeled data is limited. For instance, MRM surpasses REFERS by 2.7% and +3.1% on average when the labeling ratios are 1% and 10%, respectively. These comparisons again +verify the effectiveness of MRM over previous report-supervised pre-training counterparts. +We also investigate the impacts of pre-training on the real-world scenario with extremely limited +specialist supervision. In Table 3, we compare the performance of a range of pre-training method- +ologies on two binary classification tasks, which are distinguishing COVID-19 from non-COVID-19 +pneumonia, and differentiating between viral pneumonia and bacterial pneumonia. Again, MRM +outperforms self-supervised, transfer learning, and report-supervised pre-training methodologies by +substantial margins. Compared to REFERS, MRM brings nearly 4% and 11% improvements to +two tasks, respectively, further enhancing the practicability of the diagnosis system trained with +extremely limited supervision. +4.5 +ABLATION ANALYSIS +Advantages over single-task pre-training paradigm. First of all, we remove the two masked mod- +eling objectives and super-resolution restoration task, resulting in substantial performance drops. +These results verify the necessity of using masked modeling and super-resolution restoration in +MRM. After removing the masked report modeling objective, MRM only acquires supervision +signals from self-supervision. Thus, the whole framework degenerates into a self-supervised pre- +training methodology. From Table 4, we observe that removing LR leads to dramatic performance +degradation in different labeling ratios. Moreover, we find that introducing the masked report mod- +eling is greatly helpful to the fine-tuning performance with limited labeling resources (1% and +10%). For instance, adding LR brings about 5-percent improvements to MRM. Similarly, remov- +ing the masked radiograph modeling also leads to notable performance drops in all labeling ratios. +These results demonstrate the necessity of introducing multi-task objectives in masked modeling +pre-training. +8 + +Published as a conference paper at ICLR 2023 +Is super-resolution restoration helpful? As Table 4 shows, the proposed super-resolution restora- +tion provides consistent performance gains in different labeling ratios. The underlying reason may +be that the low to high resolution restoration process helps preserve more local information into +latent representations, which enhances the transferable ability to downstream tasks. +Would it be beneficial to introduce multi-modal information to image restoration? We investi- +gate this question by adding non-masked report token embeddings to image patch embeddings and +passing the resulting hybrid features to the image decoder. As Table 4 shows, introducing multi- +modal information to masked image restoration does not improve the fine-tuning performance. We +leave the exploration of the reason behind to future work. +Ablations on multi-modal fusion and hyper-parameters. In Table 5a, we present the experimental +results of using different strategies for radiograph-report fusion. We find that global average pooling +(GAP) outperforms global maximum pooling (GMP) by 2.2% when access to labeled data is quite +limited while achieving comparable results as the labeling ratio increases. We also investigate the +impact of applying different masking ratios to input radiographs (cf. Table 5b). Specifically, we find +that a ratio of 75% performs the best on the extremely small labeling ratio (i.e., 1%), while a ratio +of 50% achieves slightly better results when the labeling ratio becomes larger. In MRM, we set the +default masking ratio for radiographs to 75% because this operation leads to fewer input patches, +accelerating the pre-training process and reducing the memory cost. The insight behind applying a +high masking ratio (i.e., 75%) is that it addresses the heavy spatial redundancy of radiographs. By +applying a high masking ratio, we reduce the redundancy and create a surrogate task, requiring the +model to understand the high-level semantics holistically. We also perform ablative experiments to +investigate the influence of λ in Eq. 3, whose results are displayed in Table 5c. We see that λ = 1 +is an optimal choice, while a smaller λ value (i.e, 0.5) performs better than a larger value (i.e., 2.0), +indicating that the MLM objective may play a more important role than the MIM objective during +the pre-training stage. +4.6 +IMPLEMENTATION DETAILS. +Our code is implemented using PyTorch 1.8.2 (Paszke et al., 2019). +The pre-training experi- +ments were conducted on 4 GeForce RTX 3080Ti GPUs, and the training time is about 2 days +for 200 epochs, requiring 12GB memory from each GPU. The training batch size is 256. We use +AdamW (Loshchilov & Hutter, 2017) as the default optimizer, where the initial learning rate is +1.5e−4, weight decay is 0.05, β1 is 0.9, and β2 is 0.95. The MSE and cross-entropy losses are used +for masked image and language modeling, respectively. In practice, we set λ in Eq. 3 to 1. +For fine-tuning on SIIM, we train the segmentation network on 4 GeForce RTX 3080Ti GPUs. +AdamW is the default optimizer, where the initial learning rate is 2e−5, weight decay is 0.05, β1 is +0.9, and β2 is 0.999. For fine-tuning on other datasets, we train the classification network on a single +GeForce RTX 3080Ti GPU, where the default optimizer is SGD with momentum 0.9. The training +cost is a mix of focal and dice losses. +For fine-tuning on CheXpert, RSNA Pneumonia, NIH ChestX-ray, and COVID-19 Image Data Col- +lection, we adopt the cross-entropy loss. We search the best initial learning rate from 3e-2, 3e-3, and +5e-4 to get the best performance on validation set. +For both the pre-training and the fine-tuning of image classification task, the network is ”warmed +up” by increasing the learning rate linearly to the set value, and then learning rate is decreased using +the cosine decay schedule. +5 +CONCLUSION +We present masked record modeling (MRM) for radiograph representation learning. MRM for- +malizes the radiograph understanding and radiology report comprehension as two complementary +masked modeling objectives. With MRM pre-training, we achieve better results on well-established +datasets. Specifically, MRM outperforms previous self- and report-supervised counterparts by large +margins when the labeled data is extremely limited. 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M3AE (Geng et al., 2022) and MLIM (Arici et al., 2021) aim to learn joint image- +text representations. MaskVLM (Kwon et al., 2022) is developed for improving vision+language +tasks, such as image-text retrieval, visual question answering, and natural language for visual rea- +soning. These characteristics differ from our methodology, which aims to learn a representation for +radiographs only for disease diagnosis even though both radiographs and reports are used during +training. +Implementation. As aforementioned, M3AE (Geng et al., 2022) and MLIM (Arici et al., 2021) +implement unified multi-modal transformers that take imaging and textual as inputs. As a result, +they require much more pre-training data, which is often an order of magnitude larger than ours. +This is not practical and applicable in the medical field, where access to the data is quite limited. +MaskVLM (Kwon et al., 2022) applies masked modeling and contrastive learning to image-text +pairs, where a binary matching task is employed to tell whether an image and a text are aligned or +not. Besides, MaskVLM builds cross-modality encoders to incorporate multi-modal information. In +contrast, our MRM is much simpler. MRM uses masking modeling as the only training objective, +and a simple GAP-addition workflow is proposed for fusing multi-modal information. +A.2 +CONFIGURATIONS OF NETWORKS +The image encoder a ViT-like encoder (Dosovitskiy et al., 2020) that includes a patch embedding +layer followed by twelve transformer blocks. The architecture of the image decoder is very similar +to that of the encoder. The decoder architecture consists of a decoder embedding layer, four trans- +former blocks, and one fully-connected layer to predict masked patches. The numbers of attention +heads in the image encoder and decoder are twelve and six, respectively. We add a 2D sin-cos posi- +tional embedding to input patches. The report decoder is a light-weight transformer that includes an +embedding layer and six transformer blocks, followed by a one-layer fully-connected predictor. The +number of attention heads in the report decoder is six. We adopt a learnable positional embedding +in the report decoder. +A.3 +RECONSTRUCTION ANALYSIS +Fig. 3 presents example results on MIMIC-CXR. We find that even with high masking ratios, MRM +can still produce satisfactory reconstructions, though some details are missing. Specifically, the re- +constructed reports are surprisingly close to the ground truths, which we attribute to the introduction +of hybrid multi-modal representations. For instance, MRM can tell the position of rib fractures based +on the radiograph input. Besides, we obtain some interesting mistakes. In the first example, MRM +recognizes the clips over the left lung as calcifications (potentially in nipple shadow). This observa- +tion again shows that the report reconstructions rely on the image input as the clips are masked in +the input radiograph. +A.4 +SEGMENTATION ANALYSIS +In this section, we visualize the segmentation results of GLoRIA (Huang et al., 2021), +REFERS (Zhou et al., 2022), and our MRM. +13 + +Published as a conference paper at ICLR 2023 +[CLS] final [MASK] [MASK] [MASK] chest [MASK] pa and [MASK] ) indication : ___f with [MASK] onset [MASK] // [MASK] for infection [MASK] : [MASK] [MASK] [MASK] +[MASK] [MASK] [MASK] none . [MASK] : [MASK] is [MASK] focal [MASK] [MASK] pleural [MASK] or pneumothorax [MASK] [MASK] [MASK] [MASK] that [MASK] likely +represent nipple shadows . [MASK] [MASK] silhouette [MASK] [MASK] . [MASK] [MASK] over the left [MASK] , potentially [MASK] [MASK] [MASK] . the [MASK] upper +abdomen [MASK] [MASK] . [MASK] [MASK] of [MASK] [MASK] left sixth [MASK] seventh [MASK] are [MASK] [MASK] [MASK] : no acute cardiopulmonary process . +[CLS] final report examination : chest ( pa and lat ) indication : ___f with new onset confusion // eval for infection technique : chest pa and lateral comparison : +none . findings : there is no focal consolidation , pleural effusion or pneumothorax . bilateral nodular opacities that most likely represent nipple shadows . the +cardiomediastinal silhouette is normal . calcifications project over the left lung , potentially a nipple shadow . the imaged upper abdomen is unremarkable . healed +fractures of the posterior left sixth and seventh ribs are noted . impression : no acute cardiopulmonary process . +[CLS] final report examination : chest ( pa and lat ) indication : ___f with new onset ascites // eval for infection technique : chest pa and lateral comparison : +none . findings : there is no focal consolidation , pleural effusion or pneumothorax . bilateral nodular opacities that most likely represent nipple shadows . the +cardiomediastinal silhouette is normal . clips project over the left lung , potentially within the breast . the imaged upper abdomen is unremarkable . chronic +deformity of the posterior left sixth and seventh ribs are noted . impression : no acute cardiopulmonary process . +[CLS] [MASK] report [MASK] [MASK] [MASK] ( [MASK] [MASK] lat ) [MASK] : [MASK] [MASK] [MASK] with shortness [MASK] [MASK] [MASK] : [MASK] [MASK] and [MASK] +comparison : [MASK] findings : [MASK] cardiac , mediastinal and hilar [MASK] are normal . [MASK] [MASK] [MASK] normal. lungs are [MASK] . [MASK] pleural [MASK] +[MASK] pneumothorax [MASK] [MASK] . multiple clips [MASK] again seen [MASK] [MASK] the left [MASK] . remote [MASK] - sided rib fractures are also re - [MASK] . +impression : [MASK] [MASK] cardiopulmonary abnormality . +[CLS] final report examination : chest ( pa and lat ) indication : history : ___f with shortness of breath technique : chest pa and lateral comparison : ___ findings : +the cardiac , mediastinal and hilar contours are normal . pulmonary vasculature is normal . lungs are clear . no pleural effusion or pneumothorax is present . +multiple clips are again seen projecting over the left axilla . remote left- sided rib fractures are also re - demonstrated . impression : no acute cardiopulmonary +abnormality . +[CLS] final report examination : chest ( pa and lat ) indication : history : ___f with shortness of breath technique : chest pa and lateral comparison : ___ findings : +the cardiac , mediastinal and hilar contours are normal . pulmonary vasculature is normal . lungs are clear . no pleural effusion or pneumothorax is present . +multiple clips are again seen projecting over the left breast . remote left - sided rib fractures are also re - demonstrated . impression : no acute cardiopulmonary +abnormality . +Inputs +Recons. +GT +Inputs +Recons. +GT +Figure 3: Example results on MIMIC-CXR. For each triplet, we show the masked radiograph and +report (Inputs), our MRM reconstruction (Recons.), and the ground truth (GT). The masking ra- +tios are 75% (radiograph) and 50% (report). Predicted and corresponding ground truth words are +highlighted in pink and green , respectively. +14 + +OPublished as a conference paper at ICLR 2023 +Radiograph +GLoRIA* +REFERS +Our MRM +GT +15 + +ECGVPublished as a conference paper at ICLR 2023 +Radiograph +GLoRIA* +REFERS +Our MRM +GT +Figure 4: Segmentation results of MRM, GLoRIA, and REFERS. GT stands for the ground truth +masks. * means the GLoRIA implementation is based on ViT-B/16, the same backbone as used in +REFERS and MRM. +16 + +LPORTABLE \ No newline at end of file diff --git a/ZNFPT4oBgHgl3EQfuTUu/content/tmp_files/load_file.txt b/ZNFPT4oBgHgl3EQfuTUu/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..65037432eab850586020d4289c071235dc89d0e6 --- /dev/null +++ b/ZNFPT4oBgHgl3EQfuTUu/content/tmp_files/load_file.txt @@ -0,0 +1,1236 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf,len=1235 +page_content='Published as a conference paper at ICLR 2023 ADVANCING RADIOGRAPH REPRESENTATION LEARN- ING WITH MASKED RECORD MODELING Hong-Yu Zhou1,2∗ Chenyu Lian1∗ Liansheng Wang1 Yizhou Yu2 1School of Informatics, Xiamen University 2Department of Computer Science, The University of Hong Kong whuzhouhongyu@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='com, dopaminel@foxmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='com, lswang@xmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='cn, yizhouy@acm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='org ABSTRACT Modern studies in radiograph representation learning (R2L) rely on either self- supervision to encode invariant semantics or associated radiology reports to in- corporate medical expertise, while the complementarity between them is barely noticed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' To explore this, we formulate the self- and report-completion as two com- plementary objectives and present a unified framework based on masked record modeling (MRM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' In practice, MRM reconstructs masked image patches and masked report tokens following a multi-task scheme to learn knowledge-enhanced semantic representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' With MRM pre-training, we obtain pre-trained mod- els that can be well transferred to various radiography tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Specifically, we find that MRM offers superior performance in label-efficient fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' For in- stance, MRM achieves 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='5% mean AUC on CheXpert using 1% labeled data, outperforming previous R2L methods with 100% labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' On NIH ChestX-ray, MRM outperforms the best performing counterpart by about 3% under small la- beling ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Besides, MRM surpasses self- and report-supervised pre-training in identifying the pneumonia type and the pneumothorax area, sometimes by large margins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Code and models are available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='com/RL4M/ MRM-pytorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' 1 INTRODUCTION Findings: [MASK] cardiac, mediastinal and hilar [MASK] are normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' [MASK] [MASK] [MASK] normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' lungs are [MASK] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' … … Inputs Report Radiograph Paired Representations Findings: the cardiac, mediastinal and hilar contours are normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' pulmonary vasculature is normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' lungs are clear .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Outputs Figure 1: Illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' MRM learns trans- ferable radiograph representations via recon- structing masked records, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', masked radio- graph patches and masked reports tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Radiograph representation learning (R2L) has been among the core problems of medical image analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Previously, downstream radiograph analysis tasks counts on pre-trained models on ImageNet (Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2009) or large X-ray datasets (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Irvin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Bus- tos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2020) to alleviate the shortage of expert labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' The emergence of self-supervised repre- sentation learning (Doersch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Agrawal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Wang & Gupta, 2015) provides a choice to conduct pre-training with negligible human inter- vention by exploiting self-supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' However, the self-supervised paradigm ignores the introduction of medical expertise (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', anatomy), reducing its trans- ferability to downstream tasks with limited label in- formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' On the other hand, free-text radiology reports written by experienced radiologists often contain rich do- main knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' To leverage this, researchers developed automated rule-based labelers (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Irvin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2019) to extract structured labels from unstructured texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Nevertheless, ∗Work done while visiting Xiamen University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' First two authors contributed equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='13155v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='CV] 30 Jan 2023 Published as a conference paper at ICLR 2023 these labelers have several practical limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' First, some procedures of the label extraction work- flow, such as rulemaking and natural language processing, still require the intensive involvement of experts and engineers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Besides, the developed labelers can hardly adapt to new scenarios due to the fixed rules and lexicons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Against this background, report-supervised R2L was proposed (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2020) to acquire super- vision from radiology reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' In practice, this paradigm leverages words and sentences in free-text reports as supervision to guide deep neural networks to learn radiograph representations, outper- forming the archetypical label- and self-supervised pre-training by observable margins in various downstream tasks (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' The report-supervised R2L highlights the importance of the incorporation of domain knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' This differs from the self-supervised paradigm, which focuses on learning invariant semantic representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Nonetheless, current stud- ies view the self- and report-supervised R2L as separate, discrete choices, preventing their combina- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Driven by this analysis, we present a unified framework based on masked record modeling (MRM), where the self- and report-completion tasks are modeled as two complementary objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Specif- ically, masked image reconstruction integrates semantics into pre-trained models, while masked report restoration facilitates the incorporation of medical expertise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' As a result, MRM learns knowledge-enhanced semantic representations that generalize well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' In practice, MRM masks ran- dom patches and tokens from the input radiograph and associated radiology report with high mask- ing ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Following a multi-task scheme, MRM asks the radiography pre-trained model to learn visual representations that can not only reconstruct the missing patches but also restore the missing tokens from the non-masked token embeddings along with mask tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' With MRM pre-training, we can train radiography models on MIMIC-CXR (Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2019) with improved generalization performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' With a pre-trained ViT-B/16 model, we achieve 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='5% mean AUC when fine-tuned on CheXpert (Irvin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2019) with only 1% labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' This outperforms all previous counterparts with 100% labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' On NIH ChestX-ray (Wang & Gupta, 2015), MRM surpasses the report-supervised paradigm by about 3% when the labeling ratios1 are 1% and 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' On pneumonia identification tasks, MRM outperforms self- and report-supervised baselines, sometimes by substantial margins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' These observations help verify the effectiveness of MRM in learning more transferable radiograph representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' 2 RELATED WORK 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='1 REPORT-SUPERVISED RADIOGRAPH REPRESENTATION LEARNING Recently, report-supervised learning (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Liao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Boecking et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2022) emerges as a new R2L paradigm that automatically acquires supervision from free-text radiology reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' (2020) proposed ConVIRT to contrast the radiograph features with latent embeddings of sentences in radiology reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Liao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' (2021) and Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' (2021) explored the alignment between local patches and words in the report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' (2022) presented a Transformer-based R2L framework that conducts autoregressive report modeling and study-report matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Report-supervised R2L takes the advantage of label-supervised learning, which is the incorporation of domain knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Compared to the self-supervised paradigm, report- supervised R2L lays no emphasis on learning semantically invariant representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' To address the discrepancy between them, we formalize self- and report-completion as two complementary objectives, based on which we propose to encode both semantics and medical expertise into latent representations following a multi-task scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='2 VISUAL REPRESENTATION LEARNING VIA IMAGE-LANGUAGE PRE-TRAINING Learning visual representations from image-language pairs has achieved tremendous success in nat- ural image tasks (Sariyildiz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Desai & Johnson, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Radford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Mu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Geng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Dou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Arici et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Kwon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Similar to ConVIRT (Zhang 1The labeling ratio X% means that X% of the training set from a fully annotated downstream dataset are used for supervised fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' 2 Published as a conference paper at ICLR 2023 Tokenizer ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' There is no focal consolidation, effusion, or pneumothorax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' No free air below the right hemidiaphragm is seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Masked language modeling (MLM) loss GAP Duplication Low-resolution input ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' There [MASK] no [MASK] consolidation, [MASK] , or [MASK].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' No [MASK] air [MASK] the [MASK] [MASK] is seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' There is no focal consolidation, effusion, or pneumothorax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' No free air below the right hemidiaphragm is seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Paired Rand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' mask (ratio = 50%) Rand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' mask (ratio = 75%) Image encoder + Report decoder Image decoder Sub-sampling Masked image modeling (MIM) loss High-resolution output … … Non-masked report token embed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Non-masked image patch embed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Non-masked hybrid token embed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Mask token embed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Input report Masked report generation Hybrid representations Masked report restoration Input radiograph Masked LR image generation Radiograph representations Masked HR image restoration Figure 2: OVERVIEW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' During the pre-training stage, MRM requires the image encoder to provide radiograph representations to simultaneously support the restoration of masked radiograph patches and masked associated radiology report tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' The masked language and image modeling losses are only calculated on image and report tokens highlighted in pink .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Embed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', LR, and HR stand for embeddings, low-resolution, and high-resolution, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2020), image-language contrast has been widely adopted to conduct pre-training (Radford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Mu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Dou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Nowadays, efforts have been made to train a unified encoder for vision and language data (Geng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Geng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Akin to our approach, SLIP (Mu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2021) combines SimCLR (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2020) and CLIP (Radford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2021) to train a vi- sion encoder using image-language pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' However, SLIP only slightly outperforms SimCLR in fine-tuning, while requiring large batch sizes and tens of millions of image-language pairs for pre- training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' In contrast, our MRM surpasses various self-supervised methodologies by large margins and can be pre-trained using only hundreds of thousands of radiograph-report pairs, enabling effec- tive medical visual representation learning with limited annotations and computing resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' 3 MASKED RECORD MODELING We propose MRM (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', Masked Record Modeling) to learn radiograph representations using record- level supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' As the name implies, MRM acquires supervision signals from both radiographs and associated radiology reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' The motivation behind is to learn knowledge-enhanced semantic latent representations by reconstructing masked radiograph patches and masked radiology report tokens in medical records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' 2 presents an overview of MRM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' We first apply random masking to each low-resolution radio- graph and its associated radiology report (with different high masking ratios).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Then, we forward the obtained non-masked image patches to the image encoder to acquire non-masked image patch embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' These embeddings serve two purposes: (i) assist non-masked report tokens to restore the masked report tokens;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' (ii) restore the high-resolution masked radiograph patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' To achieve the first goal, we add the globally averaged radiograph representation to each non-masked report token embedding and pass the resulting hybrid representations to the report decoder for masked re- port restoration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' As for the second purpose, we conduct a novel patch restoration task to explicitly encode more local details into radiograph representations by reconstructing high-resolution patches from low-resolution inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='1 REPORT COMPREHENSION Masked report generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' In our scenario, each radiology report is associated with a radiograph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' To convert the free-text report into tokens, we use WordPiece (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2016) as the default to- kenizer, whose vocabulary has approximately 30k tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' After tokenization, we randomly mask 3 2CPublished as a conference paper at ICLR 2023 a number of report tokens with [MASK].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Compared to BERT (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2018) that randomly masks 15% tokens, we use a 50% probability of masking each token in the report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' The insight behind the use of a higher masking ratio is that we want the model to lean more upon the image embeddings to finish the report-completion task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Hybrid representations for storing multi-modal information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' We then transform non-masked report tokens into token embeddings using a simple lookup table2, which stores randomly initialized embeddings of a fixed dictionary and size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' In practice, we retrieve embeddings using indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Then, the global embedding of the associated radiograph is added to each non-masked token embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' The resulting non-masked hybrid embeddings are supposed to include the multi-modal information from the radiograph and associated radiology report, which ought to be helpful for restoring the masked tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Masked report restoration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' To reconstruct the masked tokens, we forward latent embeddings of both hybrid tokens and mask tokens to the report decoder (a light-weight transformer model), where fixed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', unlearnable) positional embeddings are added to encode the position information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' We train the report decoder using the masked language modeling objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='2 RADIOGRAPH UNDERSTANDING Masked image generation with low resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' We propose to learn radiograph representations by reconstructing high-resolution radiograph patches from low-resolution inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' The motivation behind is to encode more local information into latent embeddings via super-resolution imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' 2, we sub-sample each high-resolution radiograph by a factor of two to generate a low-resolution input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Following He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' (2022), we split low-resolution radiograph into non- overlapping image patches, where 75% patches are randomly masked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Radiograph representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' We add fixed unlearnable positional embeddings to linearly trans- formed non-masked image patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Next, we forward the resulting patch embeddings to the transformer-based image encoder, which produces non-masked image patch embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Then, the global average pooling (GAP) is applied to all non-masked embeddings, whereby a global feature is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Here, we hypothesize that the image-level information brought by the global feature is helpful to the restoration of masked report tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Based on this hypothesis, we duplicate and add the global feature to each non-masked report token embedding, producing the hybrid token embeddings that encode the multi-modal information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Masked image restoration with high resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Non-masked image and mask token represen- tations with added positional embeddings are passed to the image decoder for the restoration of masked radiograph patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Specifically, the image decoder is required to restore a high-resolution (2× the input resolution) patch from each input token via a shared fully-connected (FC) layer (across all tokens).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' In practice, the proposed restoration procedure explicitly requires the learned image rep- resentations to include more local details that often matter a lot in medical diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='3 MULTI-TASK MODELING Suppose each input radiograph consists of two set IM and IN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' The masked set IM={x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' , xh} (ground truths) contains h high-resolution image patches that serve as reconstruction targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' The non-masked set IN ={s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' , sk} comprises k low-resolution patches that are treated as model in- puts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Likewise, we denote the associated radiology report using the masked set RM={u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' , up} (ground truths) and the non-masked set RN ={v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' , vq} (inputs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Here, x, s, u, and v stand for the masked image patch, non-masked image patch, masked report token, and non-masked report token, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' For model parameters, we use ΘE, ΘD, and ΘR to denote the parameters of the image encoder, image decoder, and report decoder, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' For the restoration of masked report tokens, we forward hybrid representations to the report decoder and minimize the negative log-likelihood function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' During the training stage, the objective function 2https://pytorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='org/docs/stable/generated/torch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='nn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='Embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' 4 Published as a conference paper at ICLR 2023 LR (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', the MLM loss in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' 2) of the above optimization procedure can be summarized as follows: LR(RM, RN , IN ) = − p � i=1 log P (ui | v1:q, s1:k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' ΘE, ΘR) , (1) where P stands for the conditional probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' We ignore the mask tokens for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Similarly, we can formalize the objective function of the high-resolution masked radiograph restora- tion (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' the MIM loss in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' 2) as follows: LI(IM, IN ) = MSE (fΘD(fΘE(s1:k)), x1:h) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' (2) In practice, we adopt the mean squared error (MSE) to measure the differences between the predicted and ground-truth image patches with high resolution, where all pixel values are normalized to [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' The total multi-task training objective function (to be minimized) of the multi-modal restoration is as follows: L(RM, RN , IM, IN ) = LR(RM, RN , IN ) + λLI(IM, IN ) (3) where λ is a hyper-parameter that controls the relative impacts of two objective functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' After pre-training, we can transfer the weight parameters of the image encoder (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', ΘE) to various down- stream tasks for fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' 4 EXPERIMENTS In this section, we mainly compare MRM against report- and self-supervised R2L methodologies on 5 well-established public datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Average results are reported over three training runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='1 MIMIC-CXR FOR PRE-TRAINING We conduct pre-training on MIMIC-CXR (Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2019), one of the largest X-ray datasets, that contains more than 370,000 radiograph images from over 220,000 patient studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Each radio- graph is paired with one associated radiology report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='2 DATASETS FOR FINE-TUNING We validate the transferability of learned radiograph representations on X-ray based classification and segmentation tasks via end-to-end fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Specifically, we evaluate the pre-trained model on 4 X-ray datasets in the classification tasks, which are NIH ChestX-ray (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2017), CheXpert (Irvin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2019), RSNA Pneumonia (Shih et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2019), and COVID-19 Image Data Collection (Cohen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' For the segmentation task, we fine-tune the pre-trained model on SIIM-ACR Pneumothorax Segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='3 CheXpert introduces a multi-label classification problem on chest X-rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' We follow the official guideline (Irvin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2019) and report the model performance on 5 selected pathologies, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', at- electasis, cardiomegaly, consolidation, edema, and pleural effusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Considering the official test set of CheXpert is not available to the public, we follow ConVIRT (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2020) to regard the official validation set as the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Meanwhile, we randomly sample 5,000 images from the official training set to build the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' The training/validation/test split each constitutes 218,414/5,000/234 images of the whole dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' RSNA Pneumonia defines a binary classification problem, where each chest radiograph is cat- egorized as either pneumonia or normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' We adopt the official data split, where the train- ing/validation/test set comprises 25,184/1,500/3,000 images, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' NIH ChestX-ray consists of about 112,120 frontal-view chest radiograph images, where a multi- label classification problem on 14 chest pathologies is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' The training/validation/test split each constitutes 70%/10%/20% of the whole dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' COVID-19 Image Data Collection is a relatively small dataset, which involves 900 chest radio- graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' We follow Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' (2022) to conduct fine-tuning on this small-scale dataset to investigate 3https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='kaggle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='com/c/siim-acr-pneumothorax-segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' 5 Published as a conference paper at ICLR 2023 Methods Input Size Pre-train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Data CheXpert RSNA Pneumonia SIIM 1% 10% 100% 1% 10% 100% 10% 100% Our MRM 224 MI-CXR 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='5± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='7 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='5± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='6 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='7± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='3 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='3± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='6 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='7± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='4 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='3± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='4 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='2± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='5 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='4± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='3 CNN-based ConVIRT 224 CheXpert 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='9 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='8 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='3 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='4 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='1 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='3 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='2 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='9 GLoRIA 224 CheXpert 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='6 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='8 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='1 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='1 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='0 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='6 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='9 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='4 ConVIRT 224 MI-CXR 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='0 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='1 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='1 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='8 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='5 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='7 MedKLIP† 224 MI-CXR 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='3 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='0 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='3 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='1 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='4 BioViL 480 PubMed + MI-III/CXR 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='1 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='4 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='1 Transformer-based GLoRIA∗ 224 MI-CXR 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='5± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='8 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='5± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='6 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='8± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='5 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='7± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='8 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='2± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='5 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='1± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='3 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='8± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='7 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='9± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='4 REFERS 224 MI-CXR 87.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='0± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='7 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='4± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='3 MGCA† 224 MI-CXR 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='8 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='1 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='7 89.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' In compari- son, dice scores are presented on SIIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' The best results are bolded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' MI- stands for the MIMIC dataset series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Note that ResNet-50 and ViT-B/16 are treated as the default backbones for CNN- and Transformer-based methods, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' * denotes our implementation of GLoRIA using ViT- B/16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Approaches with † leverage disease-level annotations for pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Specifically, numbers of MGCA on CheXpert and RSNA Pneumonia are linear classification results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Labeling Ratios Methods Average Atelectasis Cardiomegaly Consolidation Edema Effusion Emphysema Fibrosis Hernia Infiltration Mass Nodule Pleural Thickening Pneumonia Pneumothorax 1% Our MRM 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='4± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='6 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='0 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='1 ImageNet Pre-training 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='0 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='3 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='3 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='6 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='9 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='9 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='4 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='5 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='8 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='9 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='9 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='7 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='5 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='0 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='7 Table 2: COMPARISONS ON NIH CHESTX-RAY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Besides self-supervised and transfer learning baselines, we also present the performance of REFERS and MedKLIP (with competitive perfor- mance on CheXpert, RSNA Pneumonia, and SIIM) as references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' AUC scores are displayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' The highest AUC scores in each labeling ratio are bolded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' the effectiveness of various pre-training methodologies when the amount of annotations is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' There are two tasks included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' The first task requires the model to distinguish COVID-19 cases from non-COVID-19 pneumonia cases, where the training/validation/test set comprises 356/54/99 radiographs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' The second task is to distinguish viral pneumonia cases from bacterial pneumonia ones, where the training/validation/test set contains 297/43/86 cases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' SIIM-ACR Pneumothorax Segmentation (SIIM) aims to facilitate the development of segmen- tation models to identify pneumothorax disease in chest radiographs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' SIIM contains over 120,000 frontal-view chest X-rays with precise manual segmentation of pneumothorax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' We follow Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' (2021) to construct the training/validation/test split, where each constitutes 70%/15%/15% of the whole dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' 6 Published as a conference paper at ICLR 2023 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='3 BASELINES 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='1 REPORT-SUPERVISED METHODOLOGIES We first compare MRM against a range of pre-training approaches, which use radiology reports as supervision to learn radiograph representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' There are 4 report-supervised approaches in- volved in the baseline comparisons, which are ConVIRT (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2020), GLoRIA (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2021), BioViL (Boecking et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2022), and REFERS (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Specifically, Con- VIRT (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2020) proposed to learn medical visual representations by contrasting paired radiographs and sentences from radiology reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' GLoRIA (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2021) improved ConVIRT by contrasting radiograph sub-regions and words in the reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' BioViL (Boecking et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2022) and REFERS (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2022) incorporated masked language modeling loss into contrastive learn- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Moreover, REFERS introduced a multi-view fusion attention to better align the representations of each radiograph and its associated report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' In addition, MGCA (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2023) and Med- KLIP (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2023) were included as two recent baselines4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Apart from above baselines, we also include M3AE (Geng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2022), a recent masked multi-modal pre-training method aside from the application to medical data, for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' In experiments, we fine-tune pre-trained models of MRM and other report-supervised methods on CheXpert (classification), RSNA Pneumonia (classification), and SIIM (segmentation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='2 SELF-SUPERVISED AND TRANSFER LEARNING METHODS Besides report-supervised approaches, we also include self-supervised and transfer learning ap- proaches in our comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Specifically, Context Restoration (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2019), Model Gen- esis (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2021), and TransVW (Haghighi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2021) are based on predictive SSL, while C2L (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2020) was developed on top of contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' In addition, MRM is also compared to ImageNet pre-training (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' In practice, we conduct the comparisons with self-supervised and transfer learning approaches on NIH ChestX-ray and COVID-19 Image Data Collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Methods COVID-19 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Others Viral vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Bacterial Our MRM 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='8± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='4 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='5± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='3 REFERS 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='1 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='4 Model Genesis 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='0 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='8 C2L 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='8 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='0 Context Restoration 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='6 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='8 TransVW 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='1 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='5 ImageNet Pre-training 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='1 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='3 Table 3: COMPARISONS ON COVID- 19 IMAGE DATA COLLECTION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' The best results are bolded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Methods 1% 10% 100% Our MRM 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='4 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='0 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='9 Masked modeling & Super-resolution restoration 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='9 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='2 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='3 Masked report modeling (LR) 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='7 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='3 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='1 Masked radiograph modeling (LI) 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='7 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='2 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='7 Super-resolution restoration 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='8 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='7 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='7 + Hybrid features for image restoration 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='9 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='6 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='7 Table 4: ABLATIONS ON NIH CHESTX-RAY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' AUC scores of three different labeling ratios are reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' The best results are bolded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='4 RESULTS 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='1 COMPARISONS WITH REPORT-SUPERVISED BASELINES In Table 1, we present the comparative results with report-supervised methodologies on CheXpert, RSNA Pneumonia, and SIIM (segmentation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Specifically, we investigate the performance when fine-tuning with limited and full supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' We provide reconstruction examples and segmentation results in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' From Table 1, we observe no obvious performance gap between CNN- and Transformer-based report-supervised pre-training methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' For instance, after implementing GLoRIA on MIMIC-CXR with ViT-B/16, we observe performance drops and improvements on CheXpert and RSNA Pneumo- nia, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' This contrast demonstrates that replacing CNN with Transformer may not bring performance gains to report-supervised pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Among all baselines, REFERS is the best per- forming approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Nonetheless, MRM consistently outperforms various baselines on all three datasets under different labeling ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Specifically, MRM maintains more advantages over previous pre-training method- 4MGCA (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2023) and MedKLIP (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2023) were released after the submission deadline of ICLR 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' We added results in the camera ready for better comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' 7 Published as a conference paper at ICLR 2023 Fusion 1% 10% 100% GAP 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='4 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='0 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='9 GMP 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='2 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='8 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='0 (a) Multi-modal fusion strategies PI 1% 10% 100% 75% 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='4 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='0 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='9 50% 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='8 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='4 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='1 0% 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='4 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='0 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='4 (b) Masking ratios for radiographs λ 1% 10% 100% 1 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='4 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='0 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='9 2 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='6 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='5 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='5 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='0 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='7 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='9 (c) Loss controlling factor λ Table 5: Ablation studies on choices of multi-modal fusion and hyper-parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' GAP and GMP stand for global average and maximum pooling, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Experiments are performed on NIH ChestX-ray under various labeling ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' The best results are bolded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' ologies when fine-tuning with limited annotations, which is quite meaningful for medical image analysis as large amounts of specialist annotations (from radiologists or clinicians) are usually hard to access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' It is worth noting that MRM achieves 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='5% when using only 1% labeled data on CheX- pert, better than previous counterparts with 100% annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' We see that M3AE generally performs worse than our MRM in all labeling ratios, especially under extremely limited data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' The underperformance of M3AE may be attributed to the fact that it requires a large amount of multi-modal data to learn transferable joint representations for images and texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='2 COMPARISONS WITH SELF-SUPERVISED AND TRANSFER LEARNING BASELINES Table 2 presents the average and per-class classification results on NIH ChestX-ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Compared to self-supervised learning baselines, MRM achieves large improvements in almost every chest pathol- ogy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' On average, MRM outperforms C2L (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2020) and TransVW (Haghighi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2021), the best two self-supervised pre-training methodologies, by about 8% when the amount of available labeled data is extremely limited (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', the labeling ratio is 1%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Similarly, we also observe remark- able improvements when comparing MRM to ImageNet Pre-training (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' These phenomena demonstrate that the radiograph representations learned by MRM are more transferable than those from previous self-supervised and transfer learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Compared to REFERS (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', the best performing report-supervised baseline in Table 1), MRM still provides notable improvements in most chest pathologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Specifically, MRM is more advantageous when the amount of labeled data is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' For instance, MRM surpasses REFERS by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='7% and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='1% on average when the labeling ratios are 1% and 10%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' These comparisons again verify the effectiveness of MRM over previous report-supervised pre-training counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' We also investigate the impacts of pre-training on the real-world scenario with extremely limited specialist supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' In Table 3, we compare the performance of a range of pre-training method- ologies on two binary classification tasks, which are distinguishing COVID-19 from non-COVID-19 pneumonia, and differentiating between viral pneumonia and bacterial pneumonia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Again, MRM outperforms self-supervised, transfer learning, and report-supervised pre-training methodologies by substantial margins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Compared to REFERS, MRM brings nearly 4% and 11% improvements to two tasks, respectively, further enhancing the practicability of the diagnosis system trained with extremely limited supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='5 ABLATION ANALYSIS Advantages over single-task pre-training paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' First of all, we remove the two masked mod- eling objectives and super-resolution restoration task, resulting in substantial performance drops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' These results verify the necessity of using masked modeling and super-resolution restoration in MRM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' After removing the masked report modeling objective, MRM only acquires supervision signals from self-supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Thus, the whole framework degenerates into a self-supervised pre- training methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' From Table 4, we observe that removing LR leads to dramatic performance degradation in different labeling ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Moreover, we find that introducing the masked report mod- eling is greatly helpful to the fine-tuning performance with limited labeling resources (1% and 10%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' For instance, adding LR brings about 5-percent improvements to MRM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Similarly, remov- ing the masked radiograph modeling also leads to notable performance drops in all labeling ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' These results demonstrate the necessity of introducing multi-task objectives in masked modeling pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' 8 Published as a conference paper at ICLR 2023 Is super-resolution restoration helpful?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' As Table 4 shows, the proposed super-resolution restora- tion provides consistent performance gains in different labeling ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' The underlying reason may be that the low to high resolution restoration process helps preserve more local information into latent representations, which enhances the transferable ability to downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Would it be beneficial to introduce multi-modal information to image restoration?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' We investi- gate this question by adding non-masked report token embeddings to image patch embeddings and passing the resulting hybrid features to the image decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' As Table 4 shows, introducing multi- modal information to masked image restoration does not improve the fine-tuning performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' We leave the exploration of the reason behind to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Ablations on multi-modal fusion and hyper-parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' In Table 5a, we present the experimental results of using different strategies for radiograph-report fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' We find that global average pooling (GAP) outperforms global maximum pooling (GMP) by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='2% when access to labeled data is quite limited while achieving comparable results as the labeling ratio increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' We also investigate the impact of applying different masking ratios to input radiographs (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Table 5b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Specifically, we find that a ratio of 75% performs the best on the extremely small labeling ratio (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 1%), while a ratio of 50% achieves slightly better results when the labeling ratio becomes larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' In MRM, we set the default masking ratio for radiographs to 75% because this operation leads to fewer input patches, accelerating the pre-training process and reducing the memory cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' The insight behind applying a high masking ratio (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 75%) is that it addresses the heavy spatial redundancy of radiographs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' By applying a high masking ratio, we reduce the redundancy and create a surrogate task, requiring the model to understand the high-level semantics holistically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' We also perform ablative experiments to investigate the influence of λ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' 3, whose results are displayed in Table 5c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' We see that λ = 1 is an optimal choice, while a smaller λ value (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='e, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='5) performs better than a larger value (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='0), indicating that the MLM objective may play a more important role than the MIM objective during the pre-training stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='6 IMPLEMENTATION DETAILS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Our code is implemented using PyTorch 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='2 (Paszke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' The pre-training experi- ments were conducted on 4 GeForce RTX 3080Ti GPUs, and the training time is about 2 days for 200 epochs, requiring 12GB memory from each GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' The training batch size is 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' We use AdamW (Loshchilov & Hutter, 2017) as the default optimizer, where the initial learning rate is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='5e−4, weight decay is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='05, β1 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='9, and β2 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' The MSE and cross-entropy losses are used for masked image and language modeling, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' In practice, we set λ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' 3 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' For fine-tuning on SIIM, we train the segmentation network on 4 GeForce RTX 3080Ti GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' AdamW is the default optimizer, where the initial learning rate is 2e−5, weight decay is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='05, β1 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='9, and β2 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' For fine-tuning on other datasets, we train the classification network on a single GeForce RTX 3080Ti GPU, where the default optimizer is SGD with momentum 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' The training cost is a mix of focal and dice losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' For fine-tuning on CheXpert, RSNA Pneumonia, NIH ChestX-ray, and COVID-19 Image Data Col- lection, we adopt the cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' We search the best initial learning rate from 3e-2, 3e-3, and 5e-4 to get the best performance on validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' For both the pre-training and the fine-tuning of image classification task, the network is ”warmed up” by increasing the learning rate linearly to the set value, and then learning rate is decreased using the cosine decay schedule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' 5 CONCLUSION We present masked record modeling (MRM) for radiograph representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' MRM for- malizes the radiograph understanding and radiology report comprehension as two complementary masked modeling objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' With MRM pre-training, we achieve better results on well-established datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Specifically, MRM outperforms previous self- and report-supervised counterparts by large margins when the labeled data is extremely limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' These observations verify the effectiveness of MRM, and we hope they will enable our field to explore the use of multi-task supervision for learning more transferable visual representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' 9 Published as a conference paper at ICLR 2023 REFERENCES Pulkit Agrawal, Joao Carreira, and Jitendra Malik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Learning to see by moving.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Jiahui Yu, Zirui Wang, Vijay Vasudevan, Legg Yeung, Mojtaba Seyedhosseini, and Yonghui Wu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Coca: Contrastive captioners are image-text foundation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' arXiv preprint arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='01917, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Yuhao Zhang, Hang Jiang, Yasuhide Miura, Christopher D Manning, and Curtis P Langlotz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Con- trastive learning of medical visual representations from paired images and text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' arXiv preprint arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='00747, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Hong-Yu Zhou, Shuang Yu, Cheng Bian, Yifan Hu, Kai Ma, and Yefeng Zheng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Comparing to learn: Surpassing imagenet pretraining on radiographs by comparing image representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' In International Conference on Medical Image Computing and Computer-Assisted Intervention, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' 398–407.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Springer, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Hong-Yu Zhou, Xiaoyu Chen, Yinghao Zhang, Ruibang Luo, Liansheng Wang, and Yizhou Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Generalized radiograph representation learning via cross-supervision between images and free- text radiology reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Nature Machine Intelligence, 4(1):32–40, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Zongwei Zhou, Vatsal Sodha, Jiaxuan Pang, Michael B Gotway, and Jianming Liang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Models genesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Medical Image Analysis, 67:101840, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' 12 Published as a conference paper at ICLR 2023 A APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='1 DISCUSSION: DIFFERENCES FROM MASKED VISION-AND-LANGUAGE PRE-TRAINING BESIDES MEDICAL DATA To our knowledge, M3AE (Geng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2022), MLIM (Arici et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2021), and MaskVLM (Kwon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2022) are three recent efforts that are most closely related to ours, all of which use masked modeling for vision and language pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' In the following, we clarify the differences between our MRM and these approaches from two perspectives: motivation and implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' M3AE (Geng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2022) and MLIM (Arici et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2021) aim to learn joint image- text representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' MaskVLM (Kwon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2022) is developed for improving vision+language tasks, such as image-text retrieval, visual question answering, and natural language for visual rea- soning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' These characteristics differ from our methodology, which aims to learn a representation for radiographs only for disease diagnosis even though both radiographs and reports are used during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' As aforementioned, M3AE (Geng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2022) and MLIM (Arici et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2021) implement unified multi-modal transformers that take imaging and textual as inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' As a result, they require much more pre-training data, which is often an order of magnitude larger than ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' This is not practical and applicable in the medical field, where access to the data is quite limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' MaskVLM (Kwon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2022) applies masked modeling and contrastive learning to image-text pairs, where a binary matching task is employed to tell whether an image and a text are aligned or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Besides, MaskVLM builds cross-modality encoders to incorporate multi-modal information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' In contrast, our MRM is much simpler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' MRM uses masking modeling as the only training objective, and a simple GAP-addition workflow is proposed for fusing multi-modal information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='2 CONFIGURATIONS OF NETWORKS The image encoder a ViT-like encoder (Dosovitskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2020) that includes a patch embedding layer followed by twelve transformer blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' The architecture of the image decoder is very similar to that of the encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' The decoder architecture consists of a decoder embedding layer, four trans- former blocks, and one fully-connected layer to predict masked patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' The numbers of attention heads in the image encoder and decoder are twelve and six, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' We add a 2D sin-cos posi- tional embedding to input patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' The report decoder is a light-weight transformer that includes an embedding layer and six transformer blocks, followed by a one-layer fully-connected predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' The number of attention heads in the report decoder is six.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' We adopt a learnable positional embedding in the report decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='3 RECONSTRUCTION ANALYSIS Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' 3 presents example results on MIMIC-CXR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' We find that even with high masking ratios, MRM can still produce satisfactory reconstructions, though some details are missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Specifically, the re- constructed reports are surprisingly close to the ground truths, which we attribute to the introduction of hybrid multi-modal representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' For instance, MRM can tell the position of rib fractures based on the radiograph input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Besides, we obtain some interesting mistakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' In the first example, MRM recognizes the clips over the left lung as calcifications (potentially in nipple shadow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' This observa- tion again shows that the report reconstructions rely on the image input as the clips are masked in the input radiograph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content='4 SEGMENTATION ANALYSIS In this section, we visualize the segmentation results of GLoRIA (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2021), REFERS (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=', 2022), and our MRM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' 13 Published as a conference paper at ICLR 2023 [CLS] final [MASK] [MASK] [MASK] chest [MASK] pa and [MASK] ) indication : ___f with [MASK] onset [MASK] // [MASK] for infection [MASK] : [MASK] [MASK] [MASK] [MASK] [MASK] [MASK] none .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' [MASK] : [MASK] is [MASK] focal [MASK] [MASK] pleural [MASK] or pneumothorax [MASK] [MASK] [MASK] [MASK] that [MASK] likely represent nipple shadows .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' [MASK] [MASK] silhouette [MASK] [MASK] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' [MASK] [MASK] over the left [MASK] , potentially [MASK] [MASK] [MASK] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' the [MASK] upper abdomen [MASK] [MASK] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' [MASK] [MASK] of [MASK] [MASK] left sixth [MASK] seventh [MASK] are [MASK] [MASK] [MASK] : no acute cardiopulmonary process .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' [CLS] final report examination : chest ( pa and lat ) indication : ___f with new onset confusion // eval for infection technique : chest pa and lateral comparison : none .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' findings : there is no focal consolidation , pleural effusion or pneumothorax .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' bilateral nodular opacities that most likely represent nipple shadows .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' the cardiomediastinal silhouette is normal .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' calcifications project over the left lung , potentially a nipple shadow .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' the imaged upper abdomen is unremarkable .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' healed fractures of the posterior left sixth and seventh ribs are noted .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' impression : no acute cardiopulmonary process .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' [CLS] final report examination : chest ( pa and lat ) indication : ___f with new onset ascites // eval for infection technique : chest pa and lateral comparison : none .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' findings : there is no focal consolidation , pleural effusion or pneumothorax .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' bilateral nodular opacities that most likely represent nipple shadows .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' the cardiomediastinal silhouette is normal .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' clips project over the left lung , potentially within the breast .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' the imaged upper abdomen is unremarkable .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' chronic deformity of the posterior left sixth and seventh ribs are noted .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' impression : no acute cardiopulmonary process .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' [CLS] [MASK] report [MASK] [MASK] [MASK] ( [MASK] [MASK] lat ) [MASK] : [MASK] [MASK] [MASK] with shortness [MASK] [MASK] [MASK] : [MASK] [MASK] and [MASK] comparison : [MASK] findings : [MASK] cardiac , mediastinal and hilar [MASK] are normal .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' [MASK] [MASK] [MASK] normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' lungs are [MASK] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' [MASK] pleural [MASK] [MASK] pneumothorax [MASK] [MASK] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' multiple clips [MASK] again seen [MASK] [MASK] the left [MASK] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' remote [MASK] - sided rib fractures are also re - [MASK] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' impression : [MASK] [MASK] cardiopulmonary abnormality .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' [CLS] final report examination : chest ( pa and lat ) indication : history : ___f with shortness of breath technique : chest pa and lateral comparison : ___ findings : the cardiac , mediastinal and hilar contours are normal .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' pulmonary vasculature is normal .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' lungs are clear .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' no pleural effusion or pneumothorax is present .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' multiple clips are again seen projecting over the left axilla .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' remote left- sided rib fractures are also re - demonstrated .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' impression : no acute cardiopulmonary abnormality .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' [CLS] final report examination : chest ( pa and lat ) indication : history : ___f with shortness of breath technique : chest pa and lateral comparison : ___ findings : the cardiac , mediastinal and hilar contours are normal .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' pulmonary vasculature is normal .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' lungs are clear .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' no pleural effusion or pneumothorax is present .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' multiple clips are again seen projecting over the left breast .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' remote left - sided rib fractures are also re - demonstrated .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' impression : no acute cardiopulmonary abnormality .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Inputs Recons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' GT Inputs Recons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' GT Figure 3: Example results on MIMIC-CXR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' For each triplet, we show the masked radiograph and report (Inputs), our MRM reconstruction (Recons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' ), and the ground truth (GT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' The masking ra- tios are 75% (radiograph) and 50% (report).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' Predicted and corresponding ground truth words are highlighted in pink and green , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' 14 OPublished as a conference paper at ICLR 2023 Radiograph GLoRIA* REFERS Our MRM GT 15 ECGVPublished as a conference paper at ICLR 2023 Radiograph GLoRIA* REFERS Our MRM GT Figure 4: Segmentation results of MRM, GLoRIA, and REFERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' GT stands for the ground truth masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' * means the GLoRIA implementation is based on ViT-B/16, the same backbone as used in REFERS and MRM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} +page_content=' 16 LPORTABLE' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFPT4oBgHgl3EQfuTUu/content/2301.13155v1.pdf'} diff --git a/dNE2T4oBgHgl3EQfGAZK/vector_store/index.pkl b/dNE2T4oBgHgl3EQfGAZK/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..b2e7bb7907cc933c85b38afe8aeca2ae4a896d6b --- /dev/null +++ b/dNE2T4oBgHgl3EQfGAZK/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:cce2f09557831c7c7528f8851e454deb26cc55fe58246d25a8cfab55db3f21a3 +size 122001 diff --git a/hNAzT4oBgHgl3EQfMvs5/content/tmp_files/2301.01136v1.pdf.txt b/hNAzT4oBgHgl3EQfMvs5/content/tmp_files/2301.01136v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c03e416e7287725a37e1381b42ae5842595015b0 --- /dev/null +++ b/hNAzT4oBgHgl3EQfMvs5/content/tmp_files/2301.01136v1.pdf.txt @@ -0,0 +1,2030 @@ +Optimal control problem for Stokes system: Asymptotic +analysis via unfolding method in a perforated domain +Swati Garg and Bidhan Chandra Sardar∗ +Department of Mathematics +Indian Institute of Technology Ropar +Rupnagar-140001, Punjab, India +swati.19maz0006@iitrpr.ac.in, swatigargmks@gmail.com +bcsardar@iitrpr.ac.in, bcsardar31@gmail.com +January 4, 2023 +Abstract +This article’s subject matter is the study of the asymptotic analysis of the optimal +control problem (OCP) constrained by the stationary Stokes equations in a periodi- +cally perforated domain. We subject the interior region of it with distributive controls. +The Stokes operator considered involves the oscillating coefficients for the state equa- +tions. We characterize the optimal control and, upon employing the method of periodic +unfolding, establish the convergence of the solutions of the considered OCP to the solu- +tions of the limit OCP governed by stationary Stokes equations over a non-perforated +domain. The convergence of the cost functional is also established. +Keywords: Stokes equations, Homogenization, Optimal control, Perforated domain, Un- +folding operator +1 +Introduction +In this article, we consider the optimal control problem (OCP) governed by generalized +stationary Stokes equations in a periodically perforated domain O∗ +ε (see Section 2, on the +domain description). The size of holes in the perforated domain is of the same order as +that of the period, and the holes are allowed to intersect the boundary of the domain. The +control is applied in the interior region of the domain, and we wish to study the asymptotic +AMS subject classifications: 35B27, 35B40, 35Q93, 49J20, 76D07 +∗Corresponding author +1 +arXiv:2301.01136v1 [math.OC] 3 Jan 2023 + +analysis (homogenization) of an interior OCP subject to the constrained stationary Stokes +equations with oscillating coefficients. +One can find several works in the literature regarding the homogenization of Stokes +equations over a perforated domain. Using the multiple-scale expansion method, the au- +thors in [16] studied the homogenization of Stokes equations in a porous medium with the +Dirichlet boundary condition on the boundary of the holes. They obtained the Darcy’s law +as the limit law in the homogenized medium. +In [9], the authors considered the Stokes +system in a periodically perforated domain with non-homogeneous slip boundary conditions +depending upon some parameter γ. Upon employing the Tartar’s method of oscillating test +functions they obtained under homogenization, the limit laws, viz., Darcy’s law ( for γ < 1), +Brinkmann’s law (for γ = 1), and Stokes’s type law (for γ > 1). In [25], the author studied +a similar problem using the method of periodic unfolding in perforated domains by [10]. +Further, the type of behavior as seen in [9] was already observed in [12] by the authors while +studying the homogeneous Fourier boundary conditions for the two-dimensional Stokes equa- +tion. Likewise, in [1, 2], the author examined the Stokes equation in a perforated domain +with holes of size much smaller than the small positive parameter ε, wherein they considered +the boundary conditions on the holes to be of the Dirichlet type in [1] and the slip type +in [2]. The domain geometry, more specifically, the size of the holes, determines the kind of +limit law in these works. Also, the author in [6] employed the Γ− convergence techniques to +get comparable results. +A few works concern the homogenization of the OCPs governed by the elliptic systems +over the periodically perforated domains with different kinds of boundary conditions on the +boundary of holes (of the size of the same order as that of the period). In this regard, with +the use of different techiniques, viz., H0− convergence in [18], two-sclae convergence in [23], +and unfolding methods in [7,21], the homogenized OCPs were thus obtained over the non- +perforated domains. Further, in context to the Stokes system, the authors in [22] studied +the homogenization of the OCPs subject to the Stokes equations with Dirichlet boundary +conditions on the boundary of holes, where the size of the holes is of the same order as that of +the period. Here, the authors could obtain the homogenized system, pertaining only to the +case when the set of admissible controls was unconstrained. For more literature concerning +the homogenization of optimal control problems in perforated domains, the reader is reffered +to [13–15,19,24] and the references therein. +The present article introduces an interior OCP subject to the generalized stationary +Stokes equations in a periodically perforated domain O∗ +ε. On the boundary of holes that +do not intersect the outer boundary, the homogeneous Neumann boundary condition is +prescribed, while on the rest part of the boundary, the homogeneous Dirichlet boundary +condition is prescribed. The underlying objective of this article is to study the homogeniza- +tion of this OCP. More specifically, we consider the minimization of the L2−cost functional +(3.1), which is subject to the constrained generalized stationary Stokes equations (3.2). +The Stokes equations are generalized in the sense that we consider a second-order elliptic +linear differential operator in divergence form with oscillating coefficients, i.e., − div (Aε∇), +first studied for the fixed domain in [4, Chapter 1], instead of the classical Laplacian operator. +2 + +Here, the action of the scalar operator − div (Aε∇) is defined in a ”diagonal” manner on any +vector u = (u1, . . . , un), with components u1, . . . , un in the H1 Sobolev space. That is, for +1 ≤ i ≤ n, we have (− div (Aε∇u))i = − div (Aε∇ui). The main difficulty observed during +the homogenization was identifying the limit pressure terms appearing in the state and the +adjoint systems, which we overcame by introducing suitable corrector functions that solved +some cell problems. We thus obtained the limit OCP associated with the stationary Stokes +equation in a non-perforated domain. +The layout of this article is as follows: In the next section, we introduce the periodically +perforated domain O∗ +ε along with the notations that will be useful in the sequel. Section +3 is devoted to a detailed description of the considered OCP and the derivation of the +optimality condition, followed by the characterization of the optimal control. In Section 4, +we derive a priori estimates of the solutions to the considered OCP and its corresponding +adjoint problem. In Section 5, we recall the definition of the method of periodic unfolding +in perforated domains (see, [8,11]) and a few of its properties. Section 6, refers to the limit +(homogenized) OCP. Finally, we derive the main convergence results in Section 7. +2 +Domain description and Notation +2.1 +Domain description +Let {b1, ..., bn} be a basis of Rn (n ≥ 2), and W be the associated reference cell defined as +W = +� +w ∈ Rn | w = +n +� +i=1 +wibi, (w1, . . . , wn) ∈ (0, 1)n +� +. +Let us denote O, W, and W ∗ = W\Y by an open bounded subset of Rn, a compact subset +of W, and the perforated reference cell, respectively. It is assumed that the boundary of Y +is Lipschitz continuous and has a finite number of connected components. +Also, let ε > 0 be a sequence that converges to zero and set +T = +� +ζ ∈ Rn | ζ = +n +� +i=1 +zibi, (z1, . . . , zn) ∈ Zn +� +, +Zε = {ζ ∈ T | ε(ζ + W) ⊂ O} . +We take into account the perforated domain O∗ +ε (see Figure 1) given by O∗ +ε = O\Yε, where +Yε = ∪ζ∈T ε(ζ + Y ). Now, let us denote � +Oε as the interior of the largest union of ε(ζ + W) +cells such that ε(ζ + W) ⊂ O, while Λε ⊂ O as containing the parts from ε(ζ + W) cells +intersecting the boundary ∂O. More precisely, we write Λε = O\ � +Oε, where +� +Oε = interior +� +∪ζ∈Zε ε(ζ + W) +� +. +The associated perforated domains are defined as +� +O∗ +ε = ˆOε\Yε, +ˆΛ∗ +ε = O∗ +ε\ � +O∗ +ε. +3 + +Figure 1: +The Perforated domain O∗ +ε and the reference cell W. +Also, we denote the boundary of the perforated domain O∗ +ε as +∂O∗ +ε = Γε +1 ∪ Γε +0, +where Γε +1 = ∂ � +Oε ∩ ∂Yε and Γε +0 = ∂O∗ +ε\Γε +1, +which means that Γε +1 denotes the boundary of set of holes contained in � +Oε. +In Figure 1, � +O∗ +ε and ˆΛ∗ +ε respectively represent the dark perforated part and the remaining +part of the perforated domain O∗ +ε. While, Γε +1 and Γε +0 respectively represent the boundary +of holes contained in � +O∗ +ε and the boundary of holes contained in ˆΛ∗ +ε along with the outer +boundary ∂O. In the following, we introduce a few notations that we shall use throughout +this article. +2.2 +Notation +• Aε(x) = A( x +ε) a.e. in O, for all ε > 0. +• vε = (vε1, . . . , vεn), for any bold symbol vector function vε. +• v = (v1, . . . , vn), for any bold symbol vector function v. +• ηε denotes the outward normal unit vector to Γε +1. +• η denotes the outward normal unit vector to ∂O. +• M t denotes the transpose of any matrix M. +• �ψ is the zero extension of any function ψ outside O∗ +ε to the whole of O. +• �ψ = (� +ψ1, · · · , � +ψn), for any vector function ψ. +• |F| is the Lebesgue measure of the measurable set F. +4 + +M• Θ = |W ∗| +|W| , the proportion of the perforated reference cell W ∗ in the reference cell W. +• MW ∗(φ) is the mean value of φ on the perforated reference cell W ∗. +• MW ∗(φ) = (MW ∗(φ1), · · · , MW ∗(φn)), for vector function φ. +• {D → R}, the set of all real valued functions defined on domain D. +• D(Ω), is the space of infinitely many times differentiable functions with compact sup- +port in Ω, for any open set Ω ∈ Rn. +3 +Problem description and Optimality condition +Let us consider the following OCP associated with Stokes system: +inf +θε∈(L2(O∗ε))n +� +Jε(θε) = 1 +2 +� +O∗ε +|uε(θε) − ud|2 + τ +2 +� +O∗ε +|θε|2 +� +, +(3.1) +subject to +� +� +� +� +� +� +� +� +� +− div (Aε∇uε) + ∇pε += θε +in O∗ +ε, +div(uε) += 0 +in O∗ +ε, +ηε · Aε∇uε − pεηε += 0 +on Γε +1, +uε += 0 +on Γε +0, +(3.2) +where the desired state ud = (ud1, . . . , udn) is defined on the space (L2(O))n, θε is a control +function defined on the space (L2(O∗ +ε))n and τ > 0 is a given regularization parameter. Here, +the matrix Aε(x) = A( x +ε), where A(x) = (aij(x))1≤i,j≤n defined on the space (L∞(O))n×n +is assumed to obey the uniform ellipticity condition: there exist real constants m1, m2 > 0 +such that m1||λ||2 ≤ �n +i,j=1 aij(x)λiλj ≤ m2||λ||2 for all λ ∈ Rn, which is endowed with +an Eucledian norm denoted by || · ||. Also, we understand the action of scalar boundary +operator ηε · Aε∇ on the vector uε|Γε +1 in a ”diagonal” manner: (ηε · Aε∇uε)i = ηε · Aε∇uεi, +for 1 ≤ i ≤ n. +We introduce the function space (H1 +Γε +0(O∗ +ε))n := {φ ∈ (H1(O∗ +ε))n | φ|Γε +0 = 0}. This is a +Banach space endowed with the norm +||φ||(H1 +Γε +0(O∗ε))n := ||∇φ||(L2(O∗ε))n×n, +∀φ ∈ (H1 +Γε +0(O∗ +ε))n. +Definition 2.1. We say a pair (uε, pε) ∈ (H1 +Γε +0(O∗ +ε))n × L2(O∗ +ε) is a weak solution to (3.2) +if, for all φ ∈ (H1 +Γε +0(O∗ +ε))n, +� +O∗ε +Aε∇uε : ∇φ dx − +� +O∗ε +pε div(φ) dx = +� +O∗ε +θε · φ dx, +(3.3) +5 + +and, for all w ∈ L2(O∗ +ε), +� +O∗ε +div(uε) w dx = 0. +(3.4) +Here, (: ) and (·) represent the summation of the component-wise multiplication of the +matrix entries and the usual scalar product of vectors, respectively. +The existence of a +unique weak solution (uε(θε), pε) ∈ (H1 +Γε +0(O∗ +ε))n × L2(O∗ +ε) of the system (3.2) follows anal- +ogous to [5, Theorem IV.7.1]. Also, for each ε > 0, there exists a unique solution to the +problem (3.1) that can be proved along the same lines as in [20, Chapter 2, Theorem 1.2]. +We call the optimal solution to (3.1) by the triplet (uε, pε, θε), with uε, pε, and θε as optimal +state, pressure, and control, respectively. +Optimality Condition: The optimality condition is given by J′ +ε(θ) · (θ − θε) ≥ 0, for all +θ ∈ (L2(O∗ +ε))n (see, [20, Chapter 2, Page 48]). One can obtain the further simplification of +this condition as +� +O∗ε(vε + τ θε) · (θ − θε) ≥ 0, for all θ ∈ (L2(O∗ +ε))n (see, [20, Chapter 2]), +where the pair (vε, qε) is the solution to the following adjoint problem: +� +� +� +� +� +� +� +� +� +− div +� +At +ε∇vε +� ++ ∇qε += uε − ud +in O∗ +ε, +div(vε) += 0 +in O∗ +ε, +ηε · At +ε∇vε − qεηε += 0 +on Γε +1, +vε += 0 +on Γε +0. +(3.5) +We call vε and qε, the adjoint state and pressure, respectively. The existence of unique weak +solution (vε, qε) to (3.5) can now be proved in a way similar to that of system (3.2). +The following theorem characterizes the optimal control, the proof of which follows analogous +to standard procedure laid in [20, Chapter 2, Theorem 1.4]. +Theorem 3.1. Let +� +uε, pε, θε +� +be the optimal solution of the problem (3.1) and (vε, qε) +solves (3.5), then the optimal control is characterized by +θε = −1 +τ vε a.e. in O∗ +ε. +(3.6) +Conversely, suppose that a triplet (ˇuε, ˇpε, ˇθε) ∈ +� +H1 +Γε +0(O∗ +ε) +�n +× L2(O∗ +ε) × (L2(O∗ +ε))n and a +pair (ˇvε, ˇqε) ∈ +� +H1 +Γε +0(O∗ +ε) +�n +× L2(O∗ +ε) solves the following system: +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +− div (Aε∇ˇuε) + ∇ˇpε = −1 +τ ˇvε +in O∗ +ε, +− div +� +At +ε∇ˇvε +� ++ ∇ˇqε = ˇuε − ud +in O∗ +ε, +div(ˇuε) = 0, div(ˇvε) = 0 +in O∗ +ε, +ηε · Aε∇ˇuε − ˇpεηε = 0 +on Γε +1, +ηε · At +ε∇ˇvε − ˇqεηε = 0 +on Γε +1, +ˇvε = 0, ˇuε = 0 +on Γε +0. +6 + +Then the triplet (ˇuε, ˇpε, − 1 +τ ˇvε) is the optimal solution of (3.1). +4 +A priori estimates +This section concerns the derivation of estimates for the optimal solution to the problem +(3.1) and the associated solution to the adjoint problem (3.5). These estimates are uniform +and independent of the parameter ε. Towards attaining this aim, we first evoke the following +two lemmas: +Lemma 4.1 (Lemma A.4, [3]). There exists a constant C ∈ R+, independent of ε, such +that +||v||L2(O∗ε)n ≤ C||∇v||(L2(O∗ε))n×n, +∀ v ∈ (H1 +Γε +0(O∗ +ε))n. +Lemma 4.2 (Lemma 5.1, [12]). For each ε > 0 and qε ∈ L2(O∗ +ε), there exists gε ∈ +(H1 +Γε +0(O∗ +ε))n and a constant C ∈ R+, independent of ε, such that +div(gε) = qε and ||∇gε||(L2(O∗ε))n×n ≤ C(O) ||qε||L2(O∗ε). +(4.1) +Theorem 4.3. For each ε > 0, let +� +uε, pε, θε +� +be the optimal solution of the problem (3.1) +and (vε, qε) solves the corresponding adjoint problem (3.5). Then, one has θε ∈ (H1 +Γε +0(O∗ +ε))n +and there exists a constant C ∈ R+, independent of ε such that +��¯θε +�� +(L2(O∗ε))n ≤ C, +(4.2) +∥¯uε∥(H1 +Γε +0(O∗ε))n ≤ C, +(4.3) +∥¯vε∥(H1 +Γε +0(O∗ε))n ≤ C, +(4.4) +∥¯pε∥L2(O∗ε) ≤ C, +(4.5) +∥¯qε∥L2(O∗ε) ≤ C. +(4.6) +Proof. Let uε(0) denotes the solution to (3.2) corresponding to θε = 0. In view of Lemma +4.1, one can show that ∥uε(0)∥(L2(O∗ε))n ≤ 0, i.e., uε(0) = 0 in (L2(O∗ +ε))n. Using this and +the optimality of solution (uε, pε, θε) to problem (3.1), we have +∥uε(θ) − ud∥2 +(L2(O∗ε))n + τ∥θε∥2 +(L2(O∗ε))n ≤ ∥uε(0) − ud∥2 +(L2(O∗ε))n ≤ C, +which gives estimate (4.2). Now, let us take uε as a test function in (3.3). Considering +(4.2) and the uniform ellipticity condition of matrix Aε, one obtains upon applying the +Cauchy-Schwarz inequality along with the Lemma 4.1, the following: +m1∥∇uε∥2 +(L2(O∗ε))n×n ≤ +� +O∗ε +Aε∇uε : ∇uε dx ≤ C ∥θε∥(L2(O∗ε))n∥∇uε∥(L2(O∗ε))n×n, +from which estimate (4.3) follows. +7 + +Owing to Lemma 4.2, for given pε ∈ L2(O∗ +ε), there exists gε ∈ (H1 +Γε +0(O∗ +ε))n satisying div(gε) = +pε. Corresponding to θε, taking v = gε in (3.3), we get +∥pε∥2 +L2(O∗ε) = +� +O∗ε +Aε∇uε : ∇gε dx − +� +O∗ε +θε · gε dx. +(4.7) +In view of (4.1), (4.2) and (4.3), and the uniform ellipticity condition of the matrix Aε, +one obtains from (4.7) upon employing the Cauchy-Schwarz inequality and Lemma 4.1, the +following: +∥pε∥2 +L2(O∗ε) ≤ +� +m2∥∇uε∥(L2(O∗ε))n×n + C∥θε∥(L2(O∗ε))n +� +∥∇gε∥(L2(O∗ε))n×n, +which gives the estimate (4.5). Likewise, one can easily obtain the estimates (4.4) and (4.6) +following the above discussion. Finally, from (3.6), we obtain that θε ∈ (H1 +Γε +0(O∗ +ε))n. +5 +The method of periodic unfolding for perforated do- +mains +We evokes the definition of the periodic unfolding operator and few of its properties as +stated in [8,11]. Given x ∈ Rn, we denote the greatest integer and the fractional parts of x +respectively by [x]W and {x}W. That is, [x]W = �n +j=1 kjbj be the unique integer combination +of periods and {x}W = x − [x]W. In particular, we have for ε > 0, +x = ε +��x +ε +� +W + +�x +ε +� +W +� +, +∀ x ∈ Rn. +Definition 5.1. The unfolding operator T ∗ +ε : {O∗ +ε → R} → {O × W ∗ → R} is defined as +T ∗ +ε (u) (x, y) = +� +u +� +ε +� x1 +ε +� +W + εy +� +a.e. +for (x, y) ∈ � +Oε × W ∗, +0 +a.e. +for (x, y) ∈ Λε × W ∗. +Also, for any domain D ⊇ O∗ +ε and vector u = (u1, · · · , un) ∈ ({D → R})n, we define its +unfolding by +T ∗ +ε (u) := (T ∗ +ε (u1), · · · , T ∗ +ε (un)). +Proposition 5.2. In the following there are the properties of the unfolding operator: +(i) T ∗ +ε is linear and continuous from L2(O∗ +ε) to L2(O × W ∗). +(ii) Let u, v ∈ L2(O∗ +ε). Then T ∗ +ε (uv) = T ∗ +ε (u) T ∗ +ε (v) . +(iii) +Let u ∈ L2 (O) . Then T ∗ +ε (u) → u strongly in L2 (O × W ∗) . +(iv) +Let u ∈ L1 (O∗ +ε) . Then +� +� +O∗ε +u(x) dx = +� +O∗ε +u(x) dx − +� +ˆΛ∗ε +u(x) dx = +1 +|W ∗| +� +O×W ∗ T ∗ +ε (u)(x, y) dxdy. +8 + +(v) +For each ε > 0, let {uε} ∈ L2 (O) and uε → u strongly in L2 (O) . +Then T ∗ +ε (uε) → u strongly in L2 (O × W ∗) . +(vi) Let v ∈ L2 (W ∗) be a W-periodic function and vε(x) = v +� x +ε +� +. Then, +T ∗ +ε (vε) (x, y) = +� +v(y) +a.e. for (x, y) ∈ � +Oε × W ∗, +0 +a.e. for (x, y) ∈ Λε × W ∗. +(vii) Let fε ∈ L2 (O∗ +ε) be uniformly bounded. Then, there exists f ∈ L2(O × W ∗) such that +T ∗ +ε (fε) ⇀ f weakly in L2(O × W ∗), and +�fε ⇀ +1 +|W| +� +W ∗ f(·, y) dy weakly in L2(O). +Proposition 5.3. Let O ⊂ Rn be bounded with Lipschitz boundary. Let fε ∈ H1(O∗ +ε) be +such that fε = 0 on ∂O ∩ ∂O∗ +ε and satisfy, +∥∇fε∥(L2(O∗ε))n ≤ C§. +Then, there exists f ∈ H1 +0(O) and �f ∈ L2 � +O; H1 +per (W ∗) +� +with MW ∗( �f) = 0, such that up to +a subsequence, +� +T ∗ +ε (∇fε) ⇀ ∇f + ∇y �f +weakly in (L2 (O × W ∗))n , +T ∗ +ε (fε) → f +strongly in L2 (O; H1 (W ∗)) . +6 +Limit optimal control problem +This section presents the limit (homogenized) system corresponding to the problem (3.1), +which we considered in the beginning. +Let us consider the function space +� +H1 +0(O) +�n := +� +ϕ ∈ (H1(O))n | ϕ|∂O = 0 +� +, +which is a Hilbert space for the norm +∥ϕ∥(H1 +0(O))n := ∥∇ϕ∥(L2(O))n×n +∀ ϕ ∈ (H1 +0(O))n. +We now consider the limit OCP associated with the Stokes system +inf +θ∈(L2(O))n +� +J(θ) = Θ +2 +� +O +|u − ud|2 dx + τΘ +2 +� +O +|θ|2 dx +� +, +(6.1) +§The symbol C represents a generic constant that is positive and independent of ε. +9 + +subject to +� +� +� +� +� +� +� +� +� +− +n +� +j,α,β=1 +∂ +∂xα +� +bαβ +ij +∂uj +∂xβ +� ++ ∇p += θ +in O, +div (u) += 0 +in O, +u += 0 +on ∂O, +(6.2) +where the tensor B = (bαβ +ij ) = (bαβ +ij )1≤i,j,α,β≤n is constant, elliptic, and for 1 ≤ i, j, α, β ≤ n, +is given by +bαβ +ij = aαβ +ij − +1 +|W ∗| +� +W ∗ A(y)∇y +� +P β +j − χβ +j +� +: ∇yχα +i dy, +with aαβ +ij = +1 +|W ∗| +� +W ∗ A(y)∇y +� +P β +j − χβ +j +� +: ∇yP α +i dy as the entries of the constant tensor A0, +P β +j = P β +j (y) = (0, . . . , yj, . . . , 0) with yj at the β-th position, and for 1 ≤ j, β ≤ n, the +correctors (χβ +j , Πβ +j ) ∈ (H1(W ∗))n × L2(W ∗) solves the cell problem +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +− divy +� +A(y)∇y(P β +j − χβ +j ) +� ++ ∇yΠβ +j += 0 +in W ∗, +η · A(y)∇y(P β +j − χβ +j ) − Πβ +j η += 0 +on ∂W ∗\∂W, +divy(P β +j − χβ +j ) += 0 +in W ∗, +(χβ +j , Πβ +j ) +W ∗- periodic, +MW ∗(χβ +j ) += 0. +(6.3) +The existence of this unique pair (u, p) ∈ (H1 +0(O))n × L2(O) can be found in [4, Chapter +1]. Further, the problem (6.1) is a standard one and there exists a unique weak solution to +it, one can follow the arguments introduced in [20, Chapter 2, Theorem 1.2]. We call the +triplet (u, p, θ) ∈ (H1 +0(O))n × L2(O) × (L2(O))n, the optimal solution to (6.1), with u, p, +and θ as the optimal state, pressure, and control, respectively. +Now, we introduce the limit adjoint system associated with (6.2): Find a pair (v, q) ∈ +(H1 +0(O))n × L2(O) which solves the system +� +� +� +� +� +− +n +� +i,α,β=1 +∂ +∂xβ +� +bβα +ji +∂vi +∂xα +� ++ ∇q += u − ud +in O, +div (v) += 0 +in O, +(6.4) +where the tensor Bt = (bβα +ji ) = (bβα +ji )1≤i,j,α,β≤n is constant, elliptic, and for 1 ≤ i, j, α, β ≤ n, +is given by +bβα +ji = aβα +ji − +1 +|W ∗| +� +W ∗ At(y)∇y +� +P β +j − Hβ +j +� +: ∇yHα +i dy, +with aβα +ji = +1 +|W ∗| +� +W ∗ At(y)∇y +� +P β +j − Hβ +j +� +: ∇yP α +i dy as the entries of the constant tensor +At +0. Also, for 1 ≤ j, β ≤ n, the correctors (Hβ +j , Zβ +j ) ∈ (H1(W ∗))n × L2(W ∗) solves the cell +10 + +problem +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +− divy +� +At(y)∇y(P β +j − Hβ +j ) +� ++ ∇yZβ +j += 0 +in W ∗, +η · At(y)∇y(P β +j − Hβ +j ) − Zβ +j η += 0 +on ∂W ∗\∂W, +divy(P β +j − Hβ +j ) += 0 +in W ∗, +(Hβ +j , Zβ +j ) +W ∗- periodic, +MW ∗(Hβ +j ) += 0. +(6.5) +In the following, we state a result similar to Theorem 3.1 that characterizes the optimal +control θ in terms of the adjoint state v and the proof of which follows analogous to the +standard procedure laid in [20, Chapter 2, Theorem 1.4]. +Theorem 6.1. Let +� +u, p, θ +� +be the optimal solution to (6.1) and (v, q) be the corresponding +adjoint solution to (6.4), then the optimal control is characterized by +θ = −1 +τ v a.e. in O. +(6.6) +Conversely, suppose that a triplet +(ˇu, ˇp, ˇθ) ∈ (H1 +0(O))n × L2(O) × (L2(O))n and a pair +(ˇv, ˇq) ∈ (H1 +0(O))n × L2(O), respectively, satisfy the following systems: +� +� +� +� +� +− +n +� +j,α,β=1 +∂ +∂xα +� +bαβ +ij +∂ˇuj +∂xβ +� ++ ∇ˇp += − 1 +τ ˇv +in O, +div (ˇu) += 0 +in O, +and +� +� +� +� +� +− +n +� +i,α,β=1 +∂ +∂xβ +� +bβα +ji +∂ˇvi +∂xα +� ++ ∇ˇq += ˇu − ud +in O, +div (ˇv) += 0 +in O. +Then, the triplet +�ˇu, ˇp, − 1 +τ ˇv +� +is the optimal solution to (6.1). +7 +Convergence results +We present here the key findings on the convergence analysis of the optimal solutions to the +problem (3.1) and its corresponding adjoint system (3.5) by using the method of periodic +unfolding for perforated domains described in Section 5. +Theorem 7.1. For given ε > 0, let the triplets (uε, pε, θε) and (u, p, θ), respectively, be the +11 + +optimal solutions of the problems (3.1) and (6.1). Then +T ∗ +ε (Aε) → A +strongly in (L2(O × W ∗))n×n, +(7.1a) +� +θε ⇀ Θ θ +weakly in +� +L2 (O) +�n , +(7.1b) +� +uε ⇀ Θ u +weakly in (H1 +0(O))n, +(7.1c) +� +vε ⇀ Θ v +weakly in (H1 +0(O))n, +(7.1d) +�pε ⇀ Θ +n A0∇u: I + Θ p +weakly in L2(O), +(7.1e) +�qε ⇀ Θ +n At +0∇v: I + Θ q +weakly in L2(O), +(7.1f) +where A0 is a tensor as defined in Section 6, I is the n × n identity matrix, θ is characterized +through (6.6) and the pairs (vε, qε) and (v, q) solve respectively the systems (3.5) and (6.4). +Moreover, +lim +ε→0 Jε(θε) = J(θ). +(7.2) +Proof. First, upon using Proposition 5.2 (vi) on the entries of the matrix Aε, we obtain (7.1a) +under the passage of limit ε → 0. Similarly, one can prove the convergence for the matrix +At +ε under unfolding. Next, in view of Theorem 4.3 and the fact that the triplet (uε, pε, θε) +is an optimal solution to problem (3.1), one gets uniform estimates for the sequences {θε}, +{uε}, {pε}, {vε}, and {qε} in the spaces (L2 (O∗ +ε))n, (H1 +Γε +0(O∗ +ε))n, L2 (O∗ +ε), (H1 +Γε +0(O∗ +ε))n, and +L2 (O∗ +ε), respectively. +Using the uniform estimate of the sequence {θε} in the space (L2 (O∗ +ε))n and Proposition +5.2 (i), we have the sequence {T ∗ +ε (θε)} to be uniformly bounded in the space (L2 (O × W ∗))n. +Thus, by weak compactness, there exists a subsequence not relabelled and a function ˆθ in +(L2 (O × W ∗))n, such that +T ∗ +ε (θε) ⇀ ˆθ +weakly in +� +L2 (O × W ∗) +�n . +(7.3) +Now, using Proposition 5.2 (vii) in (7.3) gives +� +θε ⇀ +1 +|W| +� +W ∗ +ˆθ(x, y) dy = Θ θ0 +weakly in +� +L2 (O) +�n , +(7.4) +where, θ0 = MW ∗(ˆθ). +Employing Proposition 5.2 (i), we have the uniform boundedness of the sequences {T ε(uε)}, +{T ε(∇uε)}, and {T ε(pε)} in the respective spaces (L2(O; H1 (W ∗)))n, (L2(O × W ∗))n×n, +and L2(O × W ∗). Further, upon employing Proposition 5.3 and Proposition 5.2 (vii), there +exist subsequences not relabelled and functions ˆu with MW ∗(ˆu) = 0, u0, and ˆp in spaces +12 + +(L2(O; H1 +per (W ∗)))n, (H1 +0(O))n, and L2(O × W ∗), respectively, such that +T ∗ +ε (uε) → u0 +strongly in (L2(O; H1 (W ∗)))n, +(7.5a) +T ∗ +ε (∇uε) ⇀ ∇u0 + ∇y ˆu +weakly in (L2(O × W ∗))n×n, +(7.5b) +� +uε ⇀ Θ u0 +weakly in (H1 +0(O))n, +(7.5c) +T ∗ +ε (pε) ⇀ ˆp +weakly in L2(O × W ∗), +(7.5d) +�pε ⇀ Θ MW ∗(ˆp) +weakly in L2(O). +(7.5e) +Likewise, for the associated adjoint counterparts, viz., vε, and qε , one obtains that there +exist subsequences not relabelled and functions ˆv with MW ∗(ˆv) = 0, v0, and ˆq in spaces +(L2(O; H1 +per (W ∗)))n, (H1 +0(O))n, and L2(O × W ∗), respectively, such that +T ∗ +ε (vε) → v0 +strongly in (L2(O; H1 (W ∗)))n, +(7.6a) +T ∗ +ε (∇vε) ⇀ ∇v0 + ∇yˆv +weakly in (L2(O × W ∗))n×n, +(7.6b) +� +vε ⇀ Θ v0 +weakly in (H1 +0(O))n, +(7.6c) +T ∗ +ε (qε) ⇀ ˆq +weakly in L2(O × W ∗), +(7.6d) +�qε ⇀ MW ∗(ˆq) +weakly in L2(O). +(7.6e) +The identification of the limit functions ˆu, ˆv, ˆp, ˆq, MW ∗(ˆp) and MW ∗(ˆq) is carried out in +subsequent steps. +Step 1: (Claim): For all ϕ ∈ (H1 +0(O))n, ψ ∈ +� +L2 � +O; H1 +per (W ∗) +��n , and w ∈ L2(O), we +claim that the ordered quadruplet (u0, ˆu, ˆp, θ0) ∈ (H1 +0(O))n ×(L2(O; H1 +per (W ∗)))n ×L2(O× +W ∗) × (L2(O))n is a unique solution to the following limit system: +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +1 +|W| +� +O×W ∗ A(y) (∇u0 + ∇y�u(x, y)) : (∇ϕ + ∇yψ) dx dy +− 1 +|W| +� +O×W ∗ ˆp(x, y) (div(ϕ) + divy(ψ)) dx dy = Θ +� +O +θ0 · ϕ dx, +and, +� +O +div(u0) w dx = 0, +(7.7) +and the ordered triplet (v0, ˆv, ˆq) ∈ (H1 +0(O))n×(L2(O; H1 +per (W ∗)))n×L2(O×W ∗) is a unique +solution to the following limit adjoint system: +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +1 +|W| +� +O×W ∗ At(y) (∇v0 + ∇y�v(x, y)) : (∇ϕ + ∇yψ) dx dy +− 1 +|W| +� +O×W ∗ ˆq(x, y) (div(ϕ) + divy(ψ)) dx dy = Θ +� +O +(u0 − ud) · ϕ dx, +and, +� +O +div(v0) w dx = 0. +(7.8) +13 + +Proof of the Claim: Towards the proof of (7.7), let us consider a test function ϕ ∈ (D(O))n +in (3.3) and use properties (i), (ii), and (iv) of Proposition 5.2 to get +1 +|W| +� +O×W ∗ T ∗ +ε (Aε) T ∗ +ε (∇uε): T ∗ +ε (∇ϕ) dx dy + +� +ˆΛ∗ε +Aε∇uε : ∇ϕ dx − +� +ˆΛ∗ε +pε div(ϕ) dx +− +1 +|W| +� +O×W ∗ T ∗ +ε (pε) T ∗ +ε (div(ϕ)) dx dy = +1 +|W| +� +O×W ∗ T ∗ +ε (θε) · T ∗ +ε (φε) dx dy + +� +ˆΛ∗ε +θε · ϕ dx. +(7.9) +Using Proposition 5.2 (iii), the fact that limε→0 |ˆΛ∗ +ε| = 0, and convergences (7.3), (7.1a), +(7.5b), (7.5d), we have under the passage of limit ε → 0 in (7.9) +1 +|W| +� +O×W ∗ A(y) (∇u0 + ∇y�u(x, y)) : ∇ϕ dx dy +− 1 +|W| +� +O×W ∗ ˆp(x, y) div(ϕ) dx dy = Θ +� +O +θ0 · ϕ dx, +(7.10) +which remains valid for every ϕ ∈ (H1 +0(O))n, by density. +Now, consider the function φε(x) = εφ(x)ξ( x +ε), where φ ∈ D(O) and ξ ∈ (H1 +per(W ∗))n. +Employing properties (ii), (iii), and (vi) of Proposition 5.2, one can easily obtain +T ∗ +ε (φε) (x, y) → 0 +strongly in (L2(O × W ∗))n, +(7.11a) +T ∗ +ε (∇φε) (x, y) → φ(x)∇yξ(y) +strongly in (L2(O × W ∗))n×n. +(7.11b) +Let us use the test function φε in (3.3) and employ properties (i), (ii), and (iv) of Proposition +5.2 to get +1 +|W| +� +O×W ∗ T ∗ +ε (Aε) T ∗ +ε (∇uε): T ∗ +ε (∇φε) dx dy + +� +ˆΛ∗ε +Aε∇uε : ∇φε dx − +� +ˆΛ∗ε +pε div(φε) dx +− 1 +|W| +� +O×W ∗ T ∗ +ε (pε) T ∗ +ε (div(φε)) dx dy = +1 +|W| +� +O×W ∗ T ∗ +ε (θε) · T ∗ +ε (φε) dx dy + +� +ˆΛ∗ε +θε · φε dx. +(7.12) +In (7.12), the absolute value of each integral over ˆΛ∗ +ε is bounded above with a bound of +order ε|ˆΛ∗ +ε| or |ˆΛ∗ +ε|. This with the fact that limε→0 |ˆΛ∗ +ε| = 0, and convergences (7.3), (7.1a), +(7.5b), (7.5d), and (7.11), gives under the passage of limit ε → 0 +1 +|W| +� +O×W ∗ A(y) (∇u0 + ∇y�u(x, y)) : ∇yψ dx dy − +1 +|W| +� +O×W ∗ ˆp(x, y) divy(ψ) dx dy = 0, +(7.13) +which remains valid for every φ ξ = ψ ∈ (L2(O; H1 +per(W ∗)))n, by density. +Further, for all w ∈ L2(O), we have +� +O∗ε +div(uε)w dx = 0. +(7.14) +14 + +Now, upon applying unfolding on (7.14) and using properties (i), (ii), and (iii) of Proposition +5.2 along with convergence (7.5b), we get under the passage of limit ε → 0 +1 +|W| +� +O×W ∗ (div(u0) + divy(ˆu)) w dx dy = 0, +which eventually gives upon using the fact that ˆu is W ∗− periodic, for all w ∈ L2(O): +� +O +div(u0)w dx = 0. +(7.15) +Finally, upon adding (7.10) with (7.13) and considering (7.15), we establish (7.7). Likewise, +one can easily establish (7.8). This settles the proof of the claim. +Step 2: First, we are going to identify the limit functions ˆu, ˆv, ˆp, and ˆq. Next, using these +identifications, we will identify MW ∗(ˆp) and MW ∗(ˆq). +Identification of ˆu, ˆv, ˆp, ˆq: Taking sucessively ϕ ≡ 0 and ψ ≡ 0 in (7.7), yields +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +− divy(A(y)∇y�u(x, y)) + ∇y�p(x, y) = divy(A(y))∇u0(x) +in O × W ∗, +− divx +�� +W ∗ A(y)(∇u0(x) + ∇y�u(x, y))dy +� ++ ∇�p(x, y) = |W ∗| θ0 +in O, +div(u0) = 0 +in O, +�u(x, ·) +is W ∗ − periodic. +(7.16) +In the first line of (7.16), we have the y-independence of ∇u0(x) and the linearity of opera- +tors, viz., divergence and gradient, which suggests �u(x, y) and �p(x, y) to be of the following +form (see, for e.g., [17, Page 15]): +� +� +� +� +� +� +� +� +� +� +� +�u(x, y) = − +n +� +j,β=1 +χβ +j (y)∂u0j +∂xβ ++ u1(x), +�p(x, y) = +n +� +j,β=1 +Πβ +j (y)∂u0j +∂xβ ++ p0(x). +(7.17) +where the ordered pair (u1, p0) ∈ (H1(O))n × L2(O), and for 1 ≤ j, β ≤ n, the pair (χβ +j , Πβ +j ) +satisfy the cell problem (6.3). Likewise we obtain for the corresponding adjoint weak formu- +lation (7.8): +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +− divy(A(y)∇y�v(x, y)) + ∇y�q(x, y) = divy(A(y))∇v0(x) +in O × W ∗, +− divx +�� +W ∗ A(y)(∇v0(x) + ∇y�v(x, y))dy +� ++ ∇�q(x, y) = |W ∗| (u0 − ud) +in O, +div(v0) = 0 +in O, +�v(x, ·) +is W ∗ − periodic, +(7.18) +15 + +and, +� +� +� +� +� +� +� +� +� +� +� +�v(x, y) = − +n +� +j,β=1 +Hβ +j (y)∂v0j +∂xβ ++ v1(x), +�q(x, y) = +n +� +j,β=1 +Zβ +j (y)∂v0j +∂xβ ++ q0(x), +(7.19) +where the ordered pair (v1, q0) ∈ (H1(O))n ×L2(O), and for 1 ≤ j, β ≤ n, the pair (Hβ +j , Zβ +j ) +satisfy the cell problem (6.5). +Identification of MW ∗(ˆp) and MW ∗(ˆq): +Choosing the test function y = (y1, . . . , yn) in +the weak formulation of (6.3), we get +n +� +i,l,k,α=1 +� +W ∗ alk +∂ +∂yk +� +P β +j − χβ +j +� +· ∂P α +i +∂yl +∂yi +∂yα +dy = n +� +W ∗ Πβ +j dy. +(7.20) +In view of (7.5e), (7.17), and (7.20), we observe that +MW ∗(ˆp) = +1 +|W ∗| +n +� +i,j,l,k,α,β=1 +� +W ∗ alk +∂ +∂yk +� +P β +j − χβ +j +� +· ∂P α +i +∂yl +∂yi +∂yα +∂u0j +∂xβ +dy + p0, +which upon using the definition of aαβ +ij , gives +MW ∗(ˆp) = +n +� +i,j,α,β=1 +aαβ +ij +∂u0j +∂xβ +∂yi +∂yα ++ p0. +(7.21) +Also, we re-write the equation (7.21) to get the identification of MW ∗(ˆp) as +MW ∗(ˆp) = A0∇u0 : I + p0. +(7.22) +Likewise, one can obtain the identification of MW ∗(ˆq) as +MW ∗(ˆq) = At +0∇v0 : I + q0. +(7.23) +Thus, from (7.5e) and (7.22); (7.6e) and (7.23), we have the following weak convergences: +�pε ⇀ Θ +n A0∇u0 : I + Θ p0 +weakly in L2(O), +(7.24a) +�qε ⇀ Θ +n At +0∇v0 : I + Θ q0 +weakly in L2(O). +(7.24b) +Step 3: (Claim): The pairs (u0, p0) and (v0, q0) solve the systems (6.2) and (6.4), respec- +tively. +Proof of the Claim: We now prove that the pair (u0, p0) solves the system (6.2). The proof +16 + +that the pair (v0, q0) solves the system (6.4) follows analogously. Substituting the values of +�u(x, y) and �p(x, y) from expression (7.17) into equation (7.10), we get +1 +|W| +n +� +l,k=1 +� +O×W ∗ alk +� +∂u0 +∂xk +− +n +� +j,β=1 +∂χβ +j +∂yk +∂u0j +∂xβ +� +∂ϕ +∂xl +dx dy − +1 +|W| +n +� +j,β=1 +� +O×W ∗ Πβ +j +∂u0j +∂xβ +div(ϕ) dx dy +−Θ +� +O +p0 div(ϕ) dx = Θ +� +O +θ0 · ϕ dx. +(7.25) +With P β +j = (0, . . . , yj, . . . , 0), we can express the terms ∂u0 +∂xk , ∂ϕ +∂xl, and div(ϕ) as +∂u0 +∂xk += +n +� +j,β=1 +∂P β +j +∂yk +∂u0j +∂xβ +, +∂ϕ +∂xl += +n +� +i,α=1 +∂P α +i +∂yl +∂ϕi +∂xα +, +div(ϕ) = +n +� +i,α=1 +divy(P α +i ) ∂ϕi +∂xα +. +Substituting these expressions in (7.25), we obtain +n +� +i,j,α,β=1 +� +O +� +1 +|W ∗| +n +� +l,k=1 +� +W ∗ alk +∂ +∂yk +� +P β +j − χβ +j +� ∂P α +i +∂yl +dy +� +∂u0j +∂xβ +∂ϕi +∂xα +dx +− +n +� +i,j,α,β=1 +� +O +� +1 +|W ∗| +� +W ∗ Πβ +j divy(P α +i ) dy +� ∂u0j +∂xβ +∂ϕi +∂xα +dx − +� +O +p0 div(ϕ) dx = +� +O +θ0 · ϕ dx. +(7.26) +Now, choosing the test function χα +i in the weak formulation of (6.3), we get upon using +the fact that divy(χα +i ) = divy(P α +i ) = δiα, where δ denotes the Kronecker delta function, the +following: +� +W ∗ A(y)∇y +� +P β +j − χβ +j +� +: ∇yχα +i dy = +� +W ∗ Πβ +j δiα dy. +(7.27) +Further, substituting (7.27) in (7.26), we obtain +n +� +i,j,α,β=1 +� +O +� +1 +|W ∗| +n +� +l,k=1 +� +W ∗ alk +∂ +∂yk +� +P β +j − χβ +j +� ∂ +∂yl +(P α +i − χα +i ) dy +� +∂u0j +∂xβ +∂ϕi +∂xα +dx +− +� +O +p0 div(ϕ) dx = +� +O +θ0 · ϕ dx. +(7.28) +Also, we can write equation (7.28) as +n +� +i,j,α,β=1 +� +O +bαβ +ij +∂u0j +∂xβ +∂ϕi +∂xα +dx − +� +O +p0 div(ϕ) dx = +� +O +θ0 · ϕ dx, +(7.29) +17 + +which holds true for all ϕ ∈ (H1 +0(O))n. Also, from equation (7.15), we have +� +O div(u)w dx = +0, for every w ∈ L2(O). This together with equation (7.29) implies that, for θ = θ0, the +pair (u0, p0) ∈ (H1 +0(O))n × L2(O) satisfies the variational formulation of the system (6.2). +Therefore, we obtain the optimality system for the minimization problem (6.1). Also, in +view of Theorem 6.1, we conclude that the triplet (u0, p0, θ0) is indeed an optimal solution to +the problem (6.1). Finally, upon considering the optimal solution’s uniqueness, we establish +that the subsequent pair of triplets are equal: +(u, p, θ) = (u0, p0, θ0). +(7.30) +Hence, upon comparing (7.5c), (7.6c), (7.24a), (7.24b), and (7.4) with (7.30), we obtain +convergences (7.1c), (7.1d), (7.1e), (7.1f), and (7.1b), respectively. +Step 4: Now, we will furnish the proof of the energy convergence for the L2−cost functional. +Choosing the test function (uε − ud) in the weak formulation of system (3.5), we get under +unfolding upon passing ε → 0 +lim +ε→0 +� +O∗ε +|uε − ud|2 dx = +1 +|W| lim +ε→0 +� +O×W ∗ T ∗ +ε (At +ε) T ∗ +ε (∇vε): T ∗ +ε (∇(uε − ud)) dx dy ++ +1 +|W| lim +ε→0 +� +O×W ∗ T ∗ +ε (qε) T ∗ +ε (div(ud)) dx dy, +which gives in view of (7.30), Proposition 5.2 (iii) and convergences (7.6a), (7.5b), and (7.6d) +lim +ε→0 +� +O∗ε +|uε − ud|2 dx = +1 +|W| +� +O×W ∗ At(y) (∇v + ∇y�v(x, y)) : ∇y(u − ud) dx dy ++ +1 +|W| +� +O×W ∗ ˆq(x, y) div(ud) dx dy. +(7.31) +Also, using (7.19) in (7.31) alongwith (7.30), we have upon simplification +lim +ε→0 +� +O∗ε +|uε − ud|2 dx = Θ +� +n +� +i,j,α,β=1 +� +O +bβα +ji +∂vi +∂xα +∂(u − ud)j +∂xβ +dx − +� +O +q div(u − ud) dx +� +. +(7.32) +Now, using the test function (u − ud) in the weak formulation of system (6.4), we get the +following upon comparing with the right hand side of equation (7.32) +lim +ε→0 +� +O∗ε +|uε − ud|2 dx = Θ +� +O +|u − ud|2 dx. +(7.33) +Furthermore, in view of (3.6), (7.6a), and (7.30), we get under unfolding upon the passage +18 + +of limit ε → 0 +lim +ε→0 +τ +2 +� +O∗ε +|θε|2 dx = lim +ε→0 +1 +2|W| +� +O×W ∗ |T ∗ +ε (θε)|2 dx dy += lim +ε→0 +1 +2τ|W| +� +O×W ∗ |T ∗ +ε (vε)|2 dx dy += +1 +2τ|W| +� +O×W ∗ |v|2 dx dy. +(7.34) +Also, since v is independent of y and comparing the right hand side of (7.34) with (6.6), we +get +lim +ε→0 +τ +2 +� +O∗ε +|θε|2 dx = Θτ +2 +� +O +|θ|2 dx. +(7.35) +Thus, from equations (7.33) and (7.35), we get (7.2). +This completes the proof of Theorem 7.1. +8 +Conclusions +We have addressed the limiting behavior of an interior OCP corresponding to Stokes equa- +tions in an nD (n ≥ 2) +periodically perforated domain +O∗ +ε via the technique of periodic +unfolding in perforated domains (see, [8, 11]). We employed the Neumann boundary con- +dition on the part of the boundary of the perforated domain. Firstly, we characterized the +optimal control in terms of the adjoint state. Secondly, we deduced the apriori optimal +bounds for control, state, pressure, and their associated adjoint state and pressure functions. +Thereafter, the limiting analysis for the considered OCP is carried out upon employing the +periodic unfolding method in perforated domains. We observed the convergence between the +optimal solution to the problem (3.1) posed on the perforated domain O∗ +ε and the optimal +solution to that of the limit problem (6.1) governed by stationary Stokes equation posed on +a non-perforated domain O. Finally, we established the convergence of energy corresponding +to L2−cost functional. +9 +Acknowledgments +The first author would like to thank the Ministry of Education, Government of India for +Prime Minister’s Research Fellowship (PMRF-2900953). The second author would like to +thank the support from Science & Engineering Research Board (SERB) (SRG/2019/000997), +Government of India. +19 + +References +[1] G. Allaire, Homog´en´eisation des ´equations de Stokes dans un domaine perfor´e de petits +trous r´epartis p´eriodiquement, C. R. Acad. Sci. Paris S´er. I Math. 309 (1989), no. 11, +741–746. +[2] +, Homogenization of the Navier-Stokes equations with a slip boundary condition, +Comm. Pure Appl. Math. 44 (1991), no. 6, 605–641. +[3] G. Allaire and F. Murat, Homogenization of the Neumann problem with nonisolated +holes, Asymptotic Anal. 7 (1993), no. 2, 81–95, With an appendix written jointly with +A. K. Nandakumar. +[4] A. Bensoussan, J.-L. Lions, and G. Papanicolaou, Asymptotic analysis for periodic struc- +tures, Studies in Mathematics and its Applications, vol. 5, North-Holland Publishing +Co., Amsterdam-New York, 1978. +[5] F. Boyer and P. 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Anal. 79 (2012), no. 3-4, 229–250. +21 + diff --git a/hNAzT4oBgHgl3EQfMvs5/content/tmp_files/load_file.txt b/hNAzT4oBgHgl3EQfMvs5/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8e1461979ff880967120d3b2e8549f37010a566a --- /dev/null +++ b/hNAzT4oBgHgl3EQfMvs5/content/tmp_files/load_file.txt @@ -0,0 +1,713 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf,len=712 +page_content='Optimal control problem for Stokes system: Asymptotic analysis via unfolding method in a perforated domain Swati Garg and Bidhan Chandra Sardar∗ Department of Mathematics Indian Institute of Technology Ropar Rupnagar-140001, Punjab, India swati.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='19maz0006@iitrpr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='in, swatigargmks@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='com bcsardar@iitrpr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='in, bcsardar31@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='com January 4, 2023 Abstract This article’s subject matter is the study of the asymptotic analysis of the optimal control problem (OCP) constrained by the stationary Stokes equations in a periodi- cally perforated domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' We subject the interior region of it with distributive controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' The Stokes operator considered involves the oscillating coefficients for the state equa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' We characterize the optimal control and, upon employing the method of periodic unfolding, establish the convergence of the solutions of the considered OCP to the solu- tions of the limit OCP governed by stationary Stokes equations over a non-perforated domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' The convergence of the cost functional is also established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Keywords: Stokes equations, Homogenization, Optimal control, Perforated domain, Un- folding operator 1 Introduction In this article, we consider the optimal control problem (OCP) governed by generalized stationary Stokes equations in a periodically perforated domain O∗ ε (see Section 2, on the domain description).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' The size of holes in the perforated domain is of the same order as that of the period, and the holes are allowed to intersect the boundary of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' The control is applied in the interior region of the domain, and we wish to study the asymptotic AMS subject classifications: 35B27, 35B40, 35Q93, 49J20, 76D07 ∗Corresponding author 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='01136v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='OC] 3 Jan 2023 analysis (homogenization) of an interior OCP subject to the constrained stationary Stokes equations with oscillating coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' One can find several works in the literature regarding the homogenization of Stokes equations over a perforated domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Using the multiple-scale expansion method, the au- thors in [16] studied the homogenization of Stokes equations in a porous medium with the Dirichlet boundary condition on the boundary of the holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' They obtained the Darcy’s law as the limit law in the homogenized medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' In [9], the authors considered the Stokes system in a periodically perforated domain with non-homogeneous slip boundary conditions depending upon some parameter γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Upon employing the Tartar’s method of oscillating test functions they obtained under homogenization, the limit laws, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=', Darcy’s law ( for γ < 1), Brinkmann’s law (for γ = 1), and Stokes’s type law (for γ > 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' In [25], the author studied a similar problem using the method of periodic unfolding in perforated domains by [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Further, the type of behavior as seen in [9] was already observed in [12] by the authors while studying the homogeneous Fourier boundary conditions for the two-dimensional Stokes equa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Likewise, in [1, 2], the author examined the Stokes equation in a perforated domain with holes of size much smaller than the small positive parameter ε, wherein they considered the boundary conditions on the holes to be of the Dirichlet type in [1] and the slip type in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' The domain geometry, more specifically, the size of the holes, determines the kind of limit law in these works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Also, the author in [6] employed the Γ− convergence techniques to get comparable results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' A few works concern the homogenization of the OCPs governed by the elliptic systems over the periodically perforated domains with different kinds of boundary conditions on the boundary of holes (of the size of the same order as that of the period).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' In this regard, with the use of different techiniques, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=', H0− convergence in [18], two-sclae convergence in [23], and unfolding methods in [7,21], the homogenized OCPs were thus obtained over the non- perforated domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Further, in context to the Stokes system, the authors in [22] studied the homogenization of the OCPs subject to the Stokes equations with Dirichlet boundary conditions on the boundary of holes, where the size of the holes is of the same order as that of the period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Here, the authors could obtain the homogenized system, pertaining only to the case when the set of admissible controls was unconstrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' For more literature concerning the homogenization of optimal control problems in perforated domains, the reader is reffered to [13–15,19,24] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' The present article introduces an interior OCP subject to the generalized stationary Stokes equations in a periodically perforated domain O∗ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' On the boundary of holes that do not intersect the outer boundary, the homogeneous Neumann boundary condition is prescribed, while on the rest part of the boundary, the homogeneous Dirichlet boundary condition is prescribed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' The underlying objective of this article is to study the homogeniza- tion of this OCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' More specifically, we consider the minimization of the L2−cost functional (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1), which is subject to the constrained generalized stationary Stokes equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' The Stokes equations are generalized in the sense that we consider a second-order elliptic linear differential operator in divergence form with oscillating coefficients, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=', − div (Aε∇), first studied for the fixed domain in [4, Chapter 1], instead of the classical Laplacian operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' 2 Here, the action of the scalar operator − div (Aε∇) is defined in a ”diagonal” manner on any vector u = (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' , un), with components u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' , un in the H1 Sobolev space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' That is, for 1 ≤ i ≤ n, we have (− div (Aε∇u))i = − div (Aε∇ui).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' The main difficulty observed during the homogenization was identifying the limit pressure terms appearing in the state and the adjoint systems, which we overcame by introducing suitable corrector functions that solved some cell problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' We thus obtained the limit OCP associated with the stationary Stokes equation in a non-perforated domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' The layout of this article is as follows: In the next section, we introduce the periodically perforated domain O∗ ε along with the notations that will be useful in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Section 3 is devoted to a detailed description of the considered OCP and the derivation of the optimality condition, followed by the characterization of the optimal control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' In Section 4, we derive a priori estimates of the solutions to the considered OCP and its corresponding adjoint problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' In Section 5, we recall the definition of the method of periodic unfolding in perforated domains (see, [8,11]) and a few of its properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Section 6, refers to the limit (homogenized) OCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Finally, we derive the main convergence results in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' 2 Domain description and Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1 Domain description Let {b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=', bn} be a basis of Rn (n ≥ 2), and W be the associated reference cell defined as W = � w ∈ Rn | w = n � i=1 wibi, (w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' , wn) ∈ (0, 1)n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Let us denote O, W, and W ∗ = W\\Y by an open bounded subset of Rn, a compact subset of W, and the perforated reference cell, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' It is assumed that the boundary of Y is Lipschitz continuous and has a finite number of connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Also, let ε > 0 be a sequence that converges to zero and set T = � ζ ∈ Rn | ζ = n � i=1 zibi, (z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' , zn) ∈ Zn � , Zε = {ζ ∈ T | ε(ζ + W) ⊂ O} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' We take into account the perforated domain O∗ ε (see Figure 1) given by O∗ ε = O\\Yε, where Yε = ∪ζ∈T ε(ζ + Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Now, let us denote � Oε as the interior of the largest union of ε(ζ + W) cells such that ε(ζ + W) ⊂ O, while Λε ⊂ O as containing the parts from ε(ζ + W) cells intersecting the boundary ∂O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' More precisely, we write Λε = O\\ � Oε, where � Oε = interior � ∪ζ∈Zε ε(ζ + W) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' The associated perforated domains are defined as � O∗ ε = ˆOε\\Yε, ˆΛ∗ ε = O∗ ε\\ � O∗ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' 3 Figure 1: The Perforated domain O∗ ε and the reference cell W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Also, we denote the boundary of the perforated domain O∗ ε as ∂O∗ ε = Γε 1 ∪ Γε 0, where Γε 1 = ∂ � Oε ∩ ∂Yε and Γε 0 = ∂O∗ ε\\Γε 1, which means that Γε 1 denotes the boundary of set of holes contained in � Oε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' In Figure 1, � O∗ ε and ˆΛ∗ ε respectively represent the dark perforated part and the remaining part of the perforated domain O∗ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' While, Γε 1 and Γε 0 respectively represent the boundary of holes contained in � O∗ ε and the boundary of holes contained in ˆΛ∗ ε along with the outer boundary ∂O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' In the following, we introduce a few notations that we shall use throughout this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='2 Notation Aε(x) = A( x ε) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' in O, for all ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' vε = (vε1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' , vεn), for any bold symbol vector function vε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' v = (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' , vn), for any bold symbol vector function v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' ηε denotes the outward normal unit vector to Γε 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' η denotes the outward normal unit vector to ∂O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' M t denotes the transpose of any matrix M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' �ψ is the zero extension of any function ψ outside O∗ ε to the whole of O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' �ψ = (� ψ1, · · · , � ψn), for any vector function ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' |F| is the Lebesgue measure of the measurable set F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' 4 M• Θ = |W ∗| |W| , the proportion of the perforated reference cell W ∗ in the reference cell W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' MW ∗(φ) is the mean value of φ on the perforated reference cell W ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' MW ∗(φ) = (MW ∗(φ1), · · · , MW ∗(φn)), for vector function φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' {D → R}, the set of all real valued functions defined on domain D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' D(Ω), is the space of infinitely many times differentiable functions with compact sup- port in Ω, for any open set Ω ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' 3 Problem description and Optimality condition Let us consider the following OCP associated with Stokes system: inf θε∈(L2(O∗ε))n � Jε(θε) = 1 2 � O∗ε |uε(θε) − ud|2 + τ 2 � O∗ε |θε|2 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1) subject to � � � � � � � � � − div (Aε∇uε) + ∇pε = θε in O∗ ε, div(uε) = 0 in O∗ ε, ηε · Aε∇uε − pεηε = 0 on Γε 1, uε = 0 on Γε 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='2) where the desired state ud = (ud1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' , udn) is defined on the space (L2(O))n, θε is a control function defined on the space (L2(O∗ ε))n and τ > 0 is a given regularization parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Here, the matrix Aε(x) = A( x ε), where A(x) = (aij(x))1≤i,j≤n defined on the space (L∞(O))n×n is assumed to obey the uniform ellipticity condition: there exist real constants m1, m2 > 0 such that m1||λ||2 ≤ �n i,j=1 aij(x)λiλj ≤ m2||λ||2 for all λ ∈ Rn, which is endowed with an Eucledian norm denoted by || · ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Also, we understand the action of scalar boundary operator ηε · Aε∇ on the vector uε|Γε 1 in a ”diagonal” manner: (ηε · Aε∇uε)i = ηε · Aε∇uεi, for 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' We introduce the function space (H1 Γε 0(O∗ ε))n := {φ ∈ (H1(O∗ ε))n | φ|Γε 0 = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' This is a Banach space endowed with the norm ||φ||(H1 Γε 0(O∗ε))n := ||∇φ||(L2(O∗ε))n×n, ∀φ ∈ (H1 Γε 0(O∗ ε))n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' We say a pair (uε, pε) ∈ (H1 Γε 0(O∗ ε))n × L2(O∗ ε) is a weak solution to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='2) if, for all φ ∈ (H1 Γε 0(O∗ ε))n, � O∗ε Aε∇uε : ∇φ dx − � O∗ε pε div(φ) dx = � O∗ε θε · φ dx, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='3) 5 and, for all w ∈ L2(O∗ ε), � O∗ε div(uε) w dx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='4) Here, (: ) and (·) represent the summation of the component-wise multiplication of the matrix entries and the usual scalar product of vectors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' The existence of a unique weak solution (uε(θε), pε) ∈ (H1 Γε 0(O∗ ε))n × L2(O∗ ε) of the system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='2) follows anal- ogous to [5, Theorem IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Also, for each ε > 0, there exists a unique solution to the problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1) that can be proved along the same lines as in [20, Chapter 2, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' We call the optimal solution to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1) by the triplet (uε, pε, θε), with uε, pε, and θε as optimal state, pressure, and control, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Optimality Condition: The optimality condition is given by J′ ε(θ) · (θ − θε) ≥ 0, for all θ ∈ (L2(O∗ ε))n (see, [20, Chapter 2, Page 48]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' One can obtain the further simplification of this condition as � O∗ε(vε + τ θε) · (θ − θε) ≥ 0, for all θ ∈ (L2(O∗ ε))n (see, [20, Chapter 2]), where the pair (vε, qε) is the solution to the following adjoint problem: � � � � � � � � � − div � At ε∇vε � + ∇qε = uε − ud in O∗ ε, div(vε) = 0 in O∗ ε, ηε · At ε∇vε − qεηε = 0 on Γε 1, vε = 0 on Γε 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='5) We call vε and qε, the adjoint state and pressure, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' The existence of unique weak solution (vε, qε) to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='5) can now be proved in a way similar to that of system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' The following theorem characterizes the optimal control, the proof of which follows analogous to standard procedure laid in [20, Chapter 2, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Let � uε, pε, θε � be the optimal solution of the problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1) and (vε, qε) solves (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='5), then the optimal control is characterized by θε = −1 τ vε a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' in O∗ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='6) Conversely,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' suppose that a triplet (ˇuε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' ˇpε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' ˇθε) ∈ � H1 Γε 0(O∗ ε) �n × L2(O∗ ε) × (L2(O∗ ε))n and a pair (ˇvε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' ˇqε) ∈ � H1 Γε 0(O∗ ε) �n × L2(O∗ ε) solves the following system: � � � � � � � � � � � � � � � � � � � � � � � − div (Aε∇ˇuε) + ∇ˇpε = −1 τ ˇvε in O∗ ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' − div � At ε∇ˇvε � + ∇ˇqε = ˇuε − ud in O∗ ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' div(ˇuε) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' div(ˇvε) = 0 in O∗ ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' ηε · Aε∇ˇuε − ˇpεηε = 0 on Γε 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' ηε · At ε∇ˇvε − ˇqεηε = 0 on Γε 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' ˇvε = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' ˇuε = 0 on Γε 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' 6 Then the triplet (ˇuε, ˇpε, − 1 τ ˇvε) is the optimal solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' 4 A priori estimates This section concerns the derivation of estimates for the optimal solution to the problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1) and the associated solution to the adjoint problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' These estimates are uniform and independent of the parameter ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Towards attaining this aim, we first evoke the following two lemmas: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1 (Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='4, [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' There exists a constant C ∈ R+, independent of ε, such that ||v||L2(O∗ε)n ≤ C||∇v||(L2(O∗ε))n×n, ∀ v ∈ (H1 Γε 0(O∗ ε))n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='2 (Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1, [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' For each ε > 0 and qε ∈ L2(O∗ ε), there exists gε ∈ (H1 Γε 0(O∗ ε))n and a constant C ∈ R+, independent of ε, such that div(gε) = qε and ||∇gε||(L2(O∗ε))n×n ≤ C(O) ||qε||L2(O∗ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1) Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' For each ε > 0, let � uε, pε, θε � be the optimal solution of the problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1) and (vε, qε) solves the corresponding adjoint problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Then, one has θε ∈ (H1 Γε 0(O∗ ε))n and there exists a constant C ∈ R+, independent of ε such that ��¯θε �� (L2(O∗ε))n ≤ C, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='2) ∥¯uε∥(H1 Γε 0(O∗ε))n ≤ C, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='3) ∥¯vε∥(H1 Γε 0(O∗ε))n ≤ C, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='4) ∥¯pε∥L2(O∗ε) ≤ C, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='5) ∥¯qε∥L2(O∗ε) ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='6) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Let uε(0) denotes the solution to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='2) corresponding to θε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' In view of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1, one can show that ∥uε(0)∥(L2(O∗ε))n ≤ 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=', uε(0) = 0 in (L2(O∗ ε))n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Using this and the optimality of solution (uε, pε, θε) to problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1), we have ∥uε(θ) − ud∥2 (L2(O∗ε))n + τ∥θε∥2 (L2(O∗ε))n ≤ ∥uε(0) − ud∥2 (L2(O∗ε))n ≤ C, which gives estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Now, let us take uε as a test function in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Considering (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='2) and the uniform ellipticity condition of matrix Aε, one obtains upon applying the Cauchy-Schwarz inequality along with the Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1, the following: m1∥∇uε∥2 (L2(O∗ε))n×n ≤ � O∗ε Aε∇uε : ∇uε dx ≤ C ∥θε∥(L2(O∗ε))n∥∇uε∥(L2(O∗ε))n×n, from which estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='3) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' 7 Owing to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='2, for given pε ∈ L2(O∗ ε), there exists gε ∈ (H1 Γε 0(O∗ ε))n satisying div(gε) = pε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Corresponding to θε, taking v = gε in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='3), we get ∥pε∥2 L2(O∗ε) = � O∗ε Aε∇uε : ∇gε dx − � O∗ε θε · gε dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='7) In view of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='2) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='3), and the uniform ellipticity condition of the matrix Aε, one obtains from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='7) upon employing the Cauchy-Schwarz inequality and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1, the following: ∥pε∥2 L2(O∗ε) ≤ � m2∥∇uε∥(L2(O∗ε))n×n + C∥θε∥(L2(O∗ε))n � ∥∇gε∥(L2(O∗ε))n×n, which gives the estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Likewise, one can easily obtain the estimates (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='4) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='6) following the above discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Finally, from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='6), we obtain that θε ∈ (H1 Γε 0(O∗ ε))n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' 5 The method of periodic unfolding for perforated do- mains We evokes the definition of the periodic unfolding operator and few of its properties as stated in [8,11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Given x ∈ Rn, we denote the greatest integer and the fractional parts of x respectively by [x]W and {x}W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' That is, [x]W = �n j=1 kjbj be the unique integer combination of periods and {x}W = x − [x]W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' In particular, we have for ε > 0, x = ε ��x ε � W + �x ε � W � , ∀ x ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' The unfolding operator T ∗ ε : {O∗ ε → R} → {O × W ∗ → R} is defined as T ∗ ε (u) (x, y) = � u � ε � x1 ε � W + εy � a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' for (x, y) ∈ � Oε × W ∗, 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' for (x, y) ∈ Λε × W ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Also, for any domain D ⊇ O∗ ε and vector u = (u1, · · · , un) ∈ ({D → R})n, we define its unfolding by T ∗ ε (u) := (T ∗ ε (u1), · · · , T ∗ ε (un)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' In the following there are the properties of the unfolding operator: (i) T ∗ ε is linear and continuous from L2(O∗ ε) to L2(O × W ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' (ii) Let u, v ∈ L2(O∗ ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Then T ∗ ε (uv) = T ∗ ε (u) T ∗ ε (v) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' (iii) Let u ∈ L2 (O) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Then T ∗ ε (u) → u strongly in L2 (O × W ∗) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' (iv) Let u ∈ L1 (O∗ ε) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Then � � O∗ε u(x) dx = � O∗ε u(x) dx − � ˆΛ∗ε u(x) dx = 1 |W ∗| � O×W ∗ T ∗ ε (u)(x, y) dxdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' 8 (v) For each ε > 0, let {uε} ∈ L2 (O) and uε → u strongly in L2 (O) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Then T ∗ ε (uε) → u strongly in L2 (O × W ∗) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' (vi) Let v ∈ L2 (W ∗) be a W-periodic function and vε(x) = v � x ε � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Then, T ∗ ε (vε) (x, y) = � v(y) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' for (x, y) ∈ � Oε × W ∗, 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' for (x, y) ∈ Λε × W ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' (vii) Let fε ∈ L2 (O∗ ε) be uniformly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Then, there exists f ∈ L2(O × W ∗) such that T ∗ ε (fε) ⇀ f weakly in L2(O × W ∗), and �fε ⇀ 1 |W| � W ∗ f(·, y) dy weakly in L2(O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Let O ⊂ Rn be bounded with Lipschitz boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Let fε ∈ H1(O∗ ε) be such that fε = 0 on ∂O ∩ ∂O∗ ε and satisfy, ∥∇fε∥(L2(O∗ε))n ≤ C§.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Then, there exists f ∈ H1 0(O) and �f ∈ L2 � O;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' H1 per (W ∗) � with MW ∗( �f) = 0, such that up to a subsequence, � T ∗ ε (∇fε) ⇀ ∇f + ∇y �f weakly in (L2 (O × W ∗))n , T ∗ ε (fε) → f strongly in L2 (O;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' H1 (W ∗)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' 6 Limit optimal control problem This section presents the limit (homogenized) system corresponding to the problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1), which we considered in the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Let us consider the function space � H1 0(O) �n := � ϕ ∈ (H1(O))n | ϕ|∂O = 0 � , which is a Hilbert space for the norm ∥ϕ∥(H1 0(O))n := ∥∇ϕ∥(L2(O))n×n ∀ ϕ ∈ (H1 0(O))n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' We now consider the limit OCP associated with the Stokes system inf θ∈(L2(O))n � J(θ) = Θ 2 � O |u − ud|2 dx + τΘ 2 � O |θ|2 dx � , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1) §The symbol C represents a generic constant that is positive and independent of ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' 9 subject to � � � � � � � � � − n � j,α,β=1 ∂ ∂xα � bαβ ij ∂uj ∂xβ � + ∇p = θ in O, div (u) = 0 in O, u = 0 on ∂O, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='2) where the tensor B = (bαβ ij ) = (bαβ ij )1≤i,j,α,β≤n is constant, elliptic, and for 1 ≤ i, j, α, β ≤ n, is given by bαβ ij = aαβ ij − 1 |W ∗| � W ∗ A(y)∇y � P β j − χβ j � : ∇yχα i dy, with aαβ ij = 1 |W ∗| � W ∗ A(y)∇y � P β j − χβ j � : ∇yP α i dy as the entries of the constant tensor A0, P β j = P β j (y) = (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' , yj, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' , 0) with yj at the β-th position, and for 1 ≤ j, β ≤ n, the correctors (χβ j , Πβ j ) ∈ (H1(W ∗))n × L2(W ∗) solves the cell problem � � � � � � � � � � � � � � � � � � � − divy � A(y)∇y(P β j − χβ j ) � + ∇yΠβ j = 0 in W ∗, η · A(y)∇y(P β j − χβ j ) − Πβ j η = 0 on ∂W ∗\\∂W, divy(P β j − χβ j ) = 0 in W ∗, (χβ j , Πβ j ) W ∗- periodic, MW ∗(χβ j ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='3) The existence of this unique pair (u, p) ∈ (H1 0(O))n × L2(O) can be found in [4, Chapter 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Further, the problem (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1) is a standard one and there exists a unique weak solution to it, one can follow the arguments introduced in [20, Chapter 2, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' We call the triplet (u, p, θ) ∈ (H1 0(O))n × L2(O) × (L2(O))n, the optimal solution to (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1), with u, p, and θ as the optimal state, pressure, and control, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Now, we introduce the limit adjoint system associated with (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='2): Find a pair (v, q) ∈ (H1 0(O))n × L2(O) which solves the system � � � � � − n � i,α,β=1 ∂ ∂xβ � bβα ji ∂vi ∂xα � + ∇q = u − ud in O, div (v) = 0 in O, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='4) where the tensor Bt = (bβα ji ) = (bβα ji )1≤i,j,α,β≤n is constant, elliptic, and for 1 ≤ i, j, α, β ≤ n, is given by bβα ji = aβα ji − 1 |W ∗| � W ∗ At(y)∇y � P β j − Hβ j � : ∇yHα i dy, with aβα ji = 1 |W ∗| � W ∗ At(y)∇y � P β j − Hβ j � : ∇yP α i dy as the entries of the constant tensor At 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Also, for 1 ≤ j, β ≤ n, the correctors (Hβ j , Zβ j ) ∈ (H1(W ∗))n × L2(W ∗) solves the cell 10 problem � � � � � � � � � � � � � � � � � � � − divy � At(y)∇y(P β j − Hβ j ) � + ∇yZβ j = 0 in W ∗, η · At(y)∇y(P β j − Hβ j ) − Zβ j η = 0 on ∂W ∗\\∂W, divy(P β j − Hβ j ) = 0 in W ∗, (Hβ j , Zβ j ) W ∗- periodic, MW ∗(Hβ j ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='5) In the following, we state a result similar to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1 that characterizes the optimal control θ in terms of the adjoint state v and the proof of which follows analogous to the standard procedure laid in [20, Chapter 2, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Let � u, p, θ � be the optimal solution to (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1) and (v, q) be the corresponding adjoint solution to (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='4), then the optimal control is characterized by θ = −1 τ v a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' in O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='6) Conversely, suppose that a triplet (ˇu, ˇp, ˇθ) ∈ (H1 0(O))n × L2(O) × (L2(O))n and a pair (ˇv, ˇq) ∈ (H1 0(O))n × L2(O), respectively, satisfy the following systems: � � � � � − n � j,α,β=1 ∂ ∂xα � bαβ ij ∂ˇuj ∂xβ � + ∇ˇp = − 1 τ ˇv in O, div (ˇu) = 0 in O, and � � � � � − n � i,α,β=1 ∂ ∂xβ � bβα ji ∂ˇvi ∂xα � + ∇ˇq = ˇu − ud in O, div (ˇv) = 0 in O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Then, the triplet �ˇu, ˇp, − 1 τ ˇv � is the optimal solution to (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' 7 Convergence results We present here the key findings on the convergence analysis of the optimal solutions to the problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1) and its corresponding adjoint system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='5) by using the method of periodic unfolding for perforated domains described in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' For given ε > 0, let the triplets (uε, pε, θε) and (u, p, θ), respectively, be the 11 optimal solutions of the problems (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Then T ∗ ε (Aε) → A strongly in (L2(O × W ∗))n×n, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1a) � θε ⇀ Θ θ weakly in � L2 (O) �n , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1b) � uε ⇀ Θ u weakly in (H1 0(O))n, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1c) � vε ⇀ Θ v weakly in (H1 0(O))n, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1d) �pε ⇀ Θ n A0∇u: I + Θ p weakly in L2(O), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1e) �qε ⇀ Θ n At 0∇v: I + Θ q weakly in L2(O), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1f) where A0 is a tensor as defined in Section 6, I is the n × n identity matrix, θ is characterized through (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='6) and the pairs (vε, qε) and (v, q) solve respectively the systems (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='5) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Moreover, lim ε→0 Jε(θε) = J(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='2) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' First, upon using Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='2 (vi) on the entries of the matrix Aε, we obtain (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1a) under the passage of limit ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Similarly, one can prove the convergence for the matrix At ε under unfolding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Next, in view of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='3 and the fact that the triplet (uε, pε, θε) is an optimal solution to problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1), one gets uniform estimates for the sequences {θε}, {uε}, {pε}, {vε}, and {qε} in the spaces (L2 (O∗ ε))n, (H1 Γε 0(O∗ ε))n, L2 (O∗ ε), (H1 Γε 0(O∗ ε))n, and L2 (O∗ ε), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Using the uniform estimate of the sequence {θε} in the space (L2 (O∗ ε))n and Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='2 (i), we have the sequence {T ∗ ε (θε)} to be uniformly bounded in the space (L2 (O × W ∗))n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Thus, by weak compactness, there exists a subsequence not relabelled and a function ˆθ in (L2 (O × W ∗))n, such that T ∗ ε (θε) ⇀ ˆθ weakly in � L2 (O × W ∗) �n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='3) Now, using Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='2 (vii) in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='3) gives � θε ⇀ 1 |W| � W ∗ ˆθ(x, y) dy = Θ θ0 weakly in � L2 (O) �n , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='4) where, θ0 = MW ∗(ˆθ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Employing Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='2 (i), we have the uniform boundedness of the sequences {T ε(uε)}, {T ε(∇uε)}, and {T ε(pε)} in the respective spaces (L2(O;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' H1 (W ∗)))n, (L2(O × W ∗))n×n, and L2(O × W ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Further, upon employing Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='3 and Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='2 (vii), there exist subsequences not relabelled and functions ˆu with MW ∗(ˆu) = 0, u0, and ˆp in spaces 12 (L2(O;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' H1 per (W ∗)))n, (H1 0(O))n, and L2(O × W ∗), respectively, such that T ∗ ε (uε) → u0 strongly in (L2(O;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' H1 (W ∗)))n, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='5a) T ∗ ε (∇uε) ⇀ ∇u0 + ∇y ˆu weakly in (L2(O × W ∗))n×n, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='5b) � uε ⇀ Θ u0 weakly in (H1 0(O))n, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='5c) T ∗ ε (pε) ⇀ ˆp weakly in L2(O × W ∗), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='5d) �pε ⇀ Θ MW ∗(ˆp) weakly in L2(O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='5e) Likewise, for the associated adjoint counterparts, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=', vε, and qε , one obtains that there exist subsequences not relabelled and functions ˆv with MW ∗(ˆv) = 0, v0, and ˆq in spaces (L2(O;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' H1 per (W ∗)))n, (H1 0(O))n, and L2(O × W ∗), respectively, such that T ∗ ε (vε) → v0 strongly in (L2(O;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' H1 (W ∗)))n, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='6a) T ∗ ε (∇vε) ⇀ ∇v0 + ∇yˆv weakly in (L2(O × W ∗))n×n, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='6b) � vε ⇀ Θ v0 weakly in (H1 0(O))n, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='6c) T ∗ ε (qε) ⇀ ˆq weakly in L2(O × W ∗), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='6d) �qε ⇀ MW ∗(ˆq) weakly in L2(O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='6e) The identification of the limit functions ˆu, ˆv, ˆp, ˆq, MW ∗(ˆp) and MW ∗(ˆq) is carried out in subsequent steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Step 1: (Claim): For all ϕ ∈ (H1 0(O))n, ψ ∈ � L2 � O;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' H1 per (W ∗) ��n , and w ∈ L2(O), we claim that the ordered quadruplet (u0, ˆu, ˆp, θ0) ∈ (H1 0(O))n ×(L2(O;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' H1 per (W ∗)))n ×L2(O× W ∗) × (L2(O))n is a unique solution to the following limit system: � � � � � � � � � � � � � � � � � � � 1 |W| � O×W ∗ A(y) (∇u0 + ∇y�u(x, y)) : (∇ϕ + ∇yψ) dx dy − 1 |W| � O×W ∗ ˆp(x, y) (div(ϕ) + divy(ψ)) dx dy = Θ � O θ0 · ϕ dx, and, � O div(u0) w dx = 0, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='7) and the ordered triplet (v0, ˆv, ˆq) ∈ (H1 0(O))n×(L2(O;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' H1 per (W ∗)))n×L2(O×W ∗) is a unique solution to the following limit adjoint system: � � � � � � � � � � � � � � � � � � � 1 |W| � O×W ∗ At(y) (∇v0 + ∇y�v(x, y)) : (∇ϕ + ∇yψ) dx dy − 1 |W| � O×W ∗ ˆq(x, y) (div(ϕ) + divy(ψ)) dx dy = Θ � O (u0 − ud) · ϕ dx, and, � O div(v0) w dx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='8) 13 Proof of the Claim: Towards the proof of (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='7), let us consider a test function ϕ ∈ (D(O))n in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='3) and use properties (i), (ii), and (iv) of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='2 to get 1 |W| � O×W ∗ T ∗ ε (Aε) T ∗ ε (∇uε): T ∗ ε (∇ϕ) dx dy + � ˆΛ∗ε Aε∇uε : ∇ϕ dx − � ˆΛ∗ε pε div(ϕ) dx − 1 |W| � O×W ∗ T ∗ ε (pε) T ∗ ε (div(ϕ)) dx dy = 1 |W| � O×W ∗ T ∗ ε (θε) · T ∗ ε (φε) dx dy + � ˆΛ∗ε θε · ϕ dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='9) Using Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='2 (iii), the fact that limε→0 |ˆΛ∗ ε| = 0, and convergences (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='3), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1a), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='5b), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='5d), we have under the passage of limit ε → 0 in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='9) 1 |W| � O×W ∗ A(y) (∇u0 + ∇y�u(x, y)) : ∇ϕ dx dy − 1 |W| � O×W ∗ ˆp(x, y) div(ϕ) dx dy = Θ � O θ0 · ϕ dx, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='10) which remains valid for every ϕ ∈ (H1 0(O))n, by density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Now, consider the function φε(x) = εφ(x)ξ( x ε), where φ ∈ D(O) and ξ ∈ (H1 per(W ∗))n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Employing properties (ii), (iii), and (vi) of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='2, one can easily obtain T ∗ ε (φε) (x, y) → 0 strongly in (L2(O × W ∗))n, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='11a) T ∗ ε (∇φε) (x, y) → φ(x)∇yξ(y) strongly in (L2(O × W ∗))n×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='11b) Let us use the test function φε in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='3) and employ properties (i), (ii), and (iv) of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='2 to get 1 |W| � O×W ∗ T ∗ ε (Aε) T ∗ ε (∇uε): T ∗ ε (∇φε) dx dy + � ˆΛ∗ε Aε∇uε : ∇φε dx − � ˆΛ∗ε pε div(φε) dx − 1 |W| � O×W ∗ T ∗ ε (pε) T ∗ ε (div(φε)) dx dy = 1 |W| � O×W ∗ T ∗ ε (θε) · T ∗ ε (φε) dx dy + � ˆΛ∗ε θε · φε dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='12) In (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='12), the absolute value of each integral over ˆΛ∗ ε is bounded above with a bound of order ε|ˆΛ∗ ε| or |ˆΛ∗ ε|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' This with the fact that limε→0 |ˆΛ∗ ε| = 0, and convergences (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='3), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1a), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='5b), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='5d), and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='11), gives under the passage of limit ε → 0 1 |W| � O×W ∗ A(y) (∇u0 + ∇y�u(x, y)) : ∇yψ dx dy − 1 |W| � O×W ∗ ˆp(x, y) divy(ψ) dx dy = 0, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='13) which remains valid for every φ ξ = ψ ∈ (L2(O;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' H1 per(W ∗)))n, by density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Further, for all w ∈ L2(O), we have � O∗ε div(uε)w dx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='14) 14 Now, upon applying unfolding on (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='14) and using properties (i), (ii), and (iii) of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='2 along with convergence (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='5b), we get under the passage of limit ε → 0 1 |W| � O×W ∗ (div(u0) + divy(ˆu)) w dx dy = 0, which eventually gives upon using the fact that ˆu is W ∗− periodic, for all w ∈ L2(O): � O div(u0)w dx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='15) Finally, upon adding (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='10) with (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='13) and considering (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='15), we establish (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Likewise, one can easily establish (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' This settles the proof of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Step 2: First, we are going to identify the limit functions ˆu, ˆv, ˆp, and ˆq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Next, using these identifications, we will identify MW ∗(ˆp) and MW ∗(ˆq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Identification of ˆu, ˆv, ˆp, ˆq: Taking sucessively ϕ ≡ 0 and ψ ≡ 0 in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='7), yields � � � � � � � � � � � � � � � − divy(A(y)∇y�u(x, y)) + ∇y�p(x, y) = divy(A(y))∇u0(x) in O × W ∗, − divx �� W ∗ A(y)(∇u0(x) + ∇y�u(x, y))dy � + ∇�p(x, y) = |W ∗| θ0 in O, div(u0) = 0 in O, �u(x, ·) is W ∗ − periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='16) In the first line of (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='16), we have the y-independence of ∇u0(x) and the linearity of opera- tors, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=', divergence and gradient, which suggests �u(x, y) and �p(x, y) to be of the following form (see, for e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=', [17, Page 15]): � � � � � � � � � � � �u(x, y) = − n � j,β=1 χβ j (y)∂u0j ∂xβ + u1(x), �p(x, y) = n � j,β=1 Πβ j (y)∂u0j ∂xβ + p0(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='17) where the ordered pair (u1, p0) ∈ (H1(O))n × L2(O), and for 1 ≤ j, β ≤ n, the pair (χβ j , Πβ j ) satisfy the cell problem (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Likewise we obtain for the corresponding adjoint weak formu- lation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='8): � � � � � � � � � � � � � � � − divy(A(y)∇y�v(x, y)) + ∇y�q(x, y) = divy(A(y))∇v0(x) in O × W ∗, − divx �� W ∗ A(y)(∇v0(x) + ∇y�v(x, y))dy � + ∇�q(x, y) = |W ∗| (u0 − ud) in O, div(v0) = 0 in O, �v(x, ·) is W ∗ − periodic, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='18) 15 and, � � � � � � � � � � � �v(x, y) = − n � j,β=1 Hβ j (y)∂v0j ∂xβ + v1(x), �q(x, y) = n � j,β=1 Zβ j (y)∂v0j ∂xβ + q0(x), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='19) where the ordered pair (v1, q0) ∈ (H1(O))n ×L2(O), and for 1 ≤ j, β ≤ n, the pair (Hβ j , Zβ j ) satisfy the cell problem (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Identification of MW ∗(ˆp) and MW ∗(ˆq): Choosing the test function y = (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' , yn) in the weak formulation of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='3), we get n � i,l,k,α=1 � W ∗ alk ∂ ∂yk � P β j − χβ j � ∂P α i ∂yl ∂yi ∂yα dy = n � W ∗ Πβ j dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='20) In view of (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='5e), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='17), and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='20), we observe that MW ∗(ˆp) = 1 |W ∗| n � i,j,l,k,α,β=1 � W ∗ alk ∂ ∂yk � P β j − χβ j � ∂P α i ∂yl ∂yi ∂yα ∂u0j ∂xβ dy + p0, which upon using the definition of aαβ ij , gives MW ∗(ˆp) = n � i,j,α,β=1 aαβ ij ∂u0j ∂xβ ∂yi ∂yα + p0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='21) Also, we re-write the equation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='21) to get the identification of MW ∗(ˆp) as MW ∗(ˆp) = A0∇u0 : I + p0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='22) Likewise, one can obtain the identification of MW ∗(ˆq) as MW ∗(ˆq) = At 0∇v0 : I + q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='23) Thus, from (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='5e) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='22);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='6e) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='23), we have the following weak convergences: �pε ⇀ Θ n A0∇u0 : I + Θ p0 weakly in L2(O), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='24a) �qε ⇀ Θ n At 0∇v0 : I + Θ q0 weakly in L2(O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='24b) Step 3: (Claim): The pairs (u0, p0) and (v0, q0) solve the systems (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='2) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='4), respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Proof of the Claim: We now prove that the pair (u0, p0) solves the system (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' The proof 16 that the pair (v0, q0) solves the system (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='4) follows analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Substituting the values of �u(x, y) and �p(x, y) from expression (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='17) into equation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='10), we get 1 |W| n � l,k=1 � O×W ∗ alk � ∂u0 ∂xk − n � j,β=1 ∂χβ j ∂yk ∂u0j ∂xβ � ∂ϕ ∂xl dx dy − 1 |W| n � j,β=1 � O×W ∗ Πβ j ∂u0j ∂xβ div(ϕ) dx dy −Θ � O p0 div(ϕ) dx = Θ � O θ0 · ϕ dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='25) With P β j = (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' , yj, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' , 0), we can express the terms ∂u0 ∂xk , ∂ϕ ∂xl, and div(ϕ) as ∂u0 ∂xk = n � j,β=1 ∂P β j ∂yk ∂u0j ∂xβ , ∂ϕ ∂xl = n � i,α=1 ∂P α i ∂yl ∂ϕi ∂xα , div(ϕ) = n � i,α=1 divy(P α i ) ∂ϕi ∂xα .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Substituting these expressions in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='25), we obtain n � i,j,α,β=1 � O � 1 |W ∗| n � l,k=1 � W ∗ alk ∂ ∂yk � P β j − χβ j � ∂P α i ∂yl dy � ∂u0j ∂xβ ∂ϕi ∂xα dx − n � i,j,α,β=1 � O � 1 |W ∗| � W ∗ Πβ j divy(P α i ) dy � ∂u0j ∂xβ ∂ϕi ∂xα dx − � O p0 div(ϕ) dx = � O θ0 · ϕ dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='26) Now, choosing the test function χα i in the weak formulation of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='3), we get upon using the fact that divy(χα i ) = divy(P α i ) = δiα, where δ denotes the Kronecker delta function, the following: � W ∗ A(y)∇y � P β j − χβ j � : ∇yχα i dy = � W ∗ Πβ j δiα dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='27) Further, substituting (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='27) in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='26), we obtain n � i,j,α,β=1 � O � 1 |W ∗| n � l,k=1 � W ∗ alk ∂ ∂yk � P β j − χβ j � ∂ ∂yl (P α i − χα i ) dy � ∂u0j ∂xβ ∂ϕi ∂xα dx − � O p0 div(ϕ) dx = � O θ0 · ϕ dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='28) Also, we can write equation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='28) as n � i,j,α,β=1 � O bαβ ij ∂u0j ∂xβ ∂ϕi ∂xα dx − � O p0 div(ϕ) dx = � O θ0 · ϕ dx, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='29) 17 which holds true for all ϕ ∈ (H1 0(O))n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Also, from equation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='15), we have � O div(u)w dx = 0, for every w ∈ L2(O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' This together with equation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='29) implies that, for θ = θ0, the pair (u0, p0) ∈ (H1 0(O))n × L2(O) satisfies the variational formulation of the system (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Therefore, we obtain the optimality system for the minimization problem (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Also, in view of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1, we conclude that the triplet (u0, p0, θ0) is indeed an optimal solution to the problem (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Finally, upon considering the optimal solution’s uniqueness, we establish that the subsequent pair of triplets are equal: (u, p, θ) = (u0, p0, θ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='30) Hence, upon comparing (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='5c), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='6c), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='24a), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='24b), and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='4) with (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='30), we obtain convergences (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1c), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1d), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1e), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1f), and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Step 4: Now, we will furnish the proof of the energy convergence for the L2−cost functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Choosing the test function (uε − ud) in the weak formulation of system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='5), we get under unfolding upon passing ε → 0 lim ε→0 � O∗ε |uε − ud|2 dx = 1 |W| lim ε→0 � O×W ∗ T ∗ ε (At ε) T ∗ ε (∇vε): T ∗ ε (∇(uε − ud)) dx dy + 1 |W| lim ε→0 � O×W ∗ T ∗ ε (qε) T ∗ ε (div(ud)) dx dy, which gives in view of (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='30), Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='2 (iii) and convergences (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='6a), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='5b), and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='6d) lim ε→0 � O∗ε |uε − ud|2 dx = 1 |W| � O×W ∗ At(y) (∇v + ∇y�v(x, y)) : ∇y(u − ud) dx dy + 1 |W| � O×W ∗ ˆq(x, y) div(ud) dx dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='31) Also, using (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='19) in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='31) alongwith (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='30), we have upon simplification lim ε→0 � O∗ε |uε − ud|2 dx = Θ � n � i,j,α,β=1 � O bβα ji ∂vi ∂xα ∂(u − ud)j ∂xβ dx − � O q div(u − ud) dx � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='32) Now, using the test function (u − ud) in the weak formulation of system (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='4), we get the following upon comparing with the right hand side of equation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='32) lim ε→0 � O∗ε |uε − ud|2 dx = Θ � O |u − ud|2 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='33) Furthermore, in view of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='6), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='6a), and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='30), we get under unfolding upon the passage 18 of limit ε → 0 lim ε→0 τ 2 � O∗ε |θε|2 dx = lim ε→0 1 2|W| � O×W ∗ |T ∗ ε (θε)|2 dx dy = lim ε→0 1 2τ|W| � O×W ∗ |T ∗ ε (vε)|2 dx dy = 1 2τ|W| � O×W ∗ |v|2 dx dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='34) Also, since v is independent of y and comparing the right hand side of (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='34) with (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='6), we get lim ε→0 τ 2 � O∗ε |θε|2 dx = Θτ 2 � O |θ|2 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='35) Thus, from equations (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='33) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='35), we get (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' This completes the proof of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' 8 Conclusions We have addressed the limiting behavior of an interior OCP corresponding to Stokes equa- tions in an nD (n ≥ 2) periodically perforated domain O∗ ε via the technique of periodic unfolding in perforated domains (see, [8, 11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' We employed the Neumann boundary con- dition on the part of the boundary of the perforated domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Firstly, we characterized the optimal control in terms of the adjoint state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Secondly, we deduced the apriori optimal bounds for control, state, pressure, and their associated adjoint state and pressure functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Thereafter, the limiting analysis for the considered OCP is carried out upon employing the periodic unfolding method in perforated domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' We observed the convergence between the optimal solution to the problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1) posed on the perforated domain O∗ ε and the optimal solution to that of the limit problem (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content='1) governed by stationary Stokes equation posed on a non-perforated domain O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Finally, we established the convergence of energy corresponding to L2−cost functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' 9 Acknowledgments The first author would like to thank the Ministry of Education, Government of India for Prime Minister’s Research Fellowship (PMRF-2900953).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' The second author would like to thank the support from Science & Engineering Research Board (SERB) (SRG/2019/000997), Government of India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' 19 References [1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfMvs5/content/2301.01136v1.pdf'} +page_content=' Allaire, Homog´en´eisation des ´equations de Stokes dans un domaine perfor´e de petits trous r´epartis p´eriodiquement, C.' metadata={'source': 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-0,0 +1,1835 @@ +Model Mixing Using Bayesian Additive +Regression Trees +John C. Yannotty, Thomas J. Santner, Richard J. Furnstahl, and Matthew T. Pratola +The Ohio State University +January 9, 2023 +Abstract +In modern computer experiment applications, one often encounters the situation where vari- +ous models of a physical system are considered, each implemented as a simulator on a computer. +An important question in such a setting is determining the best simulator, or the best combi- +nation of simulators, to use for prediction and inference. Bayesian model averaging (BMA) and +stacking are two statistical approaches used to account for model uncertainty by aggregating +a set of predictions through a simple linear combination or weighted average. Bayesian model +mixing (BMM) extends these ideas to capture the localized behavior of each simulator by defin- +ing input-dependent weights. One possibility is to define the relationship between inputs and +the weight functions using a flexible non-parametric model that learns the local strengths and +weaknesses of each simulator. This paper proposes a BMM model based on Bayesian Additive +Regression Trees (BART). The proposed methodology is applied to combine predictions from +Effective Field Theories (EFTs) associated with a motivating nuclear physics application. +Keywords: Computer Experiments; Effective Field Theories; Model stacking; Uncertainty quantifi- +cation +1 +Introduction +In statistical learning problems, one often considers a set of plausible models, each designed to +explain the system of interest. A common practice is to select a best performing model based on +some pre-specified criteria. The ensuing inference for quantities of interest is then carried out using +the selected model as if it were the true data generating mechanism. The resulting uncertainty +1 +arXiv:2301.02296v1 [stat.ME] 5 Jan 2023 + +quantification ignores any variability due to the underlying model structure (Draper, 1995). The +misrepresentation of uncertainties associated with such quantities can ultimately lead to misguided +interpretation or inappropriate decisions. Another shortcoming of the typical approach to modeling +is that the resulting inference may strongly depend on the selection criteria. In other words, different +sets of criteria could lead to noticeably different final models and inferential results. To account for +such uncertainties, one may elect to combine information across the set of models in some manner. +Any model set can be classified into one of three categories: M-closed, M-complete, and M- +open (Bernardo and Smith, 2009). The M-closed setting assumes the true model, M†, can formally +be defined and is contained within the set of models under consideration. In this setting, model +selection is appropriate because M† can be recovered from the set of models under consideration. +The M-complete setting describes the case where M† can formally be defined, however it is not +contained in the model set. +Similarly, the M-open case assumes the true model exists and is +excluded from the model set. However, in this situation M† is further assumed to be intractable and +thus cannot be formally defined. Model selection is inappropriate in the latter two cases because +one will inevitably select the wrong model to perform inference while simultaneously ignoring +the uncertainty induced by this error (Bernardo and Smith, 2009). This work is motivated by +applications in nuclear physics which tend to fall within the M-open class. +Assume a set of K models are considered when studying a particular system of interest. One +approach to account for model uncertainty is to combine the information across these K mod- +els. This may involve combining the individual point predictions or probability density functions +from each model, usually in some additive manner. Traditional approaches utilize global weight- +ing schemes, where each model is weighted by a value intended to reflect overall (global) model +performance. In the Bayesian paradigm, the classical global weighting scheme is Bayesian model +averaging (BMA) (Raftery et al., 1997), which combines the individual posterior densities from each +model using a convex combination. The BMA weights are given by the individual posterior model +probabilities, each which can be interpreted as the probability the individual model is the true data +generating one. Hence, BMA implicitly assumes the true model is contained within the model set, +which renders this method inappropriate outside of the M-closed setting (Draper, 1995). More +recent Bayesian global weighting schemes adopt a model stacking approach, where model weights +are assigned to minimize a specified posterior expected loss. This decision theory viewpoint of +global weighting can be used for combining point predictions (Le and Clarke, 2017) or probabil- +ity densities (Yao et al., 2018). Regardless of the implementation, Bayesian stacking methods are +2 + +Figure 1: Three different EFT experimental settings. Each panel displays the true physical system +(solid) and the EFTs under consideration (non-solid). +appropriate for both the M-open and M-closed settings because the weights are chosen based on +some pre-specified criteria and do not share the probabilistic interpretation of BMA. +Though global weighting methods are effective, they still might lead to poor approximations of +the true system when the individual model performance is localized. In such a case, one may wish +to select a weighting scheme that reflects the localized characteristics of the models by constructing +input-dependent weights. With input-dependent weights, one would expect an individual model +to receive a higher weight in input regions where it exhibits strong predictive performance, while +receiving a weight close to zero in regions of poor performance. +Localized weighting schemes +are more appropriate for the M-open or M-complete settings where the true model is better +characterized as a localized mixture of the model set under consideration. +This work is motivated by problems in nuclear physics modeled using a technique known as +Effective Field Theory (EFT) (Burgess, 2020; Petrov and Blechman, 2016; Georgi, 1993). EFTs +are designed to perform well in a particular subregion(s) of the input domain, yet diverge in the +rest of the input domain. +Prototypes of such models are the weak and strong coupling finite- +order expansions for the partition function of the zero-dimensional φ4 theory presented by Honda +(2014). Examples of this problem are shown in Figure 1 where the various dashed and dotted lines +represent the mean predictions from a finite-order expansion and the solid line denotes the true +physical system. One can see that these models are highly accurate descriptions of the true system +in some regions of the domain, yet they are unable to provide a globally accurate model. Most +EFT problems are examples of the M-open setting, in that the true underlying description of the +3 + +(a) +(b) +(c) +5 +5 +5 - +- +/ +4- +4- +/ +4 - +1 +3- +3- +/ +3- +M +F(x) +F(x) +F +2 - +2 +2 - +f2(x) +1 +21 +(4 (x) +(x) +1 +() +1 +1 + f+(x) +f+(x) +f+(x) +-0 +0- +0- +0.1 +0.2 +0.3 +0.4 +0.5 +0.1 +0.2 +0.3 +0.4 +0.5 +0.1 +0.2 +0.3 +0.4 +0.5 +X +X +Xsystem across the entire domain of interest is intractable and thus it is not contained within the +model set. Therefore, multiple EFTs are constructed to recover the true system across subsets of +the domain. +To demonstrate why problems falling in the M-open class may not be suited for model aver- +aging schemes, consider applying BMA to the model set involving the two expansions as shown in +Figure 1(a). The resulting posterior mean prediction from BMA still results in a poor estimate +of the true system as shown in Figure 2. Essentially, BMA selects the dashed model rather than +leveraging the localized strengths contained in the model set. +Figure 2: +The posterior mean prediction of +f†(x) when applying BMA to the 2nd order +weak and 4th order strong coupling expansions. +Given the characteristics of EFTs and the M- +open setting associated with these problems, a +simple weighted average of the predictions from +each model is insufficient for recovering the true +physical system. A better approach is to use an +input-dependent weighting scheme which lever- +ages the localized behaviors of each model to +ascertain appropriate mean prediction and un- +certainty quantification. Such an approach falls +under the general class of problems known as +Bayesian model mixing (BMM) (Yao et al., 2021). +A key challenge in BMM is to define the rela- +tionship between the inputs and the model weight +functions. This work proposes a Bayesian treed +model which specifies the weight functions as a sum-of-trees. This representation relies on a tree +basis of weak learners which are used to capture the localized model behavior. Additionally, this +flexible and non-parametric approach allows the user to avoid having to specify a more restrictive +model for the weight functions, such as a generalized linear model. Maintaining the traditional con- +jugacy properties associated with Bayesian Additive Regression Tree (BART) models, the weight +functions are regularized via a multivariate Gaussian prior. The prior is calibrated so that the +weight functions prefer the interval [0, 1] without imposing any further constraints. Additionally, +this framework includes a simple strategy for incorporating prior information about localized model +performance when available. All together, this approach highlights the localized behaviors of the +candidate models and yields significant improvements in prediction, interpretation, and uncertainty +4 + +4 +3 +2 - +(x)(t) +f+(x) +Post. Mean +0.1 +0.2 +0.3 +0.4 +0.5 +Xquantification compared to traditional model averaging methods. +The remainder of the paper is organized in the following manner. Section 2 highlights some +relevant work related to model averaging and model mixing. +Section 3 describes the essential +properties of EFTs, while Section 4 outlines the specifics of the proposed BART-based framework. +Three motivating EFT examples are presented in Section 5. Finally, Section 6 provides a detailed +discussion of the results presented throughout this work. +2 +Background +Methods to address model uncertainty have been widely studied throughout the past few +decades. The majority of work in this area strives to combine competing models through either +mean or density estimation. In either case, the combined result is generally found by taking a linear +combination of the individual predictive means or densities from the models under consideration. +The weights in this linear combination may or may not depend on the inputs for each model and are +learned using the set of training data D = {(x1, y1), . . . , (xn, yn)}. This section briefly reviews some +of the popular model averaging and model mixing techniques currently available in the literature. +Bayesian Model Averaging +A classical approach for combining models M1,. . . ,MK is Bayesian Model Averaging (Raftery +et al., 1997). Suppose Q is a quantity of interest. The posterior density of Q is defined using a +convex combination of the posterior densities under each model, π(Q | D) = �K +l=1 wl π(Q | D, Ml). +Each weight is defined in terms of its corresponding model evidence, i.e. wl = π(Ml | D) where +π(Ml | D) = +p(D | Ml)π(Ml) +�K +k=1 p(D | Mk)π(Mk) +and p(D | Ml) is the marginal likelihood of the data with respect to the lth model. Though BMA is +useful, it has been criticized for emphasizing a fit to the training data as opposed to out-of-sample +prediction, asymptotically selecting a single model (inappropriate in the M-complete and M-open +settings, e.g. Figure 2), and for tending to be sensitive to the prior model probabilities. +Bayesian Mean Stacking +One popular approach for mean estimation is Stacking. +Some of the earlier works in stacking +focused on frequentist approaches for combining point predictions (Breiman, 1996). More recent +work has extended stacking to the Bayesian paradigm (Clyde and Iversen, 2013; Le and Clarke, +2017). +Given K competing models, the stacked mean for a future observation ˜y at input ˜x is +5 + +constructed as a linear combination of individual model predictors +E[˜y | ˜x, D] = +K +� +l=1 +wl fl(˜x), +where E[˜y | ˜x, D, Ml] = fl(˜x). When the individual models are unknown, stacking is conducted in +a two-step procedure: (i) independently fitting the individual models Ml, l = 1, . . . K, given the +set of training data D, and (ii) estimating the weights w = (w1, . . . , wK)⊤ for the stacked predictor +given the fitted models. +In the first step, each model is fit and their corresponding mean predictions, ˆfl(xi), are obtained +at each of the training points. In practice, cross validation techniques are used in this stage to reduce +the risk of overfitting the stacked predictor to the data since each model is trained using the identical +dataset D. In the second step, the coefficients w = (w1, . . . , wK)⊤ are found via optimization. In the +Bayesian paradigm, the weights are selected to minimize the posterior risk for some pre-specified loss +function. For example, the squared error loss between the observed data and the stacked predictor +at new point ˜x is given by �w = argminw +� � +˜y − �K +l=1 wl ˆfl(˜x) +�2 +p(˜y | ˜x, D) d˜y. Using the observed +data, the posterior risk can be approximated with leave-one-out (LOO) cross validation techniques +which allows the stacking weights to be selected by �w = argminw +�n +i=1 +� +yi − �K +l=1 wl ˆf(−i) +l +(xi) +�2 +, +where ˆf(−i) +l +(xi) is the LOO cross-validation prediction for the ith observation using the lth model +as discussed in Le and Clarke (2017). +Additionally, one may choose to modify this loss function to impose a set of conditions on the +weights. For example, one may impose a simplex, non-negativity, or sum-to-m constraint on the +weights (Le and Clarke, 2017). Other approaches include regularization via a penalty term or a +prior (Breiman, 1996; Yang and Dunson, 2014). Such approaches can be useful when the individual +model predictions are highly correlated. +Bayesian Complete Stacking +Complete Stacking was introduced in 2018 by Yao et al. (2018) and motivated by the shortcomings +of BMA (Yao et al., 2018). Their proposed Bayesian stacking model emphasizes prediction, as the +weights are selected to minimize the Kullback-Leibler (KL) divergence between the true predictive +density and the stacked predictive density +p(˜y | ˜x) = +K +� +l=1 +wl p(˜y | ˜x, D, Ml), +where ˜y is a future observation with input ˜x. Similar to mean stacking, the leave-one-out (LOO) +cross validated predictive density can be used in place of p(˜y | ˜x, D, Ml) when the individual models +6 + +are unknown. Given training data, the weights are constrained to a K−dimensional simplex SK +and estimated as �w = argmaxw∈SK +�n +i=1 log �K +l=1 wl p(yi | xi, D(−i), Ml), where D(−i) denotes the +training set excluding the pair (xi, yi). +Bayesian Hierarchical Stacking +Hierarchical Stacking is a model mixing approach which extends Complete Stacking by defining +input-dependent weights that are estimated in a fully Bayesian manner. One way to define the +relationship between the inputs and the weight functions is through a parametric model. In this +case, one models K − 1 unconstrained weight functions, +w∗ +l (xi) = µl + +J +� +j=1 +αlj gj(xi), +which depend on the sets of hyperparameters {αlj} and {µl} along with user-specified basis func- +tions gj(xi), where j = 1, . . . , J and l = 1, . . . , K − 1. The Kth function w∗ +K(xi) is set to 0 to serve +as a baseline. Then, a softmax transformation is applied to the unconstrained weights in order to +confine each model weight to the K-dimensional simplex, namely +wl(x) = +exp +� +w∗ +l (x) +� +exp +� +w∗ +1(x) +� ++ · · · + exp +� +w∗ +K(x) +� +for l = 1, . . . , K. +The methods discussed above outline a number of strategies one can take to combine the +information across multiple models. In the setting of EFT experiments, the localized nature of +the predictions suggests an input-dependent weighing scheme like Bayesian Hierarchical Stacking +is more suitable. However, specifying the required basis functions may not be trivial. Thus, the +proposed method will adopt the notion of mean stacking within an additive tree basis model to +achieve localized weighting in a flexible and non-parametric manner. But first, the motivating EFT +experimental setting is reviewed in more detail. +3 +Towards Model Mixing with EFTs +Computer models such as EFTs are typically implemented as simulators which are motivated +by the known physics. The theoretical predictions of the physical system are approximations from +each simulator plus a discrepancy term, which is designed to account for the remaining unexplained +portions of a system. These two components may have specific properties which can be leveraged +when working with observational data. This section summarizes these details in the context of +EFTs, while a further discussion is provided in the supplementary material. +7 + +3.1 +Motivating EFT Example +Consider the EFT example where the true physical system is the partition function of the +zero-dimensional φ4 theory defined by +f†(x) = +� ∞ +−∞ +exp +� +− u2 +2 − x2u4� +du, +(1) +where x denotes the coupling constant (Honda, 2014). Two types of finite-order expansions exist +for this partition function and are given by (2) and (3) for ns or nl ≥ 1. +h(ns) +s +(x) = +ns +� +t=0 +stxt +where st = +� +� +� +� +� +� +� +√ +2Γ(t+0.5) +(t/2)! +(−4)(t/2) +t is even +0 +t is odd +(2) +h(nl) +l +(x) = +nl +� +t=0 +ltx−t +where lt = Γ(0.5t + 0.25) +2t! +� +− 1 +2 +�t +, +t = 0, ..., nl. +(3) +The weak coupling expansion in (2) is an asymptotic Taylor-like series of order ns centered about +zero. Thus, h(ns) +s +(x) will yield high fidelity predictions for smaller coupling constants and diverge +as the value increases. The reverse behavior is observed for the strong coupling expansion in (3), +h(nl) +l +(x), which is convergent. Example predictions of the physical system using these finite-order +expansions can be seen in Figure 1 and are discussed in detail in Section 3.2. +The theoretical predictions of the physical system using the weak and strong coupling expansions +are expressed using (4) and (5), respectively. +f(ns) +s +(x) = h(ns) +s +(x) + δ(ns) +s +(x) +(4) +f(nl) +l +(x) = h(nl) +l +(x) + δ(nl) +l +(x). +(5) +where the truncation errors δ(ns) +s +(x) and δ(nl) +l +(x) are modeled with Gaussian processes (GPs) (Gra- +macy, 2020; Santner et al., 2018). As described by Melendez et al. (2019), the parameters in both +truncation error models are dependent upon the evaluations of their corresponding finite-order +expansions (described in (2) and (3), respectively) over a sparse grid of points. The discrepancy +model also depends on physical quantities, Q(x) and yref(x), which are chosen based on domain +expertise. The relationship between these quantities and the discrepancy are summarized in the +supplementary material. When Q(x) and yref(x) are unknown, one can alternatively use the error +approximation described by Semposki et al. (2022). +The features present in this example from Honda (2014) are commonly found across the land- +scape of EFT problems. For instance, the physical system can be expressed as an additive model +8 + +involving a finite-order expansion and the induced truncation error. The finite-order expansions +are designed to provide high fidelity predictions in specific subregions of the domain. There exists +a subregion of the domain where none of the finite-order expansions yield accurate theoretical pre- +dictions. All together, this motivating example serves as a prototype for the EFTs that may be +encountered in a general experimental setting. +3.2 +The Model Set for EFT Experiments +One may encounter various experimental settings when working with EFTs. Such scenarios are +introduced in the context of the motivating example presented in Section 3.1. First, consider the +most basic case where the model set contains a single EFT. With one EFT, the overall predictive +accuracy of the true system is poor, despite the good performance in a localized region. +For +example, suppose the model set M contains the 2nd order weak coupling expansion f(2) +s (x). Mean +predictions constructed from (2) and (4) are shown by the dashed line in Figure 1(a). Clearly, this +model is limited to strong predictive accuracy in only the left subregion of the domain. +When available, one can consider different finite-order approximations of the same EFT. For +example, consider the 2nd, 4th, and the 6th order coupling expansions which are shown in Fig- +ure 1(c). The three models are very similar for lower coupling constants yet drastically differ in +the remainder of the domain. Despite each expansion’s poor theoretical predictions, one can still +leverage the available information to improve the overall prediction of the physical system. For +instance, the 2nd and 6th order expansions (dashed and dashed-dotted) are concave functions while +the 4th order expansion (dotted) is convex. This suggests the true physical system lies between the +expansions under consideration and can be recovered by re-weighting the corresponding predictions. +A third situation is to consider multiple EFTs. In this example, one can consider various finite- +order strong coupling expansions in addition to the weak coupling expansions. For example, a +model set can contain a finite-order weak coupling expansion (dashed) and the 4th order strong +coupling expansion (dotted) as shown in Figures 1(a) and 1(b). The addition of the strong cou- +pling expansion allows for a high fidelity prediction of the physical system to be considered in the +rightmost subregion of the domain. The model set listed in panel (a) implies the true system lies +between the two expansions. This is particularly useful in the intermediate range where neither +of the EFTs are accurate. Meanwhile, the set in panel (b) presents an interesting case where the +physical system lies below both EFTs in the intermediate range. In this case, the information in +the observational data can be leveraged to help recover the true system. +9 + +In this example, the predictions from the weak coupling expansion degrade slowly compared to +those from the strong coupling expansions. Consequently, the weak coupling expansions generally +appear to have a better overall predictive performance across the entirety of the domain. When +combining these two types of EFTs using global weighting schemes such as BMA, the resulting +prediction will favor the weak coupling expansion due to its drastic advantage in the overall model +performance. The undesirability of the BMA solution is evident in Figure 2, which demonstrates +that BMA effectively matches the 2nd order weak coupling expansion. Hence, a weighting scheme +which captures the localized behaviors of each model is preferred in the EFT setting. +The proceeding sections consider a general set of K different EFTs, which are denoted by +f1(x), . . . , fK(x). In this motivating example, fl(x) = hl(x) + δl(x) where hl(x) can denote either +a weak or strong coupling expansion of order Nl, where l = 1, . . . , K. Meanwhile, δl(x) is the +associated truncation error and is modeled by a GP as described in the supplementary material. +3.3 +A Two-Step Approach for EFTs +Similar to Bayesian mean stacking, a two-step approach is adopted for combining the predictions +across K EFTs. When constructing a two-step stacking approach, it is important to understand +the available sources of information and where each can be leveraged throughout the estimation +process. The first source of information is the observational data, Y1, ..., Yn which are assumed to +be independent random variables generated at fixed inputs x1, ..., xn according to +Yi = f†(xi) + ϵi, +ϵi +iid +∼ N(0, σ2) +where f†(xi) represents the true and unknown physical system. The information from the field +data is collected in the set D = {(x1, Y1), . . . , (xn, Yn)}. +Meanwhile, each EFT is also associated with its own set of information. For example, consider +the lth EFT denoted by fl(x). +It is assumed this EFT is accompanied by a set of simulator +runs across a fixed set of inputs xc +l1, . . . , xc +lnl. Information regarding the design of the computer +experiment for each EFT can be found in Melendez et al. (2021). The simulator runs are evaluations +of the finite-order expansion, hl(·), at the specified inputs. Using these model runs, one can extract +the set of Nl + 1 coefficients c0(·), . . . , cNl(·) at each of the fixed inputs. The training set for the lth +EFT is then defined by +Dl = +�� +xc +l1, C(xc +l1) +� +, . . . , (xc +l1, C(xc +lnl) +�� +where C(·) denotes the vector of known finite-order coefficients at the specified model input. The +10 + +resulting coefficients and the set of inputs can differ across the K models, thus the sets D1, . . . , DK +will contain different information. +The proposed two-step approach is tailored to EFTs by taking advantage of the sources of +data described above as well as the properties described in the supplementary material. +The +proposed Bayesian mean stacking approach extends the traditional methodologies by incorporating +input-dependent weights. Conditional on the values of the theoretical predictions at a given point, +f1(xi), . . . , fK(xi), this model can be defined by +Yi | f(xi), w(xi), σ2 ind +∼ N +� +f ⊤(xi)w(xi), σ2) +(6) +where f(xi) = +� +f1(xi), ..., fK(xi) +�⊤ and w(xi) = (w1(xi), ..., wK(xi))⊤. In practice, these values +are unknown and must be estimated. This problem serves as the first step of the stacking procedure. +This first step fits each model, fl(x), independently given data, Dl. As described in the supple- +mentary material, an EFT is fit using the finite-order coefficients to learn the unknown parameters +which characterize the GP assigned to the truncation error. All of this information can be extracted +from Dl which implies the set of field observations is not required to fit each model. Consequently, +the desired theoretical predictions can be estimated without using any of the information in D. +This differs from existing statistical approaches as summarized in Section 2. +Prior to learning the weights, point predictions from each EFT are required at x1, . . . , xn. The +predictions for an EFT are computed through the posterior predictive mean which is given by +ˆ +f l(xi) = ˆEπ|Dl� +fl(xi) +� +, +i = 1, . . . , n. +Additionally, the posterior variance of the predictions can be extracted if desired. This posterior +predictive distribution is a Gaussian process which can be characterized by the corresponding +mean and covariance functions as described in Melendez et al. (2019) (see also the supplementary +material). +The objective of the second stage of the stacking procedure is to estimate the weight functions +shown in (6), which is the focus of this work. Conditional on the mean predictions from the first +step, the model for the observational data becomes +Yi | ˆ +f(xi), w(xi), σ2 ind +∼ N +� ˆ +f +⊤(xi)w(xi), σ2) +where ˆ +f(xi) = +� ˆf1(xi), . . . , ˆfK(xi) +�⊤. The weight functions are then learned using the information +contained in the field data D. The next section outlines the proposed model mixing scheme which +defines the weight functions using Bayesian Additive Regression Trees (BART). +11 + +4 +Bayesian Additive Model Mixing Trees +4.1 +Bayesian Tree Models +Bayesian tree models have become increasingly popular for modeling complex and high dimen- +sional systems (Chipman et al., 1998). Bayesian additive regression trees are used to model an +unknown mean function, E[Y | x] (Chipman et al., 2010). This additive approach involves sum- +ming together the predictions made from m trees and is facilitated through a Bayesian backfitting +algorithm (Hastie and Tibshirani, 2000). Each tree Tj is characterized by its structure, comprised +of internal and terminal nodes, along with its associated set of terminal node parameters, Mj. The +internal nodes define binary partitions of the input space according to a specified splitting rule. The +prior probability a node is internal is p(η is internal) = α(1 + dη)−β where dη is the depth of the +node η, while α and β are tuning parameters. By construction, this prior penalizes tree complexity +and thus ensures each tree maintains a shallow and simple structure. Given d different predictors, +x1, ..., xd, splitting rules are of the form xv < c for v ∈ {1, ..., d} and cutpoint c from a discretized +subset of R. In the simplest approach, the predictor and cutpoint associated with each splitting +rule are randomly selected from discrete uniform distributions. The probabilities associated with +the designation of each node along with the splitting rules for internal nodes are used to define the +stochastic tree-generating prior for each tree. +The m trees are learned through MCMC, where a slight modification to each structure is +proposed at every iteration of the simulation. Generally, such modifications to the tree include +birth, death, perturb, or rotate as described by Chipman et al. (1998) and Pratola (2016). Proposals +are then accepted or rejected using a Metropolis-Hastings step. To avoid a complex reversible jump +MCMC, the algorithm depends on the integrated likelihood, which is obtained by integrating over +the terminal node parameters associated with the given tree. A closed form expression for this +density can be obtained with conditional conjugate priors for the terminal node parameters. +Given the tree structure, prior distributions can be assigned to each terminal node parameter. +In the BART model, the priors ensure each tree explains a small yet different source of variation +in the data. For continuous data, BART assigns Gaussian priors to the terminal node parameters. +Assuming the data is mean centered, the prior assigned to terminal node parameter µpj in node +ηpj is given by µpj | Tj ∼ N(0, τ 2) where τ = (ymax − ymin)/(2k√m) with tuning parameter k. +Additionally, a conjugate scaled-inverse-χ2 prior is assigned to the variance parameter. +The traditional Bayesian regression tree model can be extended to allow for a more complex +12 + +structure in the terminal nodes. Existing extensions include linear regression (Chipman et al., 2002; +Prado et al., 2021) and Gaussian processes (Gramacy and Lee, 2008). For the setting of model +mixing, this work extends BART to a multivariate Gaussian terminal node model. +4.2 +Model Mixing with BART +The weight functions w(x) = (w1(x), . . . , wK(x))⊤ are modeled using a sum-of-trees +w(xi) = +m +� +j=1 +g(xi, Tj, Mj), +(7) +where g(xi, Tj, Mj) is the K-dimensional output of the jth tree using the set of terminal node +parameters, Mj, at the input, xi. This approach defines the weight functions using tree bases which +are learned from the data. The amount of flexibility in the weight functions can be controlled by +changing the number of trees or tuning the hyperparameters in the prior distributions. +In this application of BART, each terminal node parameter is a K-dimensional vector which is +assigned a multivariate Gaussian prior. The parameter is regularized so that each tree accounts for +a small amount of variation in the weight functions. For the proceeding statements, let ηpj represent +the pth terminal on the jth tree and define its corresponding parameter by µpj = (µpj1, ..., µpjK)⊤. +Now assume the observations (x1, y1), ..., (xnp, ynp) lie in the hyper-rectangle defined by ηpj, where +np is the number of observations assigned to this subregion. The model at each terminal node +amounts to fitting a localized Bayesian linear regression with parameter vector µpj. Due to condi- +tional independence, the likelihood in this node is defined by +L(r1, ..., rnp | Tj, µpj, σ2) = (2πσ2)−np/2 exp +� +− +1 +2σ2 +np +� +i=1 +� +ri − ˆ +f +⊤(xi)µpj +�2� +where ˆ +f(xi) = +� ˆf1(xi), ..., ˆfK(xi) +�⊤ is a vector of mean predictions from each EFT and ri is the +ith residual given by ri = yi − � +q̸=j ˆ +f +⊤(xi)g(xi, Tq, Mq). +Conditional on the tree structure, Tj, the terminal node parameter, µpj is assigned a conjugate +multivariate Gaussian prior, namely +µpj | Tj +ind +∼ NK +� +β, τ 2IK +� +(8) +where β = (β1, ..., βK)⊤ is a K-dimensional mean vector and IK is the identity matrix. This prior is +non-informative in the sense that the mean is fixed regardless of how the input space is partitioned. +In model mixing, each simulator may perform strongly in one subregion of the input space but +weakly in another. This belief can be reflected in the prior distribution of µpj by allowing the +13 + +hyperparameters to depend on the partition of input space assigned to the given terminal node. +Thus, an informative prior for µpj can be constructed as +µpj | Tj +ind +∼ NK +� +βpj, τ 2IK +� +where βpj = (βpj1, ..., βpjK)⊤. This allows the prior mean to vary depending on the tree partitions +and thus reflect some sense of localized model performance. Meanwhile, the assumed covariance +structure implies the K vector components µpj1, ..., µpjK are independent apriori. +Both of the proposed priors are conjugate, which is an important choice in BART, as it allows for +a closed form expression for the marginal likelihood for the vector of residuals Rpj = (r1, .., rnp)⊤, +Additionally, the conjugate priors result in closed form expressions for the full conditional distribu- +tions of the terminal node parameters and the error variance. The derivations of these distributions +are found in the Appendix. In particular, the full conditional distribution for the pth terminal node +in Tj is given by +µpj | Rpj, Tj, σ2 ind +∼ NK +�� 1 +σ2 ˆF +⊤ +pj ˆF pj + 1 +τ 2 IK +�−1� 1 +τ 2 βpj + 1 +σ2 ˆF +⊤ +pjRpj +� +, +� 1 +σ2 ˆF +⊤ +pj ˆF pj + 1 +τ 2 IK +�−1 +� +where ˆF pj is the np × K design matrix with the ith row vector given by the vector ˆ +f +⊤(xi). Mean- +while, the full conditional distribution for σ2 is a scaled-inverse-χ2, i.e. σ2 ∼ ν′λ′/χ2 +ν′ where +ν′ = n + ν +and +λ′ = +1 +n + ν +� +n +� +i=1 +� +yi − ˆ +f +⊤(xi)w(xi) +�2 ++ νλ +� +. +4.3 +Calibrating Priors +First consider the prior for the terminal node parameters. The calibration of the hyperparam- +eters differs for the non-informative and informative priors, however both approaches are designed +to ensure that each model weight wl(x) should prefer the interval [0, 1] and be centered at a value +within this region. Moreover, the functions w1(x), . . . , wK(x) are assumed to be independent apriroi +at a fixed input. This enables the prior for each weight to be calibrated marginally. +4.3.1 +Non-Informative Prior +Consider a non-informative prior for the terminal node parameters. In this setting, +µpj | Tj +iid +∼ NK +� +β, τ 2IK +� +for the pth terminal node parameter in the jth tree. First, fix l ∈ {1, ..., K} +and i ∈ {1, ..., n} to calibrate the prior for wl(xi). Since the terminal node parameters are indepen- +dent and identically distributed with a diagonal covariance structure, the prior induced on wl(xi) +14 + +is the same for the remaining weight and input combinations. From (7) and (8), the induced prior +on the lth model weight is wl(xi) ∼ N(mβl, mτ 2). Since it is believed wl(xi) ∈ [0, 1] with high +probability, it is plausible to set mβl = 0.5. Consequently, βl = 0.5/m. Thus, each weight has an +equal chance to reach the “extreme” values of 0 or 1 regardless of the input location. The prior +standard deviation, τ, can be selected so that wl(xi) ∈ [0, 1] with high probability. To do this, a +confidence interval for wl(xi) is constructed such that +0 = 0.5 − kτ√m +and +1 = 0.5 + kτ√m. +Subtracting the first equation from the second and solving for τ yields τ = +1 +2k√m. +This calibration approach is very similar to the one proposed by Chipman et al. (2010). The +main difference is due to the context of the problem, as it is believed the weights are predominately +contained in an interval [0, 1] rather than the observed range of the data, [ymin, ymax]. Note, this +approach can also be generalized to situations where the weights are assumed to prefer the interval +[a, b] with high probability. Such a belief would imply τ = (b − a)/(2k√m). +4.3.2 +Informative Prior +In the informative setting, the prior mean directly depends on the partitions of the input +space induced by the given tree, i.e. µpj | Tj ∼ NK +� +βpj, τ 2IK +� +. This prior is tailored towards +EFTs, where the functional variance, vl(xi), indicates the severity of the truncation error. A larger +variance within a particular subregion of the domain indicates the presence of larger truncation +error meaning the EFT provides a poor approximation of the true system. +In this setting, the induced prior on the lth model weight is given by wl(xi) ∼ N(βl(xi), mτ 2). +The function, βl(xi) is interpreted as the prior mean weight function and can be defined in terms +of the sum-of-trees by +βl(xi) = +m +� +j=1 +Bj +� +b=1 +βbjl1(xi ∈ ηbj) +where Bj is the number of terminal nodes in Tj and 1(xi ∈ ηbj) is the indicator that xi is assigned +to the terminal node ηbj. +To calibrate the informative prior, first obtain an initial estimate of βl(xi) at each input +i = 1, ..., n. One strategy is to utilize a set of normalized precision weights (Phillips et al., 2021). +Thus, for each l ∈ {1, ..., K} and i ∈ {1, ..., n}, βl(xi) can be estimated by a ratio of precisions, +βl(xi) = +1/vl(xi) +1/v1(xi) + ... + 1/vK(xi) +15 + +where vl(xi) is the variance of the lth model at xi. This precision weighting scheme encodes prior +knowledge about each model’s relative strengths and weaknesses into the model mixing frame- +work. Since the prior of the terminal node parameter changes conditional on the tree structure, +each βpj is chosen separately from the other terminal node parameters. Moreover, because the +terminal node parameter µpj is assigned a diagonal covariance structure, each of the K vector com- +ponents are calibrated marginally where µpjl | Tj +ind +∼ N(βpjl, τ 2). Without loss of generality, assume +(x1, y1), . . . , (xnp, ynp) are assigned to the hyper-rectangle associated with the terminal node ηpj. +Given this partition and the initial guesses of βl(xi), an estimate of βpjl can be obtained by +βpjl = +1 +mnp +np +� +i=1 +βl(xi). +A confidence interval for each terminal node parameter can be set to have a length of 1/m in order +to ensure each tree is a weak learner. This is done by setting +1 +m = 2kτ, which implies τ = +1 +2km. +4.3.3 +Variance Prior +A conjugate scaled inverse chi-square distribution with hyperparameters ν and λ is assigned +to the error variance σ2. The value of ν controls the shape of the prior distribution; increasing ν +decreases the variability and results in a prior which is more concentrated around the mode. The +value of λ sets the scale of the distribution, and thus specifies the region of the domain which is +assigned non-negligible mass. +To calibrate the prior, first select a value of ν to reflect the desired shape of the distribution. +Common values of ν range from 3 to 10. +Before selecting a value for λ, one needs an initial +estimate of the error variance to help set the prior around a range of plausible values of σ2. Given +the model set and the corresponding point predictions at each of the training points ˆ +f l(xi), one +can use a lightly data informed prior by setting ˆσ2 = maxl=1,..,K +� +mini=1,..,n +� +yi − ˆfl(xi) +�2� +. This +crude estimate has worked well in the EFT examples discussed in Section 5 due to the small error +variances associated with the controlled experiments. Given this information, one strategy is to set +ˆσ2 to be the mean or mode of a λν/χ2 +ν distribution. Mean calibration sets +νλ +ν−2 = ˆσ2, which implies +λ = ν−2 +ν ˆσ2. Similarly, mode calibration sets +νλ +ν+2 = ˆσ2 which implies λ = ν+2 +ν ˆσ2. +In the simulation examples presented in Section 5, the mode calibration appears to perform the +best, as it typically allocates more mass around the true variance compared to the mean calibration. +Since a common belief is that each model yields accurate approximations of the true system over +16 + +some subregion of the domain, one should expect the set of minimum squared differences across +the K models will unveil reliable information about the true error variance. +5 +EFT Examples +This section applies the proposed model mixing methodology to three real EFT examples. The +first example involves mixing a concave weak coupling expansion and convex strong coupling ex- +pansion. In this setting, the true physics model lies between the expansions, hence an interpolation +of the two will adequately capture the true function. The second example involves mixing two +convex expansions, each of which overestimates the true system in the intermediate range of the +domain. This scenario demonstrates the benefits of applying prior regularization to the weight +functions rather than imposing strict conditions such as a simplex constraint. The final example +demonstrates the effectiveness of the proposed method in cases where a local expert does not exist +for all subregions of the domain. An area of the domain is said to be without a local expert if none +of the models under consideration adequately recovers the true underlying system. +An additional goal is to demonstrate the effectiveness of model mixing with both the non- +informative and informative priors. Given no prior information of the associated model discrepan- +cies, one should elect to use the non-informative prior. If such information is available, then the +informative prior can be applied. +The following examples involve data which is independently generated according to +Yi = f†(xi) + ϵi, +ϵi +iid +∼ N(0, σ2) +where i = 1, ..., 20, σ = 0.005, and f†(x) is defined in (1). The 20 training points are located at +inputs which are evenly spaced over the interval of 0.03 to 0.50. The error standard deviation of +0.005 was selected to mimic a controlled experiment setting. Each EFT model is fit using nc = 4 +evaluations of the corresponding finite-order expansion. +5.1 +Example 1: Mixing Two Expansions +First, consider mixing the second order weak coupling expansion, f(2) +s (x) with the fourth order +strong coupling expansion, f(4) +l +(x), over the domain of [0.03, 0.5] as shown in Figure 1(a). The pre- +dicted mean and corresponding posterior weight functions are shown in Figures 3 and 4 respectively. +Regardless of the prior type, the BART-based mixing approach exhibits accurate mean predictions +17 + +Figure 3: +(Example 1) The predicted mean (dark gray) and 95% credible intervals (shaded) when +mixing f(2) +s (x) (dashed) and f(4) +l +(x) (dotted). The true mean (light gray) is adequately captured +under both priors with a training set of 20 observations (points) and nc = 4 simulator evaluations. +with minimal uncertainty in the left and right portions of the domain. The uncertainty increases in +the intermediate range of the domain where neither EFT under consideration accurately predicts +the true system. The most notable difference between the two approaches can be seen in the associ- +ated uncertainty within this intermediate range, as the non-informative prior typically yields wider +95% credible intervals across this region. This is plausible as the informative prior directly incorpo- +rates knowledge related to the individual model performance into its hyperparameters. Therefore, +the posterior distribution of the weight functions is less dependent upon the data and is noticeably +influenced by the prior. Meanwhile, the results under both priors yield similar weight functions +which exhibit sigmoid-like curves. This result is intuitive of how the EFTs should be mixed, as +the weak coupling expansion is appropriate in the left portion of the domain while the strong cou- +pling is appropriate in the right region. A linear combination of these functions is useful for the +intermediate range. The posterior weight functions obtained using the informative prior generally +exhibit smoother shapes with less variability compared to the results under the non-informative +prior. +Figure 4 directly illustrates the effect of the prior on the posterior weight functions, as +the informative version enables one to gain a more precise understanding of the behavior of the +weight functions. This figure suggests two key insights. First, this mean mixing approach searches +for useful combinations of mean predictions in order to recover the true system. Clearly, various +18 + +Non-Informative Prior +InformativePrior +2.75- +2.75 +2.50 +2.50 +2.25 +2.25 +1 +2.00- +1 +2.00- +1 +1 +0.1 +0.2 +0.3 +0.4 +0.5 +0.1 +0.2 +0.3 +0.4 +0.5 +X +XFigure 4: (Example 1) The posterior estimates of the weight functions when mixing f(2) +s (x) (solid) +and f(4) +l +(x) (dashed). The respective 95% credible intervals are denoted by the shaded regions. +sets of weight functions can yield very similar final predictions. Finally, the amount of variability +observed in the final prediction of the physical system is directly related to the variability in the +weight functions. This is evident in the results obtained with the informative prior. +5.2 +Example 2: Mixing Two Convex Expansions +Now consider mixing two convex expansions which are always equal or greater than f†(x), as +shown in Figure 1(b). In this problem, a convex combination of the models in the intermediate +range will never capture the true system. Meanwhile, the BART-based approach benefits from the +flexibility of the prior regularization strategy as it is able to recover f†(x). +From Figure 5, both priors lead to accurate mean predictions of the physical system. As in +Example 1, the result under the informative prior yields a lower degree of uncertainty compared +to its non-informative counterpart, particularly across the region of [0.15, 0.40]. Meanwhile, the +posterior weight functions return very similar shapes under the two priors as displayed in Figure 6. +Similar to Example 1, the informative prior generally results in more precise estimates of the weight +functions which translates to lower uncertainty in the posterior predictions of the physical system. +19 + +Non-InformativePrior +Informative Prior +1.00- +1.00- +0.75- +0.75- +W(x) +0.50 +0.50 +0.25 +0.25 +0.00 +0.00 +0.1 +0.2 +0.3 +0.4 +0.5 +0.1 +0.2 +0.3 +0.4 +0.5 +X +XFigure 5: +(Example 2) The predicted mean (dark gray) and 95% credible intervals (shaded) when +mixing f(4) +s (x) (dashed) and f(4) +l +(x) (dotted) to predict the true system (light grey). +Figure 6: (Example 2) The posterior estimates of the weight functions when mixing f(4) +s (x) (solid) +and f(4) +l +(x) (dashed). The 95% credible intervals are denoted by the shaded regions. +5.3 +Example 3: Mixing Without a Local Expert +This example demonstrates the BART-based model’s ability to mix functions in regions where +no local expert may exist. Consider the model set containing three weak coupling expansions of +orders 2,4 and 6 as shown in Figure 1(c). In this case, no local expert exists in the right portion of +20 + +Non-InformativePrior +InformativePrior +2.8 +2.8 +2.6 +2.6 +/. +/ +2.4 +F +2.2 +2.2 +2.0 +2.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.1 +0.2 +0.3 +0.4 +0.5 +X +XNon-InformativePrior +InformativePrior +1.00 +1.00 +0.75 +0.75 +0.50 +M +0.50 +0.25 +0.25 +0.00 +0.00 +0.1 +0.2 +0.3 +0.4 +0.5 +0.1 +0.2 +0.3 +0.4 +0.5 +X +XFigure 7: (Example 3) The predicted mean (dark gray) and 95% credible intervals (shaded) when +mixing f(2) +s (x) (dashed), f(4) +s (x) (dotted), and f(6) +s (x) (dashed, dotted). The true mean (light gray) +is adequately captured under both priors. +the domain as all three function diverge away from the true function at different rates. +Despite the lack of a local expert, the mixed prediction adequately recovers the true system with +relatively small amounts of uncertainty as shown in Figure 7. The non-informative prior results in +a greater uncertainty between (0.3, 0.4), which is where the 2nd and 4th order expansions begin to +diverge at quicker rates, hence the mean prediction is more sensitive to small changes in the weight +values. Meanwhile, the result under the informative prior has minimal uncertainty in this right +portion of the domain. Both results also display subtle deviations from f†(x) in the remainder of +the domain. Overall, this example demonstrates the ability of the BART-based model to leverage +observational data as well as the information in the model set to make accurate predictions. +In this example, the posterior weight functions noticeably differ depending on the selected prior, +as shown in Figure 8. For example, the weight functions defined using the non-informative prior +indicate more weight is allocated to the 2nd and 4th order expansions across the interval (0.03, 0.15), +as evident by the location of the solid and dashed curves. Additionally, all three functions have +a relatively high degree of variability within this lower half of the domain. With the informative +prior, a larger portion of the prediction is attributed to the 6th order expansion, which has a mean +weight function around 0.6. The effects of the other two expansions are then shrunk in this lower +region. As the coupling constant x increases, the 2nd order expansion contributes more to the +21 + +Non-Informative Prior +InformativePrior +2.6- +2.6- +2.4 +2.4 +F +2.2 +2.2 +2.0 - +1 +1 +2.0 - +1 +1 +- +1 +- +1 +0.1 +0.2 +0.3 +0.4 +0.5 +0.1 +0.2 +0.3 +0.4 +0.5 +X +XFigure 8: (Example 3) The posterior estimates of the weight functions when mixing f(2) +s (x) (solid), +f(4) +s (x) (dashed), and f(6) +s (x) (dotted). The 95% credible intervals correspond to the shaded regions. +final prediction compared to the 4th and 6th order expansions. Finally, both approaches properly +identify that the 6th order expansion diverges at a much faster rate in the right portion of the +domain. Hence, its corresponding effect is shrunk to zero within this region. +This example reiterates that the BART-based approach searches for useful combinations of +models, and these combinations are not unique. It also poses a more interesting question related +to the interpretation of the weights. For example, in the interval (0.03, 0.15), the mean predictions +from each EFT are nearly identical and align closely with the true system. Given this, one may +expect each EFT is assigned a weight near 1/3, as a simple average of their predictions would be +adequate in this region. However, this is not the case regardless of the prior selected. Specifically, +the weight given to the 6th order expansion noticeably differs from the weights assigned to the 2nd +and 4th order expansions. With the non-informative prior, this likely occurs because the trees are +also regularized to be weak learners, meaning each is relatively shallow. Since the trees maintain a +shallow depth, some sense of global model performance is preserved, thus the effect of the 6th order +expansion is mitigated in this subregion. When considering joint credible regions, the case where +all weights are near 1/3 falls along the edge of the 99% credible region which suggests the simple +average of predictions is a possibility, though it is unlikely. With the informative prior, the 6th +order expansion is assigned a relatively higher weight within (0.03, 0.15) because it has a smaller +truncation error compared to the other models under consideration in this region. +22 + +Non-Informative Prior +Informative Prior +1.00- +1.00- +0.75- +0.75 +0.50 +M +0.25 +0.25- +0.00 +0.00 +0.1 +0.2 +0.3 +0.4 +0.5 +0.1 +0.2 +0.3 +0.4 +0.5 +X +x6 +Discussion +Prediction and Uncertainty Quantification +The proposed BART-based model mixing approach is able to adequately recover the underlying +system f†(x) in each of the examples presented. In general, the information from each individual +model tends to dominate the posterior predictions when a local expert is present, while the infor- +mation in the data is more influential in areas where no model aligns with the true system. For +example, the information in the data is crucial when obtaining predictions in the intermediate range +for Examples 1 and 2 or the right portion of the domain in Example 3. It should also be noted +similar performance in the mean prediction is observed when the data is not evenly spaced or when +the training set is reduced. In Examples 1 and 2, one should be cautious when extrapolating in the +left portion of the domain due to the rapid divergence of the 4th order strong coupling expansion. +However, in other settings where the rate of change for the EFT predictions is not as drastic, severe +issues when extrapolating slightly outside the domain of the training data are not expected. +For each example, small levels of uncertainty in the posterior prediction of f†(x) are observed +across areas where at least one EFT aligns with the true system. The uncertainty increases in areas +where the EFTs under consideration deviate from the true system. Due to the small observational +errors, the mixed-model is very confident the training points align with f†(x). As a result, the +credible intervals nearly touch each training point since the predictive mean function is nearly +interpolating the observations. Between training inputs, the uncertainty increases and displays a +bubble-like shape. These uncertainty bands tend to smooth out when the posterior variance shifts +towards high values of σ2 as the mixed-prediction is no longer interpolating between the points. +Prior Distributions +Regardless of the prior selected, it remains clear that one is able to obtain adequate predictive +performance and recover the true physical system with reasonable amounts of uncertainty. This +is crucial because prior information pertaining to a model’s localized performance may not always +be available. Compared to the informative prior, the results using the non-informative version +will generally result in higher degrees of uncertainty across the predicted system. This is expected +because there is less information about the weight functions present in the resulting posterior +distributions when using the non-informative prior. +The informative prior explicitly leverages the information in the truncation errors, which directly +relates to the localized predictive accuracy of each EFT. This information is used to calibrate the +23 + +prior mean of the terminal node parameters. +Thus, an EFT with relatively small truncation +error across a given partition of the domain will be assigned higher weight apriori compared to +an EFT with relatively high errors. One can control the influence of the prior by changing the +tuning parameters in the terminal node parameter model. Overall, the informative prior can be +an effective tool because it essentially guides the weight functions towards the right direction using +this additional model information. +Interpretation of Model Weight Functions +The primary objective of the weight functions is to re-scale the predictions given by each individual +model so that a linear combination of these predictions can adequately recover the true system. +Given the prior regularization method applied to the weight functions, exact interpretation of the +resulting values can be unclear. However, using this regularization perspective, one can conclude +that weight functions which fall close to zero within a particular input subregion indicate that the +corresponding model is unnecessary for the overall prediction. Meanwhile, a model which is the +unique local expert within a particular region should be weighted by values close to one. These +features are observed across all three examples. +Figure 9: +The posterior mean estimates and +95% credible intervals (shaded) of the sum of +weight functions from Examples 1 and 2 (solid +and dashed) using the informative prior. +The benefit of the proposed regularization +approach can further be understood through +the posterior distribution of the sum of the +weight functions, wsum(x) = �K +l=1 wl(x). Fig- +ure 9 illustrates the posterior of wsum(x) for +Examples 1 and 2 under the informative prior. +The posterior of wsum(x) from Example 1 +(solid) is centered very close to one with rel- +atively small amounts of uncertainty. This re- +sults because: (i) the prior regularization and +(ii) f†(x) lies between the selected EFTs, which +indicates a convex combination is appropri- +ate. Even though a sum-to-one property is not +strictly imposed, it appears to naturally occur +in this situation where an interpolation of the +competing models is appropriate. Meanwhile, the posterior of wsum(x) from Example 2 (dashed) +significantly drops below one in the intermediate range of the domain because both EFTs over- +24 + +Sum of the Weights +1.10- +1.05 +(x) + W2(x) +1.00 +0.95 +W1( +0.90 +Sum of Weights +Ex 1 +Ex 2 +0.85 +0.1 +0.2 +0.3 +0.4 +0.5 +Xestimate the true system, which renders a convex combination to be inappropriate. From these +observations, it appears the proposed model-mixing approach benefits by not imposing strict as- +sumptions, such as a simplex constraint, on the weights. +Additionally, one must use caution when interpreting the weight functions independently of +each other. With EFTs, the weight functions are generally correlated at a fixed input. This result +is intuitive in the first two examples where the predictive accuracy of the weak and strong coupling +expansions are inversely related. Thus, a joint interpretation is more appropriate in these problems. +Finally, the weight functions can also be used to better understand the M-open assumption +associated with the model set. An initial confirmation of the M-open setting can be made when the +weight functions noticeably change as a function of the inputs. This observation indicates localized +performance of each model, hence one can confirm the true system is not contained in the set. If +the weight functions are nearly constant, one may also wish to check the posterior of wsum(x) to +see if the sum of the weights is fixated close to one. Such a case may suggest model averaging with +a simplex constraint could also be an appropriate solution. This alone is not enough to confirm or +deny the M-open assumption, however it may indicate that the M-complete or M-closed labels are +possible classifications of the model set under consideration. A final case to consider is the situation +where a single model receives a weight near one while the effects of the competing models are shrunk +to zero across a subregion of the domain. This situation may indicate the model set is M-closed +conditional on the subregion of interest despite falling in the M-open case when considering the +entire domain at once. +In conclusion, this work proposes a Bayesian treed framework to mix predictions from a set of +competing models, each of which are intended to explain the physical system across a subregion +of the domain. This approach falls within the class of problems referred to as Bayesian model +mixing, as input-dependent weights are defined to reflect the localized behavior of each model. +The weight functions are modeled using a sum-of-trees and are regularized via a multivariate +Gaussian prior. The tree bases coupled with the regularization approach allows for the weights +to be learned in a flexible non-parametric manner free of strict constraints. +Using the weight +functions, predictions from the individual models are mixed via a linear combination. The success +of this mixing approach is demonstrated on three EFT examples, each of which considers models +with localized predictive performances. Leveraging the localized behavior of the individual models +leads to significant improvements in the posterior prediction and uncertainty quantification of f†(x) +compared to global weighting schemes. +25 + +Acknowledgements +The work of JCY and RJF work was supported in part by the National Science Foundation +under Agreement OAC-2004601. The work of MTP was supported in part by the National Science +Foundation under Agreements DMS-1916231, OAC-2004601, and in part by the King Abdullah +University of Science and Technology (KAUST) Office of Sponsored Research (OSR) under Award +No. OSR-2018-CRG7-3800.3. The work of TJS was supported in part by the National Science +Foundation under Agreement DMS-1564395 (The Ohio State University). +References +Bernardo, J. M. and Smith, A. F. (2009), Bayesian theory, Vol. 405, John Wiley & Sons. +Breiman, L. (1996), “Stacked regressions”, Machine learning 24(1), 49–64. +Burgess, C. P. 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(2021), “Bayesian hierarchical stacking: Some models +are (somewhere) useful”, Bayesian Analysis 1(1), 1–29. +Yao, Y., Vehtari, A., Simpson, D. and Gelman, A. (2018), “Using stacking to average Bayesian +predictive distributions”, Bayesian Analysis 13(3), 917–1007. +28 + +Appendix +Let ηpj denote the pth terminal node in the jth tree. +Without loss of generality, assume +(x1, y1), ..., (xnp, ynp) lie in the hyper-rectangle defined by ηpj. Furthermore, define each residual as +ri = yi − +� +q̸=j +ˆ +f +⊤(xi) g(xi, Tq, Mq), +i = 1, . . . , np +These are collected in an np dimensional vector Rpj = (r1, ..., rnp)⊤. Finally, let ˆF pj denote the +np × K matrix whose lth column is (f l(x1), ..., f l(xnp))⊤. Due to the independence and constant +variance assumptions, the model for the vector of residuals along with the associated priors is +defined by +Rpj | µpj, Tj, σ2 ∼ Nnp +� +ˆF pjµpj, σ2Inp +� +µpj | Tj +ind +∼ NK(βpj, Σ) +σ2 ∼ λν/χ2 +ν +where it is assumed Σ = τ 2IK. +The Marginal Likelihood +The marginal likelihood of the residuals in node ηpj is defined by +L(Rpj | Tj, σ2) = +� +L(Rpj | Tj, µpj, σ2)π(µpj | Tj) dµpj +(9) +Then, it follows, +L(Rpj | Tj, σ2) = +� +(2πσ2)−np/2 exp +� +− +1 +2σ2 (Rpj − ˆF pjµpj)⊤(Rpj − ˆF pjµpj) +� +× +(2πτ 2)−K/2 exp +� +− +1 +2τ 2 (µpj − βpj)⊤(µpj − βpj) +� +dµpj += (2πσ2)−np/2(2πτ 2)−K/2× +� � +exp +� +− +1 +2σ2 (R⊤ +pjRpj − 2µ⊤ +pj ˆF +⊤ +pjRpj + µ⊤ +pj ˆF +⊤ +pj ˆF pjµpj +� +× +exp +� +− +1 +2τ 2 (µ⊤ +pjµpj − 2µ⊤ +pjβpj + β⊤ +pjβpj) +� +dµpj +� += (2πσ2)−np/2(2πτ 2)−K/2 exp +� +− +1 +2σ2 R⊤ +pjRpj − +1 +2τ 2 β⊤ +pjβpj +� +× +� +exp +� +− 1 +2µ⊤ +pj +� 1 +σ2 ˆF +⊤ +pj ˆF pj + 1 +τ 2 IK +� +µpj + +� 1 +τ 2 βpj + 1 +σ2 ˆF +⊤ +pjRpj +�⊤ +µpj +� +dµpj. +29 + +Now let A−1 = +1 +σ2 ˆF +⊤ +pj ˆF pj + 1 +τ 2 IK and b = +� +1 +τ 2 βpj + 1 +σ2 ˆF +⊤ +pjRpj +� +. Substituting these terms into the +above expression yields +L(Rpj | Tj, σ2) = (2πσ2)−np/2(2πτ 2)−K/2 exp +� +− +1 +2σ2 R⊤ +pjRpj − +1 +2τ 2 β⊤ +pjβpj +� +× +(10) +� +exp +� +− 1 +2µ⊤ +pjA−1µpj + b⊤µpj +� +dµpj +Using Lemma B.1 from Santner et al. (2018) the integral simplifies as +� +exp +� +− 1 +2µ⊤ +pjA−1µpj + b⊤µpj +� +dµpj = (2π)K/2|A|1/2 exp +�1 +2b⊤Ab +� +. +(11) +Then, from (10) and (11), the marginal likelihood simplifies as +L(Rpj | Tj, σ2) = (2πσ2)−np/2(τ 2)−K/2|A|1/2 exp +� +− +1 +2σ2 R⊤ +pjRpj − +1 +2τ 2 β⊤ +pjβpj + 1 +2b⊤Ab +� +. += (2πσ2)−np/2(τ 2)−K/2 +���� +� 1 +σ2 ˆF +⊤ +pj ˆF pj + 1 +τ 2 IK +�−1���� +1/2 +× exp +� +− 1 +2 +� 1 +σ2 R⊤ +pjRpj + 1 +τ 2 β⊤ +pjβpj − b⊤Ab +�� +where b⊤Ab = +� +1 +τ 2 βpj + 1 +σ2 ˆF +⊤ +pjRpj +�⊤� +1 +σ2 ˆF +⊤ +pj ˆF pj + 1 +τ 2 IK +�−1� +1 +τ 2 βpj + 1 +σ2 ˆF +⊤ +pjRpj +� +. +The Posterior of µpj +Now consider the full conditional posterior distribution of the terminal node parameter µpj. +Using Bayes rule, +π(µpj | Rpj, Tj, σ2) ∝ L(Rpj | Tj, µpj, σ2)π(µpj | Tj) +A conjugate prior is assumed for µpj, thus the terms in the likelihood and prior can be rearranged +to obtain a Normal kernel for the posterior distribution. This process is summarized below. +π(µpj | Rpj, Tj, σ2) ∝ exp +� +− +1 +2σ2 (Rpj − ˆF pjµpj)⊤(Rpj − ˆF pjµpj) +� +× +exp +� +− +1 +2τ 2 (µpj − βpj)⊤(µpj − βpj) +� +∝ exp +� +− 1 +2 +� +µ⊤ +pj +� 1 +σ2 ˆF +⊤ +pj ˆF pj + 1 +τ 2 IK +� +µpj − 2µ⊤ +pj +� 1 +τ 2 βpj + 1 +σ2 ˆF +⊤ +pjRpj +��� +∝ exp +� +− 1 +2 +� +µ⊤ +pjA−1µpj − 2µ⊤ +pjA−1Ab +�� +where A−1 = +1 +σ2 ˆF +⊤ +pj ˆF pj + 1 +τ 2 IK and b = +1 +τ 2 βpj + 1 +σ2 ˆF +⊤ +pjRpj. The previous expression simplifies as +π(µpj | Rpj, Tj, σ2) ∝ exp +� +− 1 +2(µpj − Ab)⊤A−1(µpj − Ab) +� +30 + +This is the kernel of a Multivariate Gaussian distribution with mean Ab and covariance matrix A. +Thus it follows +µpj | Rpj, Tj, σ2 ind +∼ NK +� +Ab, A +� +replacing A and b with their respective definitions implies +µpj | Rpj, Tj, σ2 ind +∼ NK +�� 1 +σ2 ˆF +⊤ +pj ˆF pj + 1 +τ 2 IK +�−1� 1 +τ 2 βpj + 1 +σ2 ˆF +⊤ +pjRpj +� +, +� 1 +σ2 ˆF +⊤ +pj ˆF pj + 1 +τ 2 IK +�−1 +� +The Posterior Distribution of σ2 +Finally, consider the full conditional posterior for the error variance, which is defined by +π(σ2 | Y , T, M) ∝ L(Y | T, M, σ2)π(σ2) +where Y = (y1, ..., yn)⊤, T = {T1, ..., Tm}, and M = {M1, ..., Mm}. +Further, assume a conjugate prior for σ2, namely σ2 ∼ νλ/χ2 +ν which has a probability density +function defined by +π(σ2) = (ν/2)ν/2 +Γ(ν/2) λν/2(σ2)−(ν/2+1) exp +� +− νλ +2σ2 +� +Due to conjugacy, the full conditional distribution is given by +π(σ2 | Y , T, M) ∝ (σ2)−n/2 exp +� +− +1 +2σ2 +n +� +i=1 +� +yi − ˆ +f +⊤(xi)w(xi) +�2� +(σ2)−(ν/2+1) exp +� +− νλ +2σ2 +� +∝ (σ2)−(n/2+ν/2+1) exp +� +− +1 +2σ2 +� +n +� +i=1 +� +yi − ˆ +f +⊤(xi)w(xi) +�2 ++ νλ +�� +This is the kernel of another scaled inverse-χ2 distribution, namely σ2 ∼ ν′λ′/χ2 +ν′ where +ν′ = n + ν +and +λ′ = +1 +n + ν +� +n +� +i=1 +� +yi − ˆ +f +⊤(xi)w(xi) +�2 ++ νλ +� +31 + +Supplementary Material +An Overview of EFT +EFTs model physical systems by an infinite expansion of terms organized in order of decreasing +importance according to the power counting principle (Burgess, 2020; Petrov and Blechman, 2016; +Georgi, 1993). Exact theoretical predictions of the system are obtained by summing over these +terms. In practice, only a finite number of lower-order terms are known. Thus, the theoretical +prediction can be decomposed using a Taylor-like series which includes the known finite-order +expansion along with the induced truncation error. Predictions of experimental quantities can then +be represented using an additive model +Y (x) = f†(x) + ϵ(x) +f†(x) = h(N)(x) + δ(N)(x) +where x ∈ Rd denotes an independent variable associated with the system, h(N)(x) represents the +known finite-order expansion of degree N, δ(N)(x) is the associated truncation error, and ϵ(x) is +the random observational error. The accuracy of the finite-order expansion may vary significantly +across a subspace of the domain. For example, a finite-order expansion centered about zero may +yield a high fidelity approximation in the lower regions of the domain. However, the accuracy of +the prediction quickly degrades in higher regions of the domain. +It is further assumed the finite-order expansion can be modeled as a stochastic process. First, +the finite-order expansion can factorized as +h(N)(x) = yref(x) +N +� +k=0 +ck(x)Qk(x), +(12) +where yref(x) sets the scale of variation, c0(x), ..., cN(x) are dimensionless observable coefficients, +and Q(x) is a dimensionless expansion parameter. When the scale and expansion parameters are +known based on theoretical arguments, the coefficients c0(x), ..., cN(x) appear to behave as a set of +independent and identically distributed curves from a stochastic process. Thus, a common model +for the coefficients is a Gaussian process +ck(x) | θ ∼ GP(µ, ¯c2r(x, x′; ℓ)) +(13) +θ = (µ, ¯c2, ℓ), +32 + +where µ denotes a constant mean function and ¯c2r(x, x′; ℓ) represents the covariance function (Me- +lendez et al., 2019). A common assumption is to set µ = 0, while prior distributions can be assigned +to the remaining parameters in the model (Melendez et al., 2019). Additionally, a likelihood can be +formed by collecting nc evaluations of the finite-order expansion, h(N) = (h(N)(xc +1), . . . , h(N)(xc +nc))⊤, +at design inputs xc +1, . . . , xc +nc. These model runs are used to extract the observed finite-order coef- +ficients, which are modeled via (13). Using the priors and the likelihood based on the model runs, +the parameters in the GP are then learned through standard Bayesian inference. +The truncation error accounts for the remaining unknown terms in the series, thus δ(N)(x) is +modeled using a similar factorization +δ(N)(x) = yref(x) +∞ +� +k=N+1 +ck(x)Qk(x). +(14) +Using (13) and (14) along with properties of the multivariate Normal distributions (Ravishanker +et al., 2021), the induced prior on the truncation error term is given by +δ(N)(x) | θ, Q ∼ GP +� +mδ(x), ¯c2Rδ(x, x′; ℓ) +� +, +(15) +with mean and covariance functions +mδ(x) = yref(x)QN+1(x) +1 − Q(x)µ +(16) +Rδ(x, x′; ℓ) = yref(x)yref(x′)[Q(x)Q(x′)]N+1 +1 − Q(x)Q(x′) . +(17) +The unknown parameters in (15) - (17) originate from the coefficient model in (12). Thus, the mean +and covariance functions which characterize the discrepancy model are also learned using the set +of nc evaluations of the finite-order expansion. This is a unique property of EFTs, as observational +data is not required to learn the model discrepancy. +When the finite-order expansion is computationally inexpensive to evaluate, the induced prior +on the theoretical predictions, f(x) = h(N)(x) + δ(N)(x) is given by +f(x) | θ, Q, h(N) ∼ GP +� +mth(x), Σth(x, x′) +� +, +where mth(x) = h(N)(x) + mδ(x) and Σth(x, x′) = ¯c2Rδ(x, x′; ℓ). In the expensive case, a GP can +be used to emulate the finite-order expansion and is defined by +h(N)(x) | θ, Q ∼ GP +� +mN(x), ¯c2RN(x, x′; ℓ) +� +. +The resulting prior on the theoretical prediction is a GP with mean and covariance functions +mth(x) = mN(x) + mδ(x) and Σth(x, x′) = ¯c2RN(x, x′; ℓ) + ¯c2Rδ(x, x′; ℓ). In either case, given a set +of model runs h(N), one can obtain posterior predictions ˆf(˜x1), . . . , ˆf(˜xm) at new inputs ˜x1, . . . , ˜xm. +33 + diff --git a/hNE0T4oBgHgl3EQfXwAk/content/tmp_files/load_file.txt b/hNE0T4oBgHgl3EQfXwAk/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ac76fdfd00b35b78b04ae3e482c7a182ecc3c833 --- /dev/null +++ b/hNE0T4oBgHgl3EQfXwAk/content/tmp_files/load_file.txt @@ -0,0 +1,1074 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf,len=1073 +page_content='Model Mixing Using Bayesian Additive Regression Trees John C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Yannotty, Thomas J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Santner, Richard J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Furnstahl, and Matthew T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Pratola The Ohio State University January 9, 2023 Abstract In modern computer experiment applications, one often encounters the situation where vari- ous models of a physical system are considered, each implemented as a simulator on a computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' An important question in such a setting is determining the best simulator, or the best combi- nation of simulators, to use for prediction and inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Bayesian model averaging (BMA) and stacking are two statistical approaches used to account for model uncertainty by aggregating a set of predictions through a simple linear combination or weighted average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Bayesian model mixing (BMM) extends these ideas to capture the localized behavior of each simulator by defin- ing input-dependent weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' One possibility is to define the relationship between inputs and the weight functions using a flexible non-parametric model that learns the local strengths and weaknesses of each simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' This paper proposes a BMM model based on Bayesian Additive Regression Trees (BART).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The proposed methodology is applied to combine predictions from Effective Field Theories (EFTs) associated with a motivating nuclear physics application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Keywords: Computer Experiments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Effective Field Theories;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Model stacking;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Uncertainty quantifi- cation 1 Introduction In statistical learning problems, one often considers a set of plausible models, each designed to explain the system of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' A common practice is to select a best performing model based on some pre-specified criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The ensuing inference for quantities of interest is then carried out using the selected model as if it were the true data generating mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The resulting uncertainty 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='02296v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='ME] 5 Jan 2023 quantification ignores any variability due to the underlying model structure (Draper, 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The misrepresentation of uncertainties associated with such quantities can ultimately lead to misguided interpretation or inappropriate decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Another shortcoming of the typical approach to modeling is that the resulting inference may strongly depend on the selection criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' In other words, different sets of criteria could lead to noticeably different final models and inferential results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' To account for such uncertainties, one may elect to combine information across the set of models in some manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Any model set can be classified into one of three categories: M-closed, M-complete, and M- open (Bernardo and Smith, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The M-closed setting assumes the true model, M†, can formally be defined and is contained within the set of models under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' In this setting, model selection is appropriate because M† can be recovered from the set of models under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The M-complete setting describes the case where M† can formally be defined, however it is not contained in the model set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Similarly, the M-open case assumes the true model exists and is excluded from the model set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' However, in this situation M† is further assumed to be intractable and thus cannot be formally defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Model selection is inappropriate in the latter two cases because one will inevitably select the wrong model to perform inference while simultaneously ignoring the uncertainty induced by this error (Bernardo and Smith, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' This work is motivated by applications in nuclear physics which tend to fall within the M-open class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Assume a set of K models are considered when studying a particular system of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' One approach to account for model uncertainty is to combine the information across these K mod- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' This may involve combining the individual point predictions or probability density functions from each model, usually in some additive manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Traditional approaches utilize global weight- ing schemes, where each model is weighted by a value intended to reflect overall (global) model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' In the Bayesian paradigm, the classical global weighting scheme is Bayesian model averaging (BMA) (Raftery et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=', 1997), which combines the individual posterior densities from each model using a convex combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The BMA weights are given by the individual posterior model probabilities, each which can be interpreted as the probability the individual model is the true data generating one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Hence, BMA implicitly assumes the true model is contained within the model set, which renders this method inappropriate outside of the M-closed setting (Draper, 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' More recent Bayesian global weighting schemes adopt a model stacking approach, where model weights are assigned to minimize a specified posterior expected loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' This decision theory viewpoint of global weighting can be used for combining point predictions (Le and Clarke, 2017) or probabil- ity densities (Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Regardless of the implementation, Bayesian stacking methods are 2 Figure 1: Three different EFT experimental settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Each panel displays the true physical system (solid) and the EFTs under consideration (non-solid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' appropriate for both the M-open and M-closed settings because the weights are chosen based on some pre-specified criteria and do not share the probabilistic interpretation of BMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Though global weighting methods are effective, they still might lead to poor approximations of the true system when the individual model performance is localized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' In such a case, one may wish to select a weighting scheme that reflects the localized characteristics of the models by constructing input-dependent weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' With input-dependent weights, one would expect an individual model to receive a higher weight in input regions where it exhibits strong predictive performance, while receiving a weight close to zero in regions of poor performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Localized weighting schemes are more appropriate for the M-open or M-complete settings where the true model is better characterized as a localized mixture of the model set under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' This work is motivated by problems in nuclear physics modeled using a technique known as Effective Field Theory (EFT) (Burgess, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Petrov and Blechman, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Georgi, 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' EFTs are designed to perform well in a particular subregion(s) of the input domain, yet diverge in the rest of the input domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Prototypes of such models are the weak and strong coupling finite- order expansions for the partition function of the zero-dimensional φ4 theory presented by Honda (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Examples of this problem are shown in Figure 1 where the various dashed and dotted lines represent the mean predictions from a finite-order expansion and the solid line denotes the true physical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' One can see that these models are highly accurate descriptions of the true system in some regions of the domain, yet they are unable to provide a globally accurate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Most EFT problems are examples of the M-open setting, in that the true underlying description of the 3 (a) (b) (c) 5 5 5 - / 4- 4- / 4 - 1 3- 3- / 3- M F(x) F(x) F 2 - 2 2 - f2(x) 1 21 (4 (x) (x) 1 () 1 1 f+(x) f+(x) f+(x) 0 0- 0- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='5 X X Xsystem across the entire domain of interest is intractable and thus it is not contained within the model set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Therefore, multiple EFTs are constructed to recover the true system across subsets of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' To demonstrate why problems falling in the M-open class may not be suited for model aver- aging schemes, consider applying BMA to the model set involving the two expansions as shown in Figure 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The resulting posterior mean prediction from BMA still results in a poor estimate of the true system as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Essentially, BMA selects the dashed model rather than leveraging the localized strengths contained in the model set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Figure 2: The posterior mean prediction of f†(x) when applying BMA to the 2nd order weak and 4th order strong coupling expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Given the characteristics of EFTs and the M- open setting associated with these problems, a simple weighted average of the predictions from each model is insufficient for recovering the true physical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' A better approach is to use an input-dependent weighting scheme which lever- ages the localized behaviors of each model to ascertain appropriate mean prediction and un- certainty quantification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Such an approach falls under the general class of problems known as Bayesian model mixing (BMM) (Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' A key challenge in BMM is to define the rela- tionship between the inputs and the model weight functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' This work proposes a Bayesian treed model which specifies the weight functions as a sum-of-trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' This representation relies on a tree basis of weak learners which are used to capture the localized model behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Additionally, this flexible and non-parametric approach allows the user to avoid having to specify a more restrictive model for the weight functions, such as a generalized linear model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Maintaining the traditional con- jugacy properties associated with Bayesian Additive Regression Tree (BART) models, the weight functions are regularized via a multivariate Gaussian prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The prior is calibrated so that the weight functions prefer the interval [0, 1] without imposing any further constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Additionally, this framework includes a simple strategy for incorporating prior information about localized model performance when available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' All together, this approach highlights the localized behaviors of the candidate models and yields significant improvements in prediction, interpretation, and uncertainty 4 4 3 2 - (x)(t) f+(x) Post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='5 Xquantification compared to traditional model averaging methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The remainder of the paper is organized in the following manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Section 2 highlights some relevant work related to model averaging and model mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Section 3 describes the essential properties of EFTs, while Section 4 outlines the specifics of the proposed BART-based framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Three motivating EFT examples are presented in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Finally, Section 6 provides a detailed discussion of the results presented throughout this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' 2 Background Methods to address model uncertainty have been widely studied throughout the past few decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The majority of work in this area strives to combine competing models through either mean or density estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' In either case, the combined result is generally found by taking a linear combination of the individual predictive means or densities from the models under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The weights in this linear combination may or may not depend on the inputs for each model and are learned using the set of training data D = {(x1, y1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' , (xn, yn)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' This section briefly reviews some of the popular model averaging and model mixing techniques currently available in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Bayesian Model Averaging A classical approach for combining models M1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' ,MK is Bayesian Model Averaging (Raftery et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=', 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Suppose Q is a quantity of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The posterior density of Q is defined using a convex combination of the posterior densities under each model, π(Q | D) = �K l=1 wl π(Q | D, Ml).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Each weight is defined in terms of its corresponding model evidence, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' wl = π(Ml | D) where π(Ml | D) = p(D | Ml)π(Ml) �K k=1 p(D | Mk)π(Mk) and p(D | Ml) is the marginal likelihood of the data with respect to the lth model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Though BMA is useful, it has been criticized for emphasizing a fit to the training data as opposed to out-of-sample prediction, asymptotically selecting a single model (inappropriate in the M-complete and M-open settings, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Figure 2), and for tending to be sensitive to the prior model probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Bayesian Mean Stacking One popular approach for mean estimation is Stacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Some of the earlier works in stacking focused on frequentist approaches for combining point predictions (Breiman, 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' More recent work has extended stacking to the Bayesian paradigm (Clyde and Iversen, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Le and Clarke, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Given K competing models, the stacked mean for a future observation ˜y at input ˜x is 5 constructed as a linear combination of individual model predictors E[˜y | ˜x, D] = K � l=1 wl fl(˜x), where E[˜y | ˜x, D, Ml] = fl(˜x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' When the individual models are unknown, stacking is conducted in a two-step procedure: (i) independently fitting the individual models Ml, l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' K, given the set of training data D, and (ii) estimating the weights w = (w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' , wK)⊤ for the stacked predictor given the fitted models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' In the first step, each model is fit and their corresponding mean predictions, ˆfl(xi), are obtained at each of the training points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' In practice, cross validation techniques are used in this stage to reduce the risk of overfitting the stacked predictor to the data since each model is trained using the identical dataset D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' In the second step, the coefficients w = (w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' , wK)⊤ are found via optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' In the Bayesian paradigm, the weights are selected to minimize the posterior risk for some pre-specified loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' For example, the squared error loss between the observed data and the stacked predictor at new point ˜x is given by �w = argminw � � ˜y − �K l=1 wl ˆfl(˜x) �2 p(˜y | ˜x, D) d˜y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Using the observed data, the posterior risk can be approximated with leave-one-out (LOO) cross validation techniques which allows the stacking weights to be selected by �w = argminw �n i=1 � yi − �K l=1 wl ˆf(−i) l (xi) �2 , where ˆf(−i) l (xi) is the LOO cross-validation prediction for the ith observation using the lth model as discussed in Le and Clarke (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Additionally, one may choose to modify this loss function to impose a set of conditions on the weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' For example, one may impose a simplex, non-negativity, or sum-to-m constraint on the weights (Le and Clarke, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Other approaches include regularization via a penalty term or a prior (Breiman, 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Yang and Dunson, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Such approaches can be useful when the individual model predictions are highly correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Bayesian Complete Stacking Complete Stacking was introduced in 2018 by Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' (2018) and motivated by the shortcomings of BMA (Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Their proposed Bayesian stacking model emphasizes prediction, as the weights are selected to minimize the Kullback-Leibler (KL) divergence between the true predictive density and the stacked predictive density p(˜y | ˜x) = K � l=1 wl p(˜y | ˜x, D, Ml), where ˜y is a future observation with input ˜x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Similar to mean stacking, the leave-one-out (LOO) cross validated predictive density can be used in place of p(˜y | ˜x, D, Ml) when the individual models 6 are unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Given training data, the weights are constrained to a K−dimensional simplex SK and estimated as �w = argmaxw∈SK �n i=1 log �K l=1 wl p(yi | xi, D(−i), Ml), where D(−i) denotes the training set excluding the pair (xi, yi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Bayesian Hierarchical Stacking Hierarchical Stacking is a model mixing approach which extends Complete Stacking by defining input-dependent weights that are estimated in a fully Bayesian manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' One way to define the relationship between the inputs and the weight functions is through a parametric model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' In this case, one models K − 1 unconstrained weight functions, w∗ l (xi) = µl + J � j=1 αlj gj(xi), which depend on the sets of hyperparameters {αlj} and {µl} along with user-specified basis func- tions gj(xi), where j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' , J and l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' , K − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The Kth function w∗ K(xi) is set to 0 to serve as a baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Then, a softmax transformation is applied to the unconstrained weights in order to confine each model weight to the K-dimensional simplex, namely wl(x) = exp � w∗ l (x) � exp � w∗ 1(x) � + · · · + exp � w∗ K(x) � for l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' , K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The methods discussed above outline a number of strategies one can take to combine the information across multiple models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' In the setting of EFT experiments, the localized nature of the predictions suggests an input-dependent weighing scheme like Bayesian Hierarchical Stacking is more suitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' However, specifying the required basis functions may not be trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Thus, the proposed method will adopt the notion of mean stacking within an additive tree basis model to achieve localized weighting in a flexible and non-parametric manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' But first, the motivating EFT experimental setting is reviewed in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' 3 Towards Model Mixing with EFTs Computer models such as EFTs are typically implemented as simulators which are motivated by the known physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The theoretical predictions of the physical system are approximations from each simulator plus a discrepancy term, which is designed to account for the remaining unexplained portions of a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' These two components may have specific properties which can be leveraged when working with observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' This section summarizes these details in the context of EFTs, while a further discussion is provided in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='1 Motivating EFT Example Consider the EFT example where the true physical system is the partition function of the zero-dimensional φ4 theory defined by f†(x) = � ∞ −∞ exp � − u2 2 − x2u4� du, (1) where x denotes the coupling constant (Honda, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Two types of finite-order expansions exist for this partition function and are given by (2) and (3) for ns or nl ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' h(ns) s (x) = ns � t=0 stxt where st = � � � � � � � √ 2Γ(t+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='5) (t/2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' (−4)(t/2) t is even 0 t is odd (2) h(nl) l (x) = nl � t=0 ltx−t where lt = Γ(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='5t + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='25) 2t!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' � − 1 2 �t , t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=', nl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' (3) The weak coupling expansion in (2) is an asymptotic Taylor-like series of order ns centered about zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Thus, h(ns) s (x) will yield high fidelity predictions for smaller coupling constants and diverge as the value increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The reverse behavior is observed for the strong coupling expansion in (3), h(nl) l (x), which is convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Example predictions of the physical system using these finite-order expansions can be seen in Figure 1 and are discussed in detail in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The theoretical predictions of the physical system using the weak and strong coupling expansions are expressed using (4) and (5), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' f(ns) s (x) = h(ns) s (x) + δ(ns) s (x) (4) f(nl) l (x) = h(nl) l (x) + δ(nl) l (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' (5) where the truncation errors δ(ns) s (x) and δ(nl) l (x) are modeled with Gaussian processes (GPs) (Gra- macy, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Santner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' As described by Melendez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' (2019), the parameters in both truncation error models are dependent upon the evaluations of their corresponding finite-order expansions (described in (2) and (3), respectively) over a sparse grid of points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The discrepancy model also depends on physical quantities, Q(x) and yref(x), which are chosen based on domain expertise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The relationship between these quantities and the discrepancy are summarized in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' When Q(x) and yref(x) are unknown, one can alternatively use the error approximation described by Semposki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The features present in this example from Honda (2014) are commonly found across the land- scape of EFT problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' For instance, the physical system can be expressed as an additive model 8 involving a finite-order expansion and the induced truncation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The finite-order expansions are designed to provide high fidelity predictions in specific subregions of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' There exists a subregion of the domain where none of the finite-order expansions yield accurate theoretical pre- dictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' All together, this motivating example serves as a prototype for the EFTs that may be encountered in a general experimental setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='2 The Model Set for EFT Experiments One may encounter various experimental settings when working with EFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Such scenarios are introduced in the context of the motivating example presented in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' First, consider the most basic case where the model set contains a single EFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' With one EFT, the overall predictive accuracy of the true system is poor, despite the good performance in a localized region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' For example, suppose the model set M contains the 2nd order weak coupling expansion f(2) s (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Mean predictions constructed from (2) and (4) are shown by the dashed line in Figure 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Clearly, this model is limited to strong predictive accuracy in only the left subregion of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' When available, one can consider different finite-order approximations of the same EFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' For example, consider the 2nd, 4th, and the 6th order coupling expansions which are shown in Fig- ure 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The three models are very similar for lower coupling constants yet drastically differ in the remainder of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Despite each expansion’s poor theoretical predictions, one can still leverage the available information to improve the overall prediction of the physical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' For instance, the 2nd and 6th order expansions (dashed and dashed-dotted) are concave functions while the 4th order expansion (dotted) is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' This suggests the true physical system lies between the expansions under consideration and can be recovered by re-weighting the corresponding predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' A third situation is to consider multiple EFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' In this example, one can consider various finite- order strong coupling expansions in addition to the weak coupling expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' For example, a model set can contain a finite-order weak coupling expansion (dashed) and the 4th order strong coupling expansion (dotted) as shown in Figures 1(a) and 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The addition of the strong cou- pling expansion allows for a high fidelity prediction of the physical system to be considered in the rightmost subregion of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The model set listed in panel (a) implies the true system lies between the two expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' This is particularly useful in the intermediate range where neither of the EFTs are accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Meanwhile, the set in panel (b) presents an interesting case where the physical system lies below both EFTs in the intermediate range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' In this case, the information in the observational data can be leveraged to help recover the true system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' 9 In this example, the predictions from the weak coupling expansion degrade slowly compared to those from the strong coupling expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Consequently, the weak coupling expansions generally appear to have a better overall predictive performance across the entirety of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' When combining these two types of EFTs using global weighting schemes such as BMA, the resulting prediction will favor the weak coupling expansion due to its drastic advantage in the overall model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The undesirability of the BMA solution is evident in Figure 2, which demonstrates that BMA effectively matches the 2nd order weak coupling expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Hence, a weighting scheme which captures the localized behaviors of each model is preferred in the EFT setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The proceeding sections consider a general set of K different EFTs, which are denoted by f1(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' , fK(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' In this motivating example, fl(x) = hl(x) + δl(x) where hl(x) can denote either a weak or strong coupling expansion of order Nl, where l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' , K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Meanwhile, δl(x) is the associated truncation error and is modeled by a GP as described in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='3 A Two-Step Approach for EFTs Similar to Bayesian mean stacking, a two-step approach is adopted for combining the predictions across K EFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' When constructing a two-step stacking approach, it is important to understand the available sources of information and where each can be leveraged throughout the estimation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The first source of information is the observational data, Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=', Yn which are assumed to be independent random variables generated at fixed inputs x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=', xn according to Yi = f†(xi) + ϵi, ϵi iid ∼ N(0, σ2) where f†(xi) represents the true and unknown physical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The information from the field data is collected in the set D = {(x1, Y1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' , (xn, Yn)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Meanwhile, each EFT is also associated with its own set of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' For example, consider the lth EFT denoted by fl(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' It is assumed this EFT is accompanied by a set of simulator runs across a fixed set of inputs xc l1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' , xc lnl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Information regarding the design of the computer experiment for each EFT can be found in Melendez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The simulator runs are evaluations of the finite-order expansion, hl(·), at the specified inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Using these model runs, one can extract the set of Nl + 1 coefficients c0(·), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' , cNl(·) at each of the fixed inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The training set for the lth EFT is then defined by Dl = �� xc l1, C(xc l1) � , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' , (xc l1, C(xc lnl) �� where C(·) denotes the vector of known finite-order coefficients at the specified model input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The 10 resulting coefficients and the set of inputs can differ across the K models, thus the sets D1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' , DK will contain different information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The proposed two-step approach is tailored to EFTs by taking advantage of the sources of data described above as well as the properties described in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The proposed Bayesian mean stacking approach extends the traditional methodologies by incorporating input-dependent weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Conditional on the values of the theoretical predictions at a given point, f1(xi), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' , fK(xi), this model can be defined by Yi | f(xi), w(xi), σ2 ind ∼ N � f ⊤(xi)w(xi), σ2) (6) where f(xi) = � f1(xi), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=', fK(xi) �⊤ and w(xi) = (w1(xi), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=', wK(xi))⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' In practice, these values are unknown and must be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' This problem serves as the first step of the stacking procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' This first step fits each model, fl(x), independently given data, Dl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' As described in the supple- mentary material, an EFT is fit using the finite-order coefficients to learn the unknown parameters which characterize the GP assigned to the truncation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' All of this information can be extracted from Dl which implies the set of field observations is not required to fit each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Consequently, the desired theoretical predictions can be estimated without using any of the information in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' This differs from existing statistical approaches as summarized in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Prior to learning the weights, point predictions from each EFT are required at x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' , xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The predictions for an EFT are computed through the posterior predictive mean which is given by ˆ f l(xi) = ˆEπ|Dl� fl(xi) � , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Additionally, the posterior variance of the predictions can be extracted if desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' This posterior predictive distribution is a Gaussian process which can be characterized by the corresponding mean and covariance functions as described in Melendez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' (2019) (see also the supplementary material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The objective of the second stage of the stacking procedure is to estimate the weight functions shown in (6), which is the focus of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Conditional on the mean predictions from the first step, the model for the observational data becomes Yi | ˆ f(xi), w(xi), σ2 ind ∼ N � ˆ f ⊤(xi)w(xi), σ2) where ˆ f(xi) = � ˆf1(xi), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' , ˆfK(xi) �⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The weight functions are then learned using the information contained in the field data D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The next section outlines the proposed model mixing scheme which defines the weight functions using Bayesian Additive Regression Trees (BART).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' 11 4 Bayesian Additive Model Mixing Trees 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='1 Bayesian Tree Models Bayesian tree models have become increasingly popular for modeling complex and high dimen- sional systems (Chipman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=', 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Bayesian additive regression trees are used to model an unknown mean function, E[Y | x] (Chipman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' This additive approach involves sum- ming together the predictions made from m trees and is facilitated through a Bayesian backfitting algorithm (Hastie and Tibshirani, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Each tree Tj is characterized by its structure, comprised of internal and terminal nodes, along with its associated set of terminal node parameters, Mj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The internal nodes define binary partitions of the input space according to a specified splitting rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The prior probability a node is internal is p(η is internal) = α(1 + dη)−β where dη is the depth of the node η, while α and β are tuning parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' By construction, this prior penalizes tree complexity and thus ensures each tree maintains a shallow and simple structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Given d different predictors, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=', xd, splitting rules are of the form xv < c for v ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=', d} and cutpoint c from a discretized subset of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' In the simplest approach, the predictor and cutpoint associated with each splitting rule are randomly selected from discrete uniform distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The probabilities associated with the designation of each node along with the splitting rules for internal nodes are used to define the stochastic tree-generating prior for each tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The m trees are learned through MCMC, where a slight modification to each structure is proposed at every iteration of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Generally, such modifications to the tree include birth, death, perturb, or rotate as described by Chipman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' (1998) and Pratola (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Proposals are then accepted or rejected using a Metropolis-Hastings step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' To avoid a complex reversible jump MCMC, the algorithm depends on the integrated likelihood, which is obtained by integrating over the terminal node parameters associated with the given tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' A closed form expression for this density can be obtained with conditional conjugate priors for the terminal node parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Given the tree structure, prior distributions can be assigned to each terminal node parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' In the BART model, the priors ensure each tree explains a small yet different source of variation in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' For continuous data, BART assigns Gaussian priors to the terminal node parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Assuming the data is mean centered, the prior assigned to terminal node parameter µpj in node ηpj is given by µpj | Tj ∼ N(0, τ 2) where τ = (ymax − ymin)/(2k√m) with tuning parameter k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Additionally, a conjugate scaled-inverse-χ2 prior is assigned to the variance parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The traditional Bayesian regression tree model can be extended to allow for a more complex 12 structure in the terminal nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Existing extensions include linear regression (Chipman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=', 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Prado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=', 2021) and Gaussian processes (Gramacy and Lee, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' For the setting of model mixing, this work extends BART to a multivariate Gaussian terminal node model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='2 Model Mixing with BART The weight functions w(x) = (w1(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' , wK(x))⊤ are modeled using a sum-of-trees w(xi) = m � j=1 g(xi, Tj, Mj), (7) where g(xi, Tj, Mj) is the K-dimensional output of the jth tree using the set of terminal node parameters, Mj, at the input, xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' This approach defines the weight functions using tree bases which are learned from the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The amount of flexibility in the weight functions can be controlled by changing the number of trees or tuning the hyperparameters in the prior distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' In this application of BART, each terminal node parameter is a K-dimensional vector which is assigned a multivariate Gaussian prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The parameter is regularized so that each tree accounts for a small amount of variation in the weight functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' For the proceeding statements, let ηpj represent the pth terminal on the jth tree and define its corresponding parameter by µpj = (µpj1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=', µpjK)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Now assume the observations (x1, y1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=', (xnp, ynp) lie in the hyper-rectangle defined by ηpj, where np is the number of observations assigned to this subregion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The model at each terminal node amounts to fitting a localized Bayesian linear regression with parameter vector µpj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Due to condi- tional independence, the likelihood in this node is defined by L(r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=', rnp | Tj, µpj, σ2) = (2πσ2)−np/2 exp � − 1 2σ2 np � i=1 � ri − ˆ f ⊤(xi)µpj �2� where ˆ f(xi) = � ˆf1(xi), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=', ˆfK(xi) �⊤ is a vector of mean predictions from each EFT and ri is the ith residual given by ri = yi − � q̸=j ˆ f ⊤(xi)g(xi, Tq, Mq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Conditional on the tree structure, Tj, the terminal node parameter, µpj is assigned a conjugate multivariate Gaussian prior, namely µpj | Tj ind ∼ NK � β, τ 2IK � (8) where β = (β1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=', βK)⊤ is a K-dimensional mean vector and IK is the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' This prior is non-informative in the sense that the mean is fixed regardless of how the input space is partitioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' In model mixing, each simulator may perform strongly in one subregion of the input space but weakly in another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' This belief can be reflected in the prior distribution of µpj by allowing the 13 hyperparameters to depend on the partition of input space assigned to the given terminal node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Thus, an informative prior for µpj can be constructed as µpj | Tj ind ∼ NK � βpj, τ 2IK � where βpj = (βpj1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=', βpjK)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' This allows the prior mean to vary depending on the tree partitions and thus reflect some sense of localized model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Meanwhile, the assumed covariance structure implies the K vector components µpj1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=', µpjK are independent apriori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Both of the proposed priors are conjugate, which is an important choice in BART, as it allows for a closed form expression for the marginal likelihood for the vector of residuals Rpj = (r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='., rnp)⊤, Additionally, the conjugate priors result in closed form expressions for the full conditional distribu- tions of the terminal node parameters and the error variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The derivations of these distributions are found in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' In particular, the full conditional distribution for the pth terminal node in Tj is given by µpj | Rpj, Tj, σ2 ind ∼ NK �� 1 σ2 ˆF ⊤ pj ˆF pj + 1 τ 2 IK �−1� 1 τ 2 βpj + 1 σ2 ˆF ⊤ pjRpj � , � 1 σ2 ˆF ⊤ pj ˆF pj + 1 τ 2 IK �−1 � where ˆF pj is the np × K design matrix with the ith row vector given by the vector ˆ f ⊤(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Mean- while, the full conditional distribution for σ2 is a scaled-inverse-χ2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' σ2 ∼ ν′λ′/χ2 ν′ where ν′ = n + ν and λ′ = 1 n + ν � n � i=1 � yi − ˆ f ⊤(xi)w(xi) �2 + νλ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='3 Calibrating Priors First consider the prior for the terminal node parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The calibration of the hyperparam- eters differs for the non-informative and informative priors, however both approaches are designed to ensure that each model weight wl(x) should prefer the interval [0, 1] and be centered at a value within this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Moreover, the functions w1(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' , wK(x) are assumed to be independent apriroi at a fixed input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' This enables the prior for each weight to be calibrated marginally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='1 Non-Informative Prior Consider a non-informative prior for the terminal node parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' In this setting, µpj | Tj iid ∼ NK � β, τ 2IK � for the pth terminal node parameter in the jth tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' First, fix l ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=', K} and i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=', n} to calibrate the prior for wl(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Since the terminal node parameters are indepen- dent and identically distributed with a diagonal covariance structure, the prior induced on wl(xi) 14 is the same for the remaining weight and input combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' From (7) and (8), the induced prior on the lth model weight is wl(xi) ∼ N(mβl, mτ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Since it is believed wl(xi) ∈ [0, 1] with high probability, it is plausible to set mβl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Consequently, βl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='5/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Thus, each weight has an equal chance to reach the “extreme” values of 0 or 1 regardless of the input location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The prior standard deviation, τ, can be selected so that wl(xi) ∈ [0, 1] with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' To do this, a confidence interval for wl(xi) is constructed such that 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='5 − kτ√m and 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='5 + kτ√m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Subtracting the first equation from the second and solving for τ yields τ = 1 2k√m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' This calibration approach is very similar to the one proposed by Chipman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The main difference is due to the context of the problem, as it is believed the weights are predominately contained in an interval [0, 1] rather than the observed range of the data, [ymin, ymax].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Note, this approach can also be generalized to situations where the weights are assumed to prefer the interval [a, b] with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Such a belief would imply τ = (b − a)/(2k√m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='2 Informative Prior In the informative setting, the prior mean directly depends on the partitions of the input space induced by the given tree, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' µpj | Tj ∼ NK � βpj, τ 2IK � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' This prior is tailored towards EFTs, where the functional variance, vl(xi), indicates the severity of the truncation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' A larger variance within a particular subregion of the domain indicates the presence of larger truncation error meaning the EFT provides a poor approximation of the true system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' In this setting, the induced prior on the lth model weight is given by wl(xi) ∼ N(βl(xi), mτ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The function, βl(xi) is interpreted as the prior mean weight function and can be defined in terms of the sum-of-trees by βl(xi) = m � j=1 Bj � b=1 βbjl1(xi ∈ ηbj) where Bj is the number of terminal nodes in Tj and 1(xi ∈ ηbj) is the indicator that xi is assigned to the terminal node ηbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' To calibrate the informative prior, first obtain an initial estimate of βl(xi) at each input i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=', n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' One strategy is to utilize a set of normalized precision weights (Phillips et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Thus, for each l ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=', K} and i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=', n}, βl(xi) can be estimated by a ratio of precisions, βl(xi) = 1/vl(xi) 1/v1(xi) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' + 1/vK(xi) 15 where vl(xi) is the variance of the lth model at xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' This precision weighting scheme encodes prior knowledge about each model’s relative strengths and weaknesses into the model mixing frame- work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Since the prior of the terminal node parameter changes conditional on the tree structure, each βpj is chosen separately from the other terminal node parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Moreover, because the terminal node parameter µpj is assigned a diagonal covariance structure, each of the K vector com- ponents are calibrated marginally where µpjl | Tj ind ∼ N(βpjl, τ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Without loss of generality, assume (x1, y1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' , (xnp, ynp) are assigned to the hyper-rectangle associated with the terminal node ηpj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Given this partition and the initial guesses of βl(xi), an estimate of βpjl can be obtained by βpjl = 1 mnp np � i=1 βl(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' A confidence interval for each terminal node parameter can be set to have a length of 1/m in order to ensure each tree is a weak learner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' This is done by setting 1 m = 2kτ, which implies τ = 1 2km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='3 Variance Prior A conjugate scaled inverse chi-square distribution with hyperparameters ν and λ is assigned to the error variance σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The value of ν controls the shape of the prior distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' increasing ν decreases the variability and results in a prior which is more concentrated around the mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The value of λ sets the scale of the distribution, and thus specifies the region of the domain which is assigned non-negligible mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' To calibrate the prior, first select a value of ν to reflect the desired shape of the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Common values of ν range from 3 to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Before selecting a value for λ, one needs an initial estimate of the error variance to help set the prior around a range of plausible values of σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Given the model set and the corresponding point predictions at each of the training points ˆ f l(xi), one can use a lightly data informed prior by setting ˆσ2 = maxl=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='.,K � mini=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='.,n � yi − ˆfl(xi) �2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' This crude estimate has worked well in the EFT examples discussed in Section 5 due to the small error variances associated with the controlled experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Given this information, one strategy is to set ˆσ2 to be the mean or mode of a λν/χ2 ν distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Mean calibration sets νλ ν−2 = ˆσ2, which implies λ = ν−2 ν ˆσ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Similarly, mode calibration sets νλ ν+2 = ˆσ2 which implies λ = ν+2 ν ˆσ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' In the simulation examples presented in Section 5, the mode calibration appears to perform the best, as it typically allocates more mass around the true variance compared to the mean calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Since a common belief is that each model yields accurate approximations of the true system over 16 some subregion of the domain, one should expect the set of minimum squared differences across the K models will unveil reliable information about the true error variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' 5 EFT Examples This section applies the proposed model mixing methodology to three real EFT examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The first example involves mixing a concave weak coupling expansion and convex strong coupling ex- pansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' In this setting, the true physics model lies between the expansions, hence an interpolation of the two will adequately capture the true function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The second example involves mixing two convex expansions, each of which overestimates the true system in the intermediate range of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' This scenario demonstrates the benefits of applying prior regularization to the weight functions rather than imposing strict conditions such as a simplex constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The final example demonstrates the effectiveness of the proposed method in cases where a local expert does not exist for all subregions of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' An area of the domain is said to be without a local expert if none of the models under consideration adequately recovers the true underlying system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' An additional goal is to demonstrate the effectiveness of model mixing with both the non- informative and informative priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Given no prior information of the associated model discrepan- cies, one should elect to use the non-informative prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' If such information is available, then the informative prior can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The following examples involve data which is independently generated according to Yi = f†(xi) + ϵi, ϵi iid ∼ N(0, σ2) where i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=', 20, σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='005, and f†(x) is defined in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The 20 training points are located at inputs which are evenly spaced over the interval of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='03 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The error standard deviation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='005 was selected to mimic a controlled experiment setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Each EFT model is fit using nc = 4 evaluations of the corresponding finite-order expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='1 Example 1: Mixing Two Expansions First, consider mixing the second order weak coupling expansion, f(2) s (x) with the fourth order strong coupling expansion, f(4) l (x), over the domain of [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='03, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='5] as shown in Figure 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The pre- dicted mean and corresponding posterior weight functions are shown in Figures 3 and 4 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Regardless of the prior type, the BART-based mixing approach exhibits accurate mean predictions 17 Figure 3: (Example 1) The predicted mean (dark gray) and 95% credible intervals (shaded) when mixing f(2) s (x) (dashed) and f(4) l (x) (dotted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The true mean (light gray) is adequately captured under both priors with a training set of 20 observations (points) and nc = 4 simulator evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' with minimal uncertainty in the left and right portions of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The uncertainty increases in the intermediate range of the domain where neither EFT under consideration accurately predicts the true system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The most notable difference between the two approaches can be seen in the associ- ated uncertainty within this intermediate range, as the non-informative prior typically yields wider 95% credible intervals across this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' This is plausible as the informative prior directly incorpo- rates knowledge related to the individual model performance into its hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Therefore, the posterior distribution of the weight functions is less dependent upon the data and is noticeably influenced by the prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Meanwhile, the results under both priors yield similar weight functions which exhibit sigmoid-like curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' This result is intuitive of how the EFTs should be mixed, as the weak coupling expansion is appropriate in the left portion of the domain while the strong cou- pling is appropriate in the right region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' A linear combination of these functions is useful for the intermediate range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The posterior weight functions obtained using the informative prior generally exhibit smoother shapes with less variability compared to the results under the non-informative prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Figure 4 directly illustrates the effect of the prior on the posterior weight functions, as the informative version enables one to gain a more precise understanding of the behavior of the weight functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' This figure suggests two key insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' First, this mean mixing approach searches for useful combinations of mean predictions in order to recover the true system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Clearly, various 18 Non-Informative Prior InformativePrior 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='75- 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='25 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='00- 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='00- 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='5 X XFigure 4: (Example 1) The posterior estimates of the weight functions when mixing f(2) s (x) (solid) and f(4) l (x) (dashed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The respective 95% credible intervals are denoted by the shaded regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' sets of weight functions can yield very similar final predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Finally, the amount of variability observed in the final prediction of the physical system is directly related to the variability in the weight functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' This is evident in the results obtained with the informative prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='2 Example 2: Mixing Two Convex Expansions Now consider mixing two convex expansions which are always equal or greater than f†(x), as shown in Figure 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' In this problem, a convex combination of the models in the intermediate range will never capture the true system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Meanwhile, the BART-based approach benefits from the flexibility of the prior regularization strategy as it is able to recover f†(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' From Figure 5, both priors lead to accurate mean predictions of the physical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' As in Example 1, the result under the informative prior yields a lower degree of uncertainty compared to its non-informative counterpart, particularly across the region of [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='15, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Meanwhile, the posterior weight functions return very similar shapes under the two priors as displayed in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Similar to Example 1, the informative prior generally results in more precise estimates of the weight functions which translates to lower uncertainty in the posterior predictions of the physical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' 19 Non-InformativePrior Informative Prior 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='00- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='00- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='75- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='75- W(x) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='5 X XFigure 5: (Example 2) The predicted mean (dark gray) and 95% credible intervals (shaded) when mixing f(4) s (x) (dashed) and f(4) l (x) (dotted) to predict the true system (light grey).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Figure 6: (Example 2) The posterior estimates of the weight functions when mixing f(4) s (x) (solid) and f(4) l (x) (dashed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The 95% credible intervals are denoted by the shaded regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='3 Example 3: Mixing Without a Local Expert This example demonstrates the BART-based model’s ability to mix functions in regions where no local expert may exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Consider the model set containing three weak coupling expansions of orders 2,4 and 6 as shown in Figure 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' In this case, no local expert exists in the right portion of 20 Non-InformativePrior InformativePrior 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='6 /.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' / 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='4 F 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='5 X XNon-InformativePrior InformativePrior 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='50 M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='5 X XFigure 7: (Example 3) The predicted mean (dark gray) and 95% credible intervals (shaded) when mixing f(2) s (x) (dashed), f(4) s (x) (dotted), and f(6) s (x) (dashed, dotted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The true mean (light gray) is adequately captured under both priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' the domain as all three function diverge away from the true function at different rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Despite the lack of a local expert, the mixed prediction adequately recovers the true system with relatively small amounts of uncertainty as shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The non-informative prior results in a greater uncertainty between (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='4), which is where the 2nd and 4th order expansions begin to diverge at quicker rates, hence the mean prediction is more sensitive to small changes in the weight values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Meanwhile, the result under the informative prior has minimal uncertainty in this right portion of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Both results also display subtle deviations from f†(x) in the remainder of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Overall, this example demonstrates the ability of the BART-based model to leverage observational data as well as the information in the model set to make accurate predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' In this example, the posterior weight functions noticeably differ depending on the selected prior, as shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' For example, the weight functions defined using the non-informative prior indicate more weight is allocated to the 2nd and 4th order expansions across the interval (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='03, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='15), as evident by the location of the solid and dashed curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Additionally, all three functions have a relatively high degree of variability within this lower half of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' With the informative prior, a larger portion of the prediction is attributed to the 6th order expansion, which has a mean weight function around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The effects of the other two expansions are then shrunk in this lower region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' As the coupling constant x increases, the 2nd order expansion contributes more to the 21 Non-Informative Prior InformativePrior 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='6- 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='6- 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='4 F 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='0 - 1 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='0 - 1 1 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='5 X XFigure 8: (Example 3) The posterior estimates of the weight functions when mixing f(2) s (x) (solid), f(4) s (x) (dashed), and f(6) s (x) (dotted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The 95% credible intervals correspond to the shaded regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' final prediction compared to the 4th and 6th order expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Finally, both approaches properly identify that the 6th order expansion diverges at a much faster rate in the right portion of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Hence, its corresponding effect is shrunk to zero within this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' This example reiterates that the BART-based approach searches for useful combinations of models, and these combinations are not unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' It also poses a more interesting question related to the interpretation of the weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' For example, in the interval (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='03, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='15), the mean predictions from each EFT are nearly identical and align closely with the true system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Given this, one may expect each EFT is assigned a weight near 1/3, as a simple average of their predictions would be adequate in this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' However, this is not the case regardless of the prior selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Specifically, the weight given to the 6th order expansion noticeably differs from the weights assigned to the 2nd and 4th order expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' With the non-informative prior, this likely occurs because the trees are also regularized to be weak learners, meaning each is relatively shallow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Since the trees maintain a shallow depth, some sense of global model performance is preserved, thus the effect of the 6th order expansion is mitigated in this subregion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' When considering joint credible regions, the case where all weights are near 1/3 falls along the edge of the 99% credible region which suggests the simple average of predictions is a possibility, though it is unlikely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' With the informative prior, the 6th order expansion is assigned a relatively higher weight within (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='03, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='15) because it has a smaller truncation error compared to the other models under consideration in this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' 22 Non-Informative Prior Informative Prior 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='00- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='00- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='75- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='50 M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='25- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='5 X x6 Discussion Prediction and Uncertainty Quantification The proposed BART-based model mixing approach is able to adequately recover the underlying system f†(x) in each of the examples presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' In general, the information from each individual model tends to dominate the posterior predictions when a local expert is present, while the infor- mation in the data is more influential in areas where no model aligns with the true system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' For example, the information in the data is crucial when obtaining predictions in the intermediate range for Examples 1 and 2 or the right portion of the domain in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' It should also be noted similar performance in the mean prediction is observed when the data is not evenly spaced or when the training set is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' In Examples 1 and 2, one should be cautious when extrapolating in the left portion of the domain due to the rapid divergence of the 4th order strong coupling expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' However, in other settings where the rate of change for the EFT predictions is not as drastic, severe issues when extrapolating slightly outside the domain of the training data are not expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' For each example, small levels of uncertainty in the posterior prediction of f†(x) are observed across areas where at least one EFT aligns with the true system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The uncertainty increases in areas where the EFTs under consideration deviate from the true system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Due to the small observational errors, the mixed-model is very confident the training points align with f†(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' As a result, the credible intervals nearly touch each training point since the predictive mean function is nearly interpolating the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Between training inputs, the uncertainty increases and displays a bubble-like shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' These uncertainty bands tend to smooth out when the posterior variance shifts towards high values of σ2 as the mixed-prediction is no longer interpolating between the points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Prior Distributions Regardless of the prior selected, it remains clear that one is able to obtain adequate predictive performance and recover the true physical system with reasonable amounts of uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' This is crucial because prior information pertaining to a model’s localized performance may not always be available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Compared to the informative prior, the results using the non-informative version will generally result in higher degrees of uncertainty across the predicted system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' This is expected because there is less information about the weight functions present in the resulting posterior distributions when using the non-informative prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The informative prior explicitly leverages the information in the truncation errors, which directly relates to the localized predictive accuracy of each EFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' This information is used to calibrate the 23 prior mean of the terminal node parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Thus, an EFT with relatively small truncation error across a given partition of the domain will be assigned higher weight apriori compared to an EFT with relatively high errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' One can control the influence of the prior by changing the tuning parameters in the terminal node parameter model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Overall, the informative prior can be an effective tool because it essentially guides the weight functions towards the right direction using this additional model information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Interpretation of Model Weight Functions The primary objective of the weight functions is to re-scale the predictions given by each individual model so that a linear combination of these predictions can adequately recover the true system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Given the prior regularization method applied to the weight functions, exact interpretation of the resulting values can be unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' However, using this regularization perspective, one can conclude that weight functions which fall close to zero within a particular input subregion indicate that the corresponding model is unnecessary for the overall prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Meanwhile, a model which is the unique local expert within a particular region should be weighted by values close to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' These features are observed across all three examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Figure 9: The posterior mean estimates and 95% credible intervals (shaded) of the sum of weight functions from Examples 1 and 2 (solid and dashed) using the informative prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The benefit of the proposed regularization approach can further be understood through the posterior distribution of the sum of the weight functions, wsum(x) = �K l=1 wl(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Fig- ure 9 illustrates the posterior of wsum(x) for Examples 1 and 2 under the informative prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The posterior of wsum(x) from Example 1 (solid) is centered very close to one with rel- atively small amounts of uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' This re- sults because: (i) the prior regularization and (ii) f†(x) lies between the selected EFTs, which indicates a convex combination is appropri- ate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Even though a sum-to-one property is not strictly imposed, it appears to naturally occur in this situation where an interpolation of the competing models is appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Meanwhile, the posterior of wsum(x) from Example 2 (dashed) significantly drops below one in the intermediate range of the domain because both EFTs over- 24 Sum of the Weights 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='10- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='05 (x) + W2(x) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='95 W1( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='90 Sum of Weights Ex 1 Ex 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='5 Xestimate the true system, which renders a convex combination to be inappropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' From these observations, it appears the proposed model-mixing approach benefits by not imposing strict as- sumptions, such as a simplex constraint, on the weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Additionally, one must use caution when interpreting the weight functions independently of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' With EFTs, the weight functions are generally correlated at a fixed input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' This result is intuitive in the first two examples where the predictive accuracy of the weak and strong coupling expansions are inversely related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Thus, a joint interpretation is more appropriate in these problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Finally, the weight functions can also be used to better understand the M-open assumption associated with the model set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' An initial confirmation of the M-open setting can be made when the weight functions noticeably change as a function of the inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' This observation indicates localized performance of each model, hence one can confirm the true system is not contained in the set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' If the weight functions are nearly constant, one may also wish to check the posterior of wsum(x) to see if the sum of the weights is fixated close to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Such a case may suggest model averaging with a simplex constraint could also be an appropriate solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' This alone is not enough to confirm or deny the M-open assumption, however it may indicate that the M-complete or M-closed labels are possible classifications of the model set under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' A final case to consider is the situation where a single model receives a weight near one while the effects of the competing models are shrunk to zero across a subregion of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' This situation may indicate the model set is M-closed conditional on the subregion of interest despite falling in the M-open case when considering the entire domain at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' In conclusion, this work proposes a Bayesian treed framework to mix predictions from a set of competing models, each of which are intended to explain the physical system across a subregion of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' This approach falls within the class of problems referred to as Bayesian model mixing, as input-dependent weights are defined to reflect the localized behavior of each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The weight functions are modeled using a sum-of-trees and are regularized via a multivariate Gaussian prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The tree bases coupled with the regularization approach allows for the weights to be learned in a flexible non-parametric manner free of strict constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Using the weight functions, predictions from the individual models are mixed via a linear combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The success of this mixing approach is demonstrated on three EFT examples, each of which considers models with localized predictive performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Leveraging the localized behavior of the individual models leads to significant improvements in the posterior prediction and uncertainty quantification of f†(x) compared to global weighting schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' 25 Acknowledgements The work of JCY and RJF work was supported in part by the National Science Foundation under Agreement OAC-2004601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The 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distributions”, Bayesian Analysis 13(3), 917–1007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' 28 Appendix Let ηpj denote the pth terminal node in the jth tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Without loss of generality, assume (x1, y1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=', (xnp, ynp) lie in the hyper-rectangle defined by ηpj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Furthermore, define each residual as ri = yi − � q̸=j ˆ f ⊤(xi) g(xi, Tq, Mq), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' , np These are collected in an np dimensional vector Rpj = (r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=', rnp)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Finally, let ˆF pj denote the np × K matrix whose lth column is (f l(x1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=', f l(xnp))⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Due to the independence and constant variance assumptions, the model for the vector of residuals along with the associated priors is defined by Rpj | µpj, Tj, σ2 ∼ Nnp � ˆF pjµpj, σ2Inp � µpj | Tj ind ∼ NK(βpj, Σ) σ2 ∼ λν/χ2 ν where it is assumed Σ = τ 2IK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The Marginal Likelihood The marginal likelihood of the residuals in node ηpj is defined by L(Rpj | Tj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' σ2) = � L(Rpj | Tj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' µpj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' σ2)π(µpj | Tj) dµpj (9) Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' it follows,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' L(Rpj | Tj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' σ2) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='(2πσ2)−np/2 exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='2σ2 (Rpj − ˆF pjµpj)⊤(Rpj − ˆF pjµpj) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='(2πτ 2)−K/2 exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='2τ 2 (µpj − βpj)⊤(µpj − βpj) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='dµpj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='= (2πσ2)−np/2(2πτ 2)−K/2× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='2σ2 (R⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='pjRpj − 2µ⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='pj ˆF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='pjRpj + µ⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='pj ˆF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='pj ˆF pjµpj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='2τ 2 (µ⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='pjµpj − 2µ⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='pjβpj + β⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='pjβpj) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='dµpj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='= (2πσ2)−np/2(2πτ 2)−K/2 exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='2σ2 R⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='pjRpj − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='2τ 2 β⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='pjβpj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='2µ⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='pj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='σ2 ˆF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='pj ˆF pj + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='τ 2 IK ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='µpj + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='τ 2 βpj + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='σ2 ˆF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='pjRpj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='�⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='µpj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='dµpj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' 29 Now let A−1 = 1 σ2 ˆF ⊤ pj ˆF pj + 1 τ 2 IK and b = � 1 τ 2 βpj + 1 σ2 ˆF ⊤ pjRpj � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Substituting these terms into the above expression yields L(Rpj | Tj, σ2) = (2πσ2)−np/2(2πτ 2)−K/2 exp � − 1 2σ2 R⊤ pjRpj − 1 2τ 2 β⊤ pjβpj � × (10) � exp � − 1 2µ⊤ pjA−1µpj + b⊤µpj � dµpj Using Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='1 from Santner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' (2018) the integral simplifies as � exp � − 1 2µ⊤ pjA−1µpj + b⊤µpj � dµpj = (2π)K/2|A|1/2 exp �1 2b⊤Ab � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' (11) Then, from (10) and (11), the marginal likelihood simplifies as L(Rpj | Tj, σ2) = (2πσ2)−np/2(τ 2)−K/2|A|1/2 exp � − 1 2σ2 R⊤ pjRpj − 1 2τ 2 β⊤ pjβpj + 1 2b⊤Ab � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' = (2πσ2)−np/2(τ 2)−K/2 ���� � 1 σ2 ˆF ⊤ pj ˆF pj + 1 τ 2 IK �−1���� 1/2 × exp � − 1 2 � 1 σ2 R⊤ pjRpj + 1 τ 2 β⊤ pjβpj − b⊤Ab �� where b⊤Ab = � 1 τ 2 βpj + 1 σ2 ˆF ⊤ pjRpj �⊤� 1 σ2 ˆF ⊤ pj ˆF pj + 1 τ 2 IK �−1� 1 τ 2 βpj + 1 σ2 ˆF ⊤ pjRpj � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The Posterior of µpj Now consider the full conditional posterior distribution of the terminal node parameter µpj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Using Bayes rule, π(µpj | Rpj, Tj, σ2) ∝ L(Rpj | Tj, µpj, σ2)π(µpj | Tj) A conjugate prior is assumed for µpj, thus the terms in the likelihood and prior can be rearranged to obtain a Normal kernel for the posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' This process is summarized below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' π(µpj | Rpj, Tj, σ2) ∝ exp � − 1 2σ2 (Rpj − ˆF pjµpj)⊤(Rpj − ˆF pjµpj) � × exp � − 1 2τ 2 (µpj − βpj)⊤(µpj − βpj) � ∝ exp � − 1 2 � µ⊤ pj � 1 σ2 ˆF ⊤ pj ˆF pj + 1 τ 2 IK � µpj − 2µ⊤ pj � 1 τ 2 βpj + 1 σ2 ˆF ⊤ pjRpj ��� ∝ exp � − 1 2 � µ⊤ pjA−1µpj − 2µ⊤ pjA−1Ab �� where A−1 = 1 σ2 ˆF ⊤ pj ˆF pj + 1 τ 2 IK and b = 1 τ 2 βpj + 1 σ2 ˆF ⊤ pjRpj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The previous expression simplifies as π(µpj | Rpj, Tj, σ2) ∝ exp � − 1 2(µpj − Ab)⊤A−1(µpj − Ab) � 30 This is the kernel of a Multivariate Gaussian distribution with mean Ab and covariance matrix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Thus it follows µpj | Rpj, Tj, σ2 ind ∼ NK � Ab, A � replacing A and b with their respective definitions implies µpj | Rpj, Tj, σ2 ind ∼ NK �� 1 σ2 ˆF ⊤ pj ˆF pj + 1 τ 2 IK �−1� 1 τ 2 βpj + 1 σ2 ˆF ⊤ pjRpj � , � 1 σ2 ˆF ⊤ pj ˆF pj + 1 τ 2 IK �−1 � The Posterior Distribution of σ2 Finally, consider the full conditional posterior for the error variance, which is defined by π(σ2 | Y , T, M) ∝ L(Y | T, M, σ2)π(σ2) where Y = (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=', yn)⊤, T = {T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=', Tm}, and M = {M1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=', Mm}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Further,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' assume a conjugate prior for σ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' namely σ2 ∼ νλ/χ2 ν which has a probability density function defined by π(σ2) = (ν/2)ν/2 Γ(ν/2) λν/2(σ2)−(ν/2+1) exp � − νλ 2σ2 � Due to conjugacy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' the full conditional distribution is given by π(σ2 | Y ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' M) ∝ (σ2)−n/2 exp � − 1 2σ2 n � i=1 � yi − ˆ f ⊤(xi)w(xi) �2� (σ2)−(ν/2+1) exp � − νλ 2σ2 � ∝ (σ2)−(n/2+ν/2+1) exp � − 1 2σ2 � n � i=1 � yi − ˆ f ⊤(xi)w(xi) �2 + νλ �� This is the kernel of another scaled inverse-χ2 distribution,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' namely σ2 ∼ ν′λ′/χ2 ν′ where ν′ = n + ν and λ′ = 1 n + ν � n � i=1 � yi − ˆ f ⊤(xi)w(xi) �2 + νλ � 31 Supplementary Material An Overview of EFT EFTs model physical systems by an infinite expansion of terms organized in order of decreasing importance according to the power counting principle (Burgess,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Petrov and Blechman, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Georgi, 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Exact theoretical predictions of the system are obtained by summing over these terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' In practice, only a finite number of lower-order terms are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Thus, the theoretical prediction can be decomposed using a Taylor-like series which includes the known finite-order expansion along with the induced truncation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Predictions of experimental quantities can then be represented using an additive model Y (x) = f†(x) + ϵ(x) f†(x) = h(N)(x) + δ(N)(x) where x ∈ Rd denotes an independent variable associated with the system, h(N)(x) represents the known finite-order expansion of degree N, δ(N)(x) is the associated truncation error, and ϵ(x) is the random observational error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The accuracy of the finite-order expansion may vary significantly across a subspace of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' For example, a finite-order expansion centered about zero may yield a high fidelity approximation in the lower regions of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' However, the accuracy of the prediction quickly degrades in higher regions of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' It is further assumed the finite-order expansion can be modeled as a stochastic process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' First, the finite-order expansion can factorized as h(N)(x) = yref(x) N � k=0 ck(x)Qk(x), (12) where yref(x) sets the scale of variation, c0(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=', cN(x) are dimensionless observable coefficients, and Q(x) is a dimensionless expansion parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' When the scale and expansion parameters are known based on theoretical arguments, the coefficients c0(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=', cN(x) appear to behave as a set of independent and identically distributed curves from a stochastic process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Thus, a common model for the coefficients is a Gaussian process ck(x) | θ ∼ GP(µ, ¯c2r(x, x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' ℓ)) (13) θ = (µ, ¯c2, ℓ), 32 where µ denotes a constant mean function and ¯c2r(x, x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' ℓ) represents the covariance function (Me- lendez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' A common assumption is to set µ = 0, while prior distributions can be assigned to the remaining parameters in the model (Melendez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Additionally, a likelihood can be formed by collecting nc evaluations of the finite-order expansion, h(N) = (h(N)(xc 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' , h(N)(xc nc))⊤, at design inputs xc 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' , xc nc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' These model runs are used to extract the observed finite-order coef- ficients, which are modeled via (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Using the priors and the likelihood based on the model runs, the parameters in the GP are then learned through standard Bayesian inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The truncation error accounts for the remaining unknown terms in the series, thus δ(N)(x) is modeled using a similar factorization δ(N)(x) = yref(x) ∞ � k=N+1 ck(x)Qk(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' (14) Using (13) and (14) along with properties of the multivariate Normal distributions (Ravishanker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=', 2021), the induced prior on the truncation error term is given by δ(N)(x) | θ, Q ∼ GP � mδ(x), ¯c2Rδ(x, x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' ℓ) � , (15) with mean and covariance functions mδ(x) = yref(x)QN+1(x) 1 − Q(x)µ (16) Rδ(x, x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' ℓ) = yref(x)yref(x′)[Q(x)Q(x′)]N+1 1 − Q(x)Q(x′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' (17) The unknown parameters in (15) - (17) originate from the coefficient model in (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' Thus, the mean and covariance functions which characterize the discrepancy model are also learned using the set of nc evaluations of the finite-order expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' This is a unique property of EFTs, as observational data is not required to learn the model discrepancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' When the finite-order expansion is computationally inexpensive to evaluate, the induced prior on the theoretical predictions, f(x) = h(N)(x) + δ(N)(x) is given by f(x) | θ, Q, h(N) ∼ GP � mth(x), Σth(x, x′) � , where mth(x) = h(N)(x) + mδ(x) and Σth(x, x′) = ¯c2Rδ(x, x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' In the expensive case, a GP can be used to emulate the finite-order expansion and is defined by h(N)(x) | θ, Q ∼ GP � mN(x), ¯c2RN(x, x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' ℓ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' The resulting prior on the theoretical prediction is a GP with mean and covariance functions mth(x) = mN(x) + mδ(x) and Σth(x, x′) = ¯c2RN(x, x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' ℓ) + ¯c2Rδ(x, x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' In either case, given a set of model runs h(N), one can obtain posterior predictions ˆf(˜x1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' , ˆf(˜xm) at new inputs ˜x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' , ˜xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} +page_content=' 33' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQfXwAk/content/2301.02296v1.pdf'} diff --git a/j9FPT4oBgHgl3EQfGDRS/content/tmp_files/2301.13002v1.pdf.txt b/j9FPT4oBgHgl3EQfGDRS/content/tmp_files/2301.13002v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2937db3dffb1b0b4eff4a285b67159c78251d0a7 --- /dev/null +++ b/j9FPT4oBgHgl3EQfGDRS/content/tmp_files/2301.13002v1.pdf.txt @@ -0,0 +1,518 @@ +arXiv:2301.13002v1 [math.FA] 30 Jan 2023 +Jordan derivations on the θ−Lau products of +Banach algebras +M. Ghasemi and M. J. Mehdipour +Abstract. In this paper, we study Jordan derivation-like maps on the θ−Lau products of +algebras. We characterize them and prove that under certain condition any Jordan derivation-like +maps on the θ−Lau products is a derivation-like map. Moreover, we investigate the concept of +centralizing for Jordan derivation-like maps on the θ−Lau products of algebras. +Mathematics Subject Classification(2020). 47B47;16W25. +Keywords: Jordan derivations, θ−Lau products, centralizing mappings. +1 +Introduction +Let A be a Banach algebra. Let us recall that a linear mapping D : A → A is called +a derivation if +D(ax) = D(a)x + aD(x) +for all a, x ∈ A. Also, D is called a Jordan derivation if for every a ∈ A +D(a2) = D(a)a + aD(a). +The set of all derivations and Jordan derivations on A are denoted by Der(A) and +DerJ(A), respectively. +Let B be a Banach algebra and θ be a nonzero multiplicative linear functional +on B. Following [16], the θ−Lau product A and B is denoted by A ×θ B and it +is the direct product A × B together with the component wise addition and the +multiplication +(a, b) ·θ (x, y) = (ax + θ(y)a + θ(b)x, by). +We note that in the case where B = C and θ is the identity map on C, the unitization +A will be obtained. We also note that if we permit θ = 0, the θ−Lau product A×θ B +is the usual direct product. Hence we disregard the possibility that θ = 0. +The θ−Lau products A×θB were first introduced by Lau [12], for certain Banach +algebras. Sanjani Monfared [16] extended this product to arbitrary Banach algebras +A and B. The θ−Lau products are significance and utility. Because, the θ−Lau +product is a strongly splitting Banach algebra extension of B by A; for the study of +extensions of Banach algebras see [3, 7]. Also, many properties are not shared by +1 + +2 +Jordan derivations on the θ−Lau products +arbitrary strongly splitting extensions, while the θ−Lau products exhibit them; see +[16]. Furthermore, the θ−Lau products are a source of examples or counterexamples; +see for instance [17]. +These reasons caused that several authors studied various +aspects of the products [6, 8, 9, 11, 17, 19]. +In this paper, we continue these +investigations and study Jordan derivation-like maps of them. +It is clear that every derivation is a Jordan derivation. But, the converse is, +in general, not true. +Here a question arises: when dose the converse hold? +In +1957, Herstein [10] proved that every Jordan derivation on a 2-torsion free prime +ring is a derivation; see also [1, 5, 14, 15]. +Bresar [4] gave a generalization of +Herstein’s result for semiprime rings. +Many attempts were made to study this +question for Jordan derivations on Banach algebras [2, 4, 18]. For example, Sinclair +[18] proved that every continuous Joradan derivation on a semisimple Banach algebra +is a derivation. Brasar [4] showed that any Jordan derivation on a semisimple Banach +algebra is continuous. So any Jordan derivation on a semisimple Banach algebra is +a derivation. It is natural to ask whether results concerning Jordan derivations on +Banach algebras hold for the θ−Lau products A ×θ B? The other question comes +to mind immediately: what happens to θ in these investigations? To answer these +questions, we consider linear mappings d : A × B → A × B satisfying +d((a, b) ·θ (a, b)) = d(a, b) ·φ (a, b) + (a, b) ·γ d(a, b) (a ∈ A and b ∈ B), +where θ, φ and γ are nonzero multiplicative linear functional on B . We denote +the set of all theses mappings by DerJ(A ×φ,γ +θ +B). In this paper, we investigate the +questions concerning Jordan derivations for elements of DerJ(A ×φ,γ +θ +B). +In this paper, we characterize elements of DerJ(A×φ,γ +θ +B) in the case where A has +a right identity. We also give a necessary and sufficient condition under which every +element of DerJ(A ×φ,γ +θ +B) is a derivation. For unitary algebra A and semisimple +Banach algebra B, we prove that if θ ̸= φ, then DerJ(A ×φ,φ +θ +B) = Der(A ×φ,φ +θ +B). +Furthermore, we investigate (η1, η2)−centralizing element of DerJ(A ×φ,γ +θ +B). +2 +Main Results +In the sequel, let A be a Banach algebra with a right identity u and right annihilator +ran(A), the set of all z ∈ A with az = 0 for all a ∈ A. Let also θ, φ and γ be nonzero +multiplicative linear functionals on any Banach algebra B. +The next Lemma is +needed to prove our results. +Lemma 2.1 Let d : A × B → A × B be a mapping with +d((a, b) ·θ (a, b)) = d(a, b) ·φ (a, b) + (a, b) ·γ d(a, b) +for all a ∈ A and b ∈ B. Then d maps A into itself and d(u, 0) ∈ ran(A). + +M. Ghasemi and M. J. Mehdipour +3 +Proof. By hypothesis, we have +d((a + u, 0) ·θ (a + u, 0)) += +d(a + u, 0) ·φ (a + u, 0) ++ +(a + u, 0) ·γ d(a + u, 0) +for all a ∈ A. So +d(a, 0) ++ +d(ua, 0) += +d(a, 0) ·φ (u, 0) + d(u, 0) ·φ (a, 0) +(1) ++ +(a, 0) ·γ d(u, 0) + (u, 0) ·γ d(a, 0) +Take a = u in (1). Then +(z, w) += +(z, w) ·φ (u, 0) + (u, 0) ·γ (z, w) += +(z + φ(w)u + uz + γ(w)u, 0), +(2) +where d(u, 0) = (z, w) for some z ∈ A and w ∈ B. Hence w = 0 and from (2) we +obtain az = 0 for all a ∈ A. So z ∈ ran(A). +Let d(a, 0) = (x0, y0) and d(ua, 0) = (x1, y1) for some x0, x1 ∈ A and y0, y1 ∈ B. +If we replace a by ua in (1), then +(x1 + x1, y1 + y1) = (x1 + φ(y1)u + za + γ(y1)u + uaz + ux1, 0), +Hence y1 = 0 and by (1), y0 = 0. Therefore d maps A into itself. +□ +The main result of this paper is the following. +Theorem 2.2 Let d : A × B → A × B be a mapping. Then d ∈ DerJ(A ×φ,γ +θ +B) if +and only if the following statements hold. +(i) There exist unique Jordan derivations dA ∈ DerJ(A) and dB ∈ DerJ(B) such +that +d(a, b) = (dA(a) + (2θ − φ − γ)(b)dA(u) − 1 +2(φ + γ)(dB(b))u, dB(b)) +for all a ∈ A and b ∈ B. +(ii) (2θ − φ − γ)(b)(dA(a) − dA(u)a) = 1 +2(φ + γ)(dB(b))(a − ua) for all a ∈ A and +b ∈ B. +(iii) (θ − φ)(b)(θ − γ)(b)dA(u) = (γ − φ)(b)(γ − φ)(dB(b))u = 0 for all a ∈ A and +b ∈ B. +Proof. +For b ∈ B, let d(0, b) = (x1, y1) and d(u, 0) = (z, 0) for some x1 ∈ A, +z ∈ ran(A) and y1 ∈ B. Then +d(u, 0) ++ +d(2θ(b)u, 0) + d(0, b2) + +4 +Jordan derivations on the θ−Lau products += +d((u, b) ·θ (u, b)) += +d(u, b) ·φ (u, b) + (u, b) ·γ d(u, b) += +d(u, 0) ·φ (u, 0) + d(u, 0) ·φ (0, b) ++ +d(0, b) ·φ (u, 0) + d(0, b) ·φ (0, b) ++ +(u, 0) ·γ d(u, 0) + (u, 0) ·γ d(0, b) ++ +(0, b) ·γ d(u, 0) + (0, b) ·γ d(0, b). +So +(2θ(b)z, 0) += +d(2θ(b)u, 0) += +d(u, 0) ·φ (0, b) + d(0, b) ·φ (u, 0) ++ +(u, 0) ·γ d(0, b) + (0, b) ·γ d(u, 0) += +(z, 0) ·φ (0, b) + (x1, y1) ·φ (u, 0) ++ +(u, 0) ·γ (x1, y1) + (0, b) ·γ (z, 0) += +(φ(b)z + x1 + φ(y1)u + ux1 + γ(y1)u + γ(b)z, 0). +This shows that +x1 = (2θ − φ − γ)(b)z − ux1 − (φ + γ)(y1)u. +(3) +If we multiply (3) by u from the left, then +ux1 = −1 +2(φ + γ)(y1)u. +From this and (3), we have +x1 = (2θ − φ − γ)(b)z − 1 +2(φ + γ)(y1)u. +Hence +d(0, b) = ((2θ − φ − γ)(b)z − 1 +2(φ + γ)(y1)u, y1). +(4) +In view of Lemma 2.1, for every a ∈ A, there exists x0 ∈ A such that d(a, 0) = (x0, 0). +This together with (4) shows that +d(a, b) += +d(a, 0) + d(0, b) += +(x0 + (2θ − φ − γ)(b)z +− +1 +2(φ + γ)(y1)u, y1). + +M. Ghasemi and M. J. Mehdipour +5 +Assume that πA : A × B → A and πB : A × B → B be canonical projections. We +define the functions dA : A → A and dB : B → B by the following rules: +dA(a) = πA(d(a, 0)) +and +dB(b) = πB(d(0, b)). +It is clear that these functions are Jordan derivations and +d(a, b) = (dA(a) + (2θ − φ − γ)(b)dA(u) − 1 +2(φ + γ)(dB(b))u, dB(b)) +(5) +for all a ∈ A and b ∈ B. Hence (i) holds. +Since d ∈ DerJ(A ×φ,γ +θ +B), for every a ∈ A and b ∈ B, we have +d((a, b) ·θ (a, b)) = d(a, b) ·φ (a, b) + (a, b) ·γ d(a, b). +(6) +From (5) and (6), we conclude that +2θ(b)dA(a) ++ +(2θ − φ − γ)(b2)dA(u) − 1 +2(φ + γ)(dB(b2))u += +(φ + γ)(b)dA(a) + (2θ − φ − γ)(b)dA(u)a ++ +1 +2(φ + γ)(dB(b))(a − ua) +(7) ++ +(φ + γ)(b)(2θ − φ − γ)(b)dA(u) +− +1 +2(φ + γ)(b)(φ + γ)(dB(b))u +for all a ∈ A and b ∈ B. Set a = 0 in (7). Then +(2θ − φ − γ)(b2)dA(u) +− +1 +2(φ + γ)(dB(b2))u += +(φ + γ)(b)(2θ − φ − γ)(b)dA(u) +(8) +− +1 +2(φ + γ)(b)(φ + γ)(dB(b))u +for all b ∈ B. Regarding (7) and (8), we infer that +(2θ − φ − γ)(b)(dA(a) − dA(u)a) = 1 +2(φ + γ)(dB(b))(a − ua). +That is, (ii) holds. Let us multiply (8) by u from the left. Then +(γ − φ)(b)(γ − φ)(dB(b))u = 0 +for all a ∈ A and b ∈ B. This together with (8) follows that +(θ − φ)(b)(θ − γ)(b)dA(u) = 0 +for all a ∈ A and b ∈ B. Hence (iii) holds. +□ +In the sequel, dA and dB are as in Theorem 2.2. + +6 +Jordan derivations on the θ−Lau products +Corollary 2.3 Let d be an element in DerJ(A ×φ,γ +θ +B). Then the following state- +ments hold. +(i) Either θ = φ = γ or d maps A into ran(A). +(ii) If either A has a unit or A is semisimple, then θ = φ = γ or d is zero on A. +Proof.In view of Theorem 2.2 (ii), for every a, x ∈ A and b ∈ B we have +(2θ − φ − γ)(b)x(dA(a) − dA(u)a) += +1 +2(φ + γ)(dB(b))x(a − ua) += +0. +This implies that for every a, x ∈ A and b ∈ B +(2θ − φ − γ)(b)xdA(a) = 0. +(9) +Suppose now that d does not map A into ran(A). Then by (9), +(2θ − φ − γ)(b) = 0 +for all b ∈ B. Hence +θ(b) = 1 +2(φ + γ)(b) +for all b ∈ B. Writing b by b2 in the above relation, we get +(φ(b) − γ(b))2 = 0 +for all b ∈ B. This implies that +θ = φ = γ. +So (i) holds. The other statement of the present result follows at once from (i). +□ +In the following, a linear mapping d : A × B → A × B is called a (θ, φ, γ)- +derivation if +d((a, b) ·θ (x, y)) = d(a, b) ·φ (x, y) + (a, b) ·γ d(x, y) +for all a, x ∈ A and b, y ∈ B. The set of all these mappings is denoted by Der(A×φ,γ +θ +B). +Theorem 2.4 Let d be an element in DerJ(A ×φ,γ +θ +B). Then d ∈ Der(A ×φ,γ +θ +B) if +and only if the following assertions hold. +(i) dA ∈ Der(A) and dB ∈ Der(B). +(ii) (θ − φ)(b)dA(a) = (θ − γ)(b)(dA(a) − dA(u)a) = 0 for all a ∈ A and b ∈ B. +(iii) φdB(b)(φ − γ)(y)u = φdB(b)(a − ua) = 0 for all a ∈ A and b, y ∈ B. +(iv) γdB = φdB on B. +Furthermore, if A is a Bnach algebra without identity, then d(a, b) = (dA(a) + +(θ − γ)(b)dA(u), dB(b)) for all a ∈ A and b ∈ B. + +M. Ghasemi and M. J. Mehdipour +7 +Proof. Let d ∈ DerJ(A×φ,γ +θ +B). According to Theorem 2.2, there exist dA ∈ DerJ(A) +and dB ∈ DerJ(B) such that +d(a, b) = (dA(a) + (2θ − φ − γ)(b)dA(u) − 1 +2(φ + γ)(dB(b))u, dB(b)), +for all a ∈ A and b ∈ B. Suppose that d ∈ Der(A ×φ,γ +θ +B). Then for all a, x ∈ A and +b, y ∈ B +d((a, b) ·θ (x, y)) = d(a, b) ·φ (x, y) + (a, b) ·γ d(x, y). +So +dA(ax) ++ +θ(b)dA(x) + θ(y)dA(a) ++ +(2θ − φ − γ)(by)dA(u) − 1 +2(φ + γ)(dB(by))u += +dA(a)x + (2θ − φ − γ)(b)dA(u)x − 1 +2(φ + γ)(dB(b))ux ++ +φ(y)dA(a) + φ(y)(2θ − φ − γ)(b)dA(u) +(10) +− +1 +2φ(y)(φ + γ)(dB(b))u + adA(x) − 1 +2(φ + γ)(dB(y))a ++ +γ(b)dA(x) + γ(b)(2θ − φ − γ)(y)dA(u) +− +1 +2γ(b)(φ + γ)(dB(y))u + γdB(y)a + φdB(b)x +and +dB(by) = dB(b)y + bdB(y). +(11) +The relation (11) shows that dB is a derivation on B. Set b = y = 0 in (10). Then +dA(ax) = dA(a)x + adA(x) +for all a, x ∈ A. Hence dA is a derivation on A. That is, (i) holds. Now, let a = x = 0 +in (10), we obtain +(2θ − φ − γ)(by)dA(u) +− +1 +2(φ + γ)(dB(by))u += +φ(y)(2θ − φ − γ)(b)dA(u) +− +1 +2φ(y)(φ + γ)(dB(b))u +(12) ++ +γ(b)(2θ − φ − γ)(y)dA(u) +− +1 +2γ(b)(φ + γ)(dB(y))u + +8 +Jordan derivations on the θ−Lau products +Subtracting (12) from (10), we arrive at +(θ − γ)(b)(dA(x) − dA(u)x) ++ +(θ − φ)(y)dA(a) += +(θ − φ)(b)dA(u)x +− +1 +2(φ + γ)(dB(b))ux +(13) ++ +1 +2(γ − φ)(dB(y))a ++ +φdB(b)x +Taking b = 0 in (13), we have +(θ − φ)(y)dA(a) = 1 +2(γ − φ)(dB(y))a +(14) +for all a ∈ A and y ∈ B. Put a = u in (14) and then multiply it by u from the left. +These imply that +φdB(b) = γdB(b) +(15) +for all b ∈ B. So (iv) holds. From this and (14) we infer that +(θ − φ)(y)dA(a) = 0 +(16) +for all a ∈ A and y ∈ B. This together with (13) and (15) shows that +(θ − γ)(b)(dA(x) − dA(u)x) = φdB(b)(x − ux) +(17) +for all x ∈ A and b ∈ B. So (12) can be written as follows. +((θ − γ)(by) +− +φ(y)(θ − γ)(b) − γ(b)(θ − γ)(y))dA(u) += +(φdB(by) − φ(y)φdB(b) − γ(b)φdB(y))u +for all b, y ∈ B. Hence +φdR(by) − φ(y)φdB(b) − γ(b)φdB(y) = 0 +by Lemma 2.1. Since dB is a derivation on B, +(φ − γ)(b)φdB(y) = 0 +(18) +for all b, y ∈ B. If φdB ̸= 0, then φ = γ and by (16) and (17), we get +(θ − γ)(b)(dA(x) − dA(u)x) = φdB(b)(x − ux) = 0 +(19) +for all x ∈ A and b ∈ B. From (16), (18) and (19) we see that the assertions (ii) +and (iii) hold. + +M. Ghasemi and M. J. Mehdipour +9 +Finally, if A is an algebra without identify, then by (iii) and (iv), +φdB = γdB = 0. +From this and (ii), we infer that +d(a, b) = (dA(a) + (θ − γ)(b)dA(u), dB(b)) +for all a ∈ A and b ∈ B. +□ +As an immediate consequence of [4], Corollary 2.3 and Theorem 2.4 we present +the following result. +Corollary 2.5 Let A be a Banach algebra with identity and B be a semisimple +Banach algebra. If θ ̸= φ, then DerJ(A ×φ,φ +θ +B) = Der(A ×φ,φ +θ +B) = Der(B). +Let us recall that a mapping T : A → A is called centralizing if for every a ∈ A +[T(a), a] ∈ Z(A), +where Z(A) is the center of A and for each a, x ∈ A +[a, x] = ax − xa. +This concept can be stated for the θ−Lau products A ×θ B as follows. For nonzero +multiplicative linear functionals η1, η2 on B, an element d ∈ DerJ(A×φ,γ +θ +B) is called +(η1, η2)−centralizing if for every a ∈ A and b ∈ B, +[d(a, b), (a, b)]η1,η2 := d(a, b) ·η1 (a, b) − (a, b) ·η2 d(a, b) ∈ Z(A) × Z(B). +Theorem 2.6 Let B be a semisimple Banach algebra. +If θ ̸= φ, then the only +(η1, η2)−centralizing element of DerJ(A ×φ,γ +θ +B) is the zero map. +Proof. Let d ∈ DerJ(A×φ,γ +θ +B) be (η1, η2)−centralizing. So there exist dA ∈ DerJ(A) +and dB ∈ DerJ(B) such that +d(a, b) = (dA(a) + (2θ − φ − γ)(b)dA(u) − 1 +2(φ + γ)(dB(b))u, dB(b)) +for all a ∈ A and b ∈ B. Since d is (η1, η2)−centralizing, dA and dB are centralizing +on A and B, respectively. It follows from [4, 13] that dB = 0 on B and +dA(u) = dA(u)u − udA(u) ∈ Z(A). +From Lemma 2.1 we infer that +dA(u) = dA(u)u = udA(u) = 0. + +10 +Jordan derivations on the θ−Lau products +This implies that +dA(a) − udA(a) ∈ Z(A) +and so +dA(a) − udA(a) = u(dA(a) − udA(a)) = 0 +for all a ∈ A. Hence by Corollary 2.3 (i), we have +dA(a) = 0 +for all a ∈ A. Therefore, d = 0. +□ +References +[1] +M. H. Ahmadi Gandomani and M. J. Mehdipour, Jordan, Jordan right and Jordan left +derivations on convolution algebras, Bull. Iranian Math. Soc., 45 (2019) 189–204. +[2] +M. H. Ahmadi Gandomani and M. J. Mehdipour, Generalized derivations on some convolution +algebras, Aequationes Math., 92 (2) (2018), 223–241. +[3] W. G. Bade, H. G. Dales and Z. A. Lykova, Algebraic and strong splittings of extensions of +Banach algebras, Mem. Amer. Math. Soc., 137 (656) (1999), 39-54. +[4] M. Bresar, Jordan derivations on semiprime rings, Proc. Amer. Math. Soc., 104 (4) (1988), +1003–1006. +[5] M. Bresar and J. 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Lau, Analysis on a class of Banach algebras with applications to harmonic analysis on +locally compact groups and semigroups, Fund. Math., 118 (3) (1983), 161–175. +[13] M. Mathieu and V. Runde, Derivations mapping into the radical II, Bull. London Math. Soc., +24 (1992), 485–487. +[14] M. J. Mehdipour and Z. Saeedi, Derivations on group algebras of a locally compact abelian +group, Monatsh. Math., 180 (2016), 595–605. + +M. Ghasemi and M. J. Mehdipour +11 +[15] +M. J. Mehdipour and Z. Saeedi, Derivations on convolution algebras, Bull. Korean Math. +Soc., 52 (2015), 123–1132. +[16] M. Sangani Monfared, On certain products of Banach algebras with applications to harmonic +analysis, Studia Math., 178 (3) (2007), 277–294. +[17] M. Sangani Monfared, Character amenability of Banach algebras, Math. Proc. Camb. Phil. +Soc., 144 (2008), 697–706. +[18] A. M. Sinclair, Jordan homomorphisms and derivations on semisimple Banach algebrfas, Proc. +Amer. Math. Soc., 24 (1970), 209–214. +[19] B. Willson, Configurations and invariant nets for amenable hypergroups and related algebras, +Trans. Amer. Math. Soc., 366 (2014), 5087–5112. +Mina Ghasemi +Department of Mathematics, +Shiraz University of Technology, +Shiraz 71555-313, Iran +e-mail: mi.ghasemi@sutech.ac.ir +Mohammad Javad Mehdipour +Department of Mathematics, +Shiraz University of Technology, +Shiraz 71555-313, Iran +e-mail: mehdipour@sutech.ac.ir + diff --git a/j9FPT4oBgHgl3EQfGDRS/content/tmp_files/load_file.txt b/j9FPT4oBgHgl3EQfGDRS/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2e3f5ac2b0ad8ab036fd0989a99f05bf75071723 --- /dev/null +++ b/j9FPT4oBgHgl3EQfGDRS/content/tmp_files/load_file.txt @@ -0,0 +1,329 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf,len=328 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content='13002v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content='FA] 30 Jan 2023 Jordan derivations on the θ−Lau products of Banach algebras M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Ghasemi and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Mehdipour Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' In this paper, we study Jordan derivation-like maps on the θ−Lau products of algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' We characterize them and prove that under certain condition any Jordan derivation-like maps on the θ−Lau products is a derivation-like map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Moreover, we investigate the concept of centralizing for Jordan derivation-like maps on the θ−Lau products of algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Mathematics Subject Classification(2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' 47B47;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content='16W25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Keywords: Jordan derivations, θ−Lau products, centralizing mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' 1 Introduction Let A be a Banach algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Let us recall that a linear mapping D : A → A is called a derivation if D(ax) = D(a)x + aD(x) for all a, x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Also, D is called a Jordan derivation if for every a ∈ A D(a2) = D(a)a + aD(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' The set of all derivations and Jordan derivations on A are denoted by Der(A) and DerJ(A), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Let B be a Banach algebra and θ be a nonzero multiplicative linear functional on B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Following [16], the θ−Lau product A and B is denoted by A ×θ B and it is the direct product A × B together with the component wise addition and the multiplication (a, b) ·θ (x, y) = (ax + θ(y)a + θ(b)x, by).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' We note that in the case where B = C and θ is the identity map on C, the unitization A will be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' We also note that if we permit θ = 0, the θ−Lau product A×θ B is the usual direct product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Hence we disregard the possibility that θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' The θ−Lau products A×θB were first introduced by Lau [12], for certain Banach algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Sanjani Monfared [16] extended this product to arbitrary Banach algebras A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' The θ−Lau products are significance and utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Because, the θ−Lau product is a strongly splitting Banach algebra extension of B by A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' for the study of extensions of Banach algebras see [3, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Also, many properties are not shared by 1 2 Jordan derivations on the θ−Lau products arbitrary strongly splitting extensions, while the θ−Lau products exhibit them;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' see [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Furthermore, the θ−Lau products are a source of examples or counterexamples;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' see for instance [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' These reasons caused that several authors studied various aspects of the products [6, 8, 9, 11, 17, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' In this paper, we continue these investigations and study Jordan derivation-like maps of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' It is clear that every derivation is a Jordan derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' But, the converse is, in general, not true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Here a question arises: when dose the converse hold?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' In 1957, Herstein [10] proved that every Jordan derivation on a 2-torsion free prime ring is a derivation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' see also [1, 5, 14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Bresar [4] gave a generalization of Herstein’s result for semiprime rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Many attempts were made to study this question for Jordan derivations on Banach algebras [2, 4, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' For example, Sinclair [18] proved that every continuous Joradan derivation on a semisimple Banach algebra is a derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Brasar [4] showed that any Jordan derivation on a semisimple Banach algebra is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' So any Jordan derivation on a semisimple Banach algebra is a derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' It is natural to ask whether results concerning Jordan derivations on Banach algebras hold for the θ−Lau products A ×θ B?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' The other question comes to mind immediately: what happens to θ in these investigations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' To answer these questions, we consider linear mappings d : A × B → A × B satisfying d((a, b) ·θ (a, b)) = d(a, b) ·φ (a, b) + (a, b) ·γ d(a, b) (a ∈ A and b ∈ B), where θ, φ and γ are nonzero multiplicative linear functional on B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' We denote the set of all theses mappings by DerJ(A ×φ,γ θ B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' In this paper, we investigate the questions concerning Jordan derivations for elements of DerJ(A ×φ,γ θ B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' In this paper, we characterize elements of DerJ(A×φ,γ θ B) in the case where A has a right identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' We also give a necessary and sufficient condition under which every element of DerJ(A ×φ,γ θ B) is a derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' For unitary algebra A and semisimple Banach algebra B, we prove that if θ ̸= φ, then DerJ(A ×φ,φ θ B) = Der(A ×φ,φ θ B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Furthermore, we investigate (η1, η2)−centralizing element of DerJ(A ×φ,γ θ B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' 2 Main Results In the sequel, let A be a Banach algebra with a right identity u and right annihilator ran(A), the set of all z ∈ A with az = 0 for all a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Let also θ, φ and γ be nonzero multiplicative linear functionals on any Banach algebra B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' The next Lemma is needed to prove our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content='1 Let d : A × B → A × B be a mapping with d((a, b) ·θ (a, b)) = d(a, b) ·φ (a, b) + (a, b) ·γ d(a, b) for all a ∈ A and b ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Then d maps A into itself and d(u, 0) ∈ ran(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Ghasemi and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Mehdipour 3 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' By hypothesis, we have d((a + u, 0) ·θ (a + u, 0)) = d(a + u, 0) ·φ (a + u, 0) + (a + u, 0) ·γ d(a + u, 0) for all a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' So d(a, 0) + d(ua, 0) = d(a, 0) ·φ (u, 0) + d(u, 0) ·φ (a, 0) (1) + (a, 0) ·γ d(u, 0) + (u, 0) ·γ d(a, 0) Take a = u in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Then (z, w) = (z, w) ·φ (u, 0) + (u, 0) ·γ (z, w) = (z + φ(w)u + uz + γ(w)u, 0), (2) where d(u, 0) = (z, w) for some z ∈ A and w ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Hence w = 0 and from (2) we obtain az = 0 for all a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' So z ∈ ran(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Let d(a, 0) = (x0, y0) and d(ua, 0) = (x1, y1) for some x0, x1 ∈ A and y0, y1 ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' If we replace a by ua in (1), then (x1 + x1, y1 + y1) = (x1 + φ(y1)u + za + γ(y1)u + uaz + ux1, 0), Hence y1 = 0 and by (1), y0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Therefore d maps A into itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' □ The main result of this paper is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content='2 Let d : A × B → A × B be a mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Then d ∈ DerJ(A ×φ,γ θ B) if and only if the following statements hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' (i) There exist unique Jordan derivations dA ∈ DerJ(A) and dB ∈ DerJ(B) such that d(a, b) = (dA(a) + (2θ − φ − γ)(b)dA(u) − 1 2(φ + γ)(dB(b))u, dB(b)) for all a ∈ A and b ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' (ii) (2θ − φ − γ)(b)(dA(a) − dA(u)a) = 1 2(φ + γ)(dB(b))(a − ua) for all a ∈ A and b ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' (iii) (θ − φ)(b)(θ − γ)(b)dA(u) = (γ − φ)(b)(γ − φ)(dB(b))u = 0 for all a ∈ A and b ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' For b ∈ B, let d(0, b) = (x1, y1) and d(u, 0) = (z, 0) for some x1 ∈ A, z ∈ ran(A) and y1 ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Then d(u, 0) + d(2θ(b)u, 0) + d(0, b2) 4 Jordan derivations on the θ−Lau products = d((u, b) ·θ (u, b)) = d(u, b) ·φ (u, b) + (u, b) ·γ d(u, b) = d(u, 0) ·φ (u, 0) + d(u, 0) ·φ (0, b) + d(0, b) ·φ (u, 0) + d(0, b) ·φ (0, b) + (u, 0) ·γ d(u, 0) + (u, 0) ·γ d(0, b) + (0, b) ·γ d(u, 0) + (0, b) ·γ d(0, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' So (2θ(b)z, 0) = d(2θ(b)u, 0) = d(u, 0) ·φ (0, b) + d(0, b) ·φ (u, 0) + (u, 0) ·γ d(0, b) + (0, b) ·γ d(u, 0) = (z, 0) ·φ (0, b) + (x1, y1) ·φ (u, 0) + (u, 0) ·γ (x1, y1) + (0, b) ·γ (z, 0) = (φ(b)z + x1 + φ(y1)u + ux1 + γ(y1)u + γ(b)z, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' This shows that x1 = (2θ − φ − γ)(b)z − ux1 − (φ + γ)(y1)u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' (3) If we multiply (3) by u from the left, then ux1 = −1 2(φ + γ)(y1)u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' From this and (3), we have x1 = (2θ − φ − γ)(b)z − 1 2(φ + γ)(y1)u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Hence d(0, b) = ((2θ − φ − γ)(b)z − 1 2(φ + γ)(y1)u, y1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' (4) In view of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content='1, for every a ∈ A, there exists x0 ∈ A such that d(a, 0) = (x0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' This together with (4) shows that d(a, b) = d(a, 0) + d(0, b) = (x0 + (2θ − φ − γ)(b)z − 1 2(φ + γ)(y1)u, y1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Ghasemi and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Mehdipour 5 Assume that πA : A × B → A and πB : A × B → B be canonical projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' We define the functions dA : A → A and dB : B → B by the following rules: dA(a) = πA(d(a, 0)) and dB(b) = πB(d(0, b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' It is clear that these functions are Jordan derivations and d(a, b) = (dA(a) + (2θ − φ − γ)(b)dA(u) − 1 2(φ + γ)(dB(b))u, dB(b)) (5) for all a ∈ A and b ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Hence (i) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Since d ∈ DerJ(A ×φ,γ θ B), for every a ∈ A and b ∈ B, we have d((a, b) ·θ (a, b)) = d(a, b) ·φ (a, b) + (a, b) ·γ d(a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' (6) From (5) and (6), we conclude that 2θ(b)dA(a) + (2θ − φ − γ)(b2)dA(u) − 1 2(φ + γ)(dB(b2))u = (φ + γ)(b)dA(a) + (2θ − φ − γ)(b)dA(u)a + 1 2(φ + γ)(dB(b))(a − ua) (7) + (φ + γ)(b)(2θ − φ − γ)(b)dA(u) − 1 2(φ + γ)(b)(φ + γ)(dB(b))u for all a ∈ A and b ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Set a = 0 in (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Then (2θ − φ − γ)(b2)dA(u) − 1 2(φ + γ)(dB(b2))u = (φ + γ)(b)(2θ − φ − γ)(b)dA(u) (8) − 1 2(φ + γ)(b)(φ + γ)(dB(b))u for all b ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Regarding (7) and (8), we infer that (2θ − φ − γ)(b)(dA(a) − dA(u)a) = 1 2(φ + γ)(dB(b))(a − ua).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' That is, (ii) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Let us multiply (8) by u from the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Then (γ − φ)(b)(γ − φ)(dB(b))u = 0 for all a ∈ A and b ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' This together with (8) follows that (θ − φ)(b)(θ − γ)(b)dA(u) = 0 for all a ∈ A and b ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Hence (iii) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' □ In the sequel, dA and dB are as in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' 6 Jordan derivations on the θ−Lau products Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content='3 Let d be an element in DerJ(A ×φ,γ θ B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Then the following state- ments hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' (i) Either θ = φ = γ or d maps A into ran(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' (ii) If either A has a unit or A is semisimple, then θ = φ = γ or d is zero on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content='In view of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content='2 (ii), for every a, x ∈ A and b ∈ B we have (2θ − φ − γ)(b)x(dA(a) − dA(u)a) = 1 2(φ + γ)(dB(b))x(a − ua) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' This implies that for every a, x ∈ A and b ∈ B (2θ − φ − γ)(b)xdA(a) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' (9) Suppose now that d does not map A into ran(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Then by (9), (2θ − φ − γ)(b) = 0 for all b ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Hence θ(b) = 1 2(φ + γ)(b) for all b ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Writing b by b2 in the above relation, we get (φ(b) − γ(b))2 = 0 for all b ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' This implies that θ = φ = γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' So (i) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' The other statement of the present result follows at once from (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' □ In the following, a linear mapping d : A × B → A × B is called a (θ, φ, γ)- derivation if d((a, b) ·θ (x, y)) = d(a, b) ·φ (x, y) + (a, b) ·γ d(x, y) for all a, x ∈ A and b, y ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' The set of all these mappings is denoted by Der(A×φ,γ θ B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content='4 Let d be an element in DerJ(A ×φ,γ θ B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Then d ∈ Der(A ×φ,γ θ B) if and only if the following assertions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' (i) dA ∈ Der(A) and dB ∈ Der(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' (ii) (θ − φ)(b)dA(a) = (θ − γ)(b)(dA(a) − dA(u)a) = 0 for all a ∈ A and b ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' (iii) φdB(b)(φ − γ)(y)u = φdB(b)(a − ua) = 0 for all a ∈ A and b, y ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' (iv) γdB = φdB on B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Furthermore, if A is a Bnach algebra without identity, then d(a, b) = (dA(a) + (θ − γ)(b)dA(u), dB(b)) for all a ∈ A and b ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Ghasemi and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Mehdipour 7 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Let d ∈ DerJ(A×φ,γ θ B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' According to Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content='2, there exist dA ∈ DerJ(A) and dB ∈ DerJ(B) such that d(a, b) = (dA(a) + (2θ − φ − γ)(b)dA(u) − 1 2(φ + γ)(dB(b))u, dB(b)), for all a ∈ A and b ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Suppose that d ∈ Der(A ×φ,γ θ B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Then for all a, x ∈ A and b, y ∈ B d((a, b) ·θ (x, y)) = d(a, b) ·φ (x, y) + (a, b) ·γ d(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' So dA(ax) + θ(b)dA(x) + θ(y)dA(a) + (2θ − φ − γ)(by)dA(u) − 1 2(φ + γ)(dB(by))u = dA(a)x + (2θ − φ − γ)(b)dA(u)x − 1 2(φ + γ)(dB(b))ux + φ(y)dA(a) + φ(y)(2θ − φ − γ)(b)dA(u) (10) − 1 2φ(y)(φ + γ)(dB(b))u + adA(x) − 1 2(φ + γ)(dB(y))a + γ(b)dA(x) + γ(b)(2θ − φ − γ)(y)dA(u) − 1 2γ(b)(φ + γ)(dB(y))u + γdB(y)a + φdB(b)x and dB(by) = dB(b)y + bdB(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' (11) The relation (11) shows that dB is a derivation on B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Set b = y = 0 in (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Then dA(ax) = dA(a)x + adA(x) for all a, x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Hence dA is a derivation on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' That is, (i) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Now,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' let a = x = 0 in (10),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' we obtain (2θ − φ − γ)(by)dA(u) − 1 2(φ + γ)(dB(by))u = φ(y)(2θ − φ − γ)(b)dA(u) − 1 2φ(y)(φ + γ)(dB(b))u (12) + γ(b)(2θ − φ − γ)(y)dA(u) − 1 2γ(b)(φ + γ)(dB(y))u 8 Jordan derivations on the θ−Lau products Subtracting (12) from (10),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' we arrive at (θ − γ)(b)(dA(x) − dA(u)x) + (θ − φ)(y)dA(a) = (θ − φ)(b)dA(u)x − 1 2(φ + γ)(dB(b))ux (13) + 1 2(γ − φ)(dB(y))a + φdB(b)x Taking b = 0 in (13),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' we have (θ − φ)(y)dA(a) = 1 2(γ − φ)(dB(y))a (14) for all a ∈ A and y ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Put a = u in (14) and then multiply it by u from the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' These imply that φdB(b) = γdB(b) (15) for all b ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' So (iv) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' From this and (14) we infer that (θ − φ)(y)dA(a) = 0 (16) for all a ∈ A and y ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' This together with (13) and (15) shows that (θ − γ)(b)(dA(x) − dA(u)x) = φdB(b)(x − ux) (17) for all x ∈ A and b ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' So (12) can be written as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' ((θ − γ)(by) − φ(y)(θ − γ)(b) − γ(b)(θ − γ)(y))dA(u) = (φdB(by) − φ(y)φdB(b) − γ(b)φdB(y))u for all b, y ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Hence φdR(by) − φ(y)φdB(b) − γ(b)φdB(y) = 0 by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Since dB is a derivation on B, (φ − γ)(b)φdB(y) = 0 (18) for all b, y ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' If φdB ̸= 0, then φ = γ and by (16) and (17), we get (θ − γ)(b)(dA(x) − dA(u)x) = φdB(b)(x − ux) = 0 (19) for all x ∈ A and b ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' From (16), (18) and (19) we see that the assertions (ii) and (iii) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Ghasemi and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Mehdipour 9 Finally, if A is an algebra without identify, then by (iii) and (iv), φdB = γdB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' From this and (ii), we infer that d(a, b) = (dA(a) + (θ − γ)(b)dA(u), dB(b)) for all a ∈ A and b ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' □ As an immediate consequence of [4], Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content='3 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content='4 we present the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content='5 Let A be a Banach algebra with identity and B be a semisimple Banach algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' If θ ̸= φ, then DerJ(A ×φ,φ θ B) = Der(A ×φ,φ θ B) = Der(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Let us recall that a mapping T : A → A is called centralizing if for every a ∈ A [T(a), a] ∈ Z(A), where Z(A) is the center of A and for each a, x ∈ A [a, x] = ax − xa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' This concept can be stated for the θ−Lau products A ×θ B as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' For nonzero multiplicative linear functionals η1, η2 on B, an element d ∈ DerJ(A×φ,γ θ B) is called (η1, η2)−centralizing if for every a ∈ A and b ∈ B, [d(a, b), (a, b)]η1,η2 := d(a, b) ·η1 (a, b) − (a, b) ·η2 d(a, b) ∈ Z(A) × Z(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content='6 Let B be a semisimple Banach algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' If θ ̸= φ, then the only (η1, η2)−centralizing element of DerJ(A ×φ,γ θ B) is the zero map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Let d ∈ DerJ(A×φ,γ θ B) be (η1, η2)−centralizing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' So there exist dA ∈ DerJ(A) and dB ∈ DerJ(B) such that d(a, b) = (dA(a) + (2θ − φ − γ)(b)dA(u) − 1 2(φ + γ)(dB(b))u, dB(b)) for all a ∈ A and b ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Since d is (η1, η2)−centralizing, dA and dB are centralizing on A and B, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' It follows from [4, 13] that dB = 0 on B and dA(u) = dA(u)u − udA(u) ∈ Z(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' From Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content='1 we infer that dA(u) = dA(u)u = udA(u) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' 10 Jordan derivations on the θ−Lau products This implies that dA(a) − udA(a) ∈ Z(A) and so dA(a) − udA(a) = u(dA(a) − udA(a)) = 0 for all a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Hence by Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content='3 (i), we have dA(a) = 0 for all a ∈ A.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=', 366 (2014), 5087–5112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content=' Mina Ghasemi Department of Mathematics, Shiraz University of Technology, Shiraz 71555-313, Iran e-mail: mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content='ghasemi@sutech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content='ir Mohammad Javad Mehdipour Department of Mathematics, Shiraz University of Technology, Shiraz 71555-313, Iran e-mail: mehdipour@sutech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} +page_content='ir' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FPT4oBgHgl3EQfGDRS/content/2301.13002v1.pdf'} diff --git a/j9FRT4oBgHgl3EQfWzfr/content/tmp_files/2301.13545v1.pdf.txt b/j9FRT4oBgHgl3EQfWzfr/content/tmp_files/2301.13545v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8681b79d8a704985b0c683dfeffc4aee605aa510 --- /dev/null +++ b/j9FRT4oBgHgl3EQfWzfr/content/tmp_files/2301.13545v1.pdf.txt @@ -0,0 +1,971 @@ +©2023 IEEE. Personal use of this material is permitted. Permission from +IEEE must be obtained for all other uses, in any current or future +media, including reprinting/republishing this material for advertising or +promotional purposes, creating new collective works, for resale or +redistribution to servers or lists, or reuse of any copyrighted component of +this work in other works. +Accepted to be published in: 2023 IEEE International Conference +on Robotics and Automation (ICRA), May 29 - June 2, 2023, London +arXiv:2301.13545v1 [cs.RO] 31 Jan 2023 + +Holistic Graph-based Motion Prediction +Daniel Grimm1, Philip Sch¨orner1, Moritz Dreßler2 and J.-Marius Z¨ollner1,2 +Abstract— Motion prediction for automated vehicles in com- +plex environments is a difficult task that is to be mastered +when automated vehicles are to be used in arbitrary situ- +ations. Many factors influence the future motion of traffic +participants starting with traffic rules and reaching from the +interaction between each other to personal habits of human +drivers. Therefore we present a novel approach for a graph- +based prediction based on a heterogeneous holistic graph +representation that combines temporal information, properties +and relations between traffic participants as well as relations +with static elements like the road network. The information +are encoded through different types of nodes and edges that +both are enriched with arbitrary features. We evaluated the +approach on the INTERACTION and the Argoverse dataset +and conducted an informative ablation study to demonstrate +the benefit of different types of information for the motion +prediction quality. +I. INTRODUCTION +Machine learning has improved in recent years and excels in domains, +where it is hard to find an explicit mathematical description of the solution. +In autonomous driving machine learning led to great improvements in +perception tasks. However, driving in crowded scenes remains challenging +for autonomous vehicles (AVs), mainly because the motion prediction +becomes harder due to the increasing number of possible interactions among +the traffic participants while paying attention to the road. +This problem is not restricted to autonomous driving and can easily be +transferred to other use cases, where autonomous systems interact and share +their space with humans, e.g. a logistic robot in a warehouse. In this work +we focus on motion prediction for AVs. +Recent works [1], [2] solve the spatio-temporal characteristic of the +problem in a two staged fusion approach. Firstly dynamic information, i.e. +the past trajectory of the traffic participants, is fused over time. Secondly +information is shared between the traffic participants and the road. [3] +propose a simultaneous temporal and spacial fusion of the past trajectories, +using a masked transformer. This allows to capture the dynamic context at +a higher resolution. However, map information is modelled as a birds eye +view image and processed via CNN. This is not optimal, because each agent +should process only surrounding road elements, that really influence their +future motion and not the complete map. Works like [1] and [4] model +the map as a homogeneous graph, therefore allowing traffic participants +to attend to areas of the map they are currently driving on. But those +approaches use the aforementioned staged fusion approach. In contrast, we +propose a heterogeneous graph for simultaneous attention to past history, +other agents’ time-discrete trajectory and map information without using +pre-fused data. In summary, the contribution of this paper includes: +• Holistic heterogeneous graph: Formulating the problem as a graph +without pre-fused data makes it possible to capture interactions at a +higher resolution. +• Modularity: More opportunities to encode expert-knowledge in the +graph via edges and their features. The modular construction of the +graph also allows for further extensions in the future. +• Benchmarking: INTERACTION and Argoverse +II. RELATED WORK +Motion prediction is an ongoing research topic in the field of autonomous +driving. In this section we provide an overview regarding graph neural +1 +FZI +Research +Center +for +Information +Technology, +76131 +Karlsruhe, +Germany. +daniel.grimm, schoerner, +zoellner@fzi.de +2 Karlsruhe Institute of Technology (KIT), Germany. +Fig. 1: Nodes in the heterogeneous graph. Map-nodes are depicted in yellow. +Different colored nodes represent agent-nodes, where nodes of the same +color belong to the same trajectory. Time context is visualized with color +fading. The high definition map (HD-Map) is depicted in light gray for +better understanding of the traffic scene. +networks (GNN) and motion prediction. As we are persuing a learned +prediction approach we are focussing on learning based approaches for +motion prediction. +A. Graph Neural Networks +Graph neural networks are used to extract information from data which +can be structured in graphs. For homogeneous graphs exist a wide variety +of operations to exchange information between nodes, e.g. GCN [5], +GraphSAGE [6], GAT [7] and Gatv2 [8], each following the message +passing scheme [9] to update the nodes in the graph. Most previous works, +like [5], [7] focus on homogeneous graphs, which is not sufficient in the +field of motion prediction, where different entities interact. Heterogeneous +graphs consist of different node and edge types [10]. [11] propose to +model attention in a heterogeneous graph in a two stage approach, called +Node-Level Attention and Semantic-Level Attention. [12] introduces ideas +of a Transformer [13] in a heterogeneous graph. The attention matrix is +calculated dependent on the edge type and node type. However edge features +are not considered. +B. Motion Prediction +The task of motion prediction is mostly formulated as a seq2seq problem. +Therefore early motion prediction models, such as [14], [15], [16] rely on +Recurrent Neural Network (RNN) structures like LSTM [17] or GRU [18]. +With the success of Convolutional Neural Networks (CNNs) in the domain +of image classification [19], [20], it became possible to use a 2D birds +eye view (BEV) image of the street layout in motion prediction. [21], [22] +and [23] encode a rich representation of the environment including road +elements, dynamic context and other traffic participants in the image. Due +to the success of Transformer [13] in Natural Language Processing, which +is also a seq2seq problem, works such as [24] and [25] adopted the attention +mechanism for motion prediction. [3] combines attention over time and other +agents in one Transformer called Agentformer. Attention is done in a fully +connected fashion, not regarding spatial distance between agents. To the +authors knowledge VectorNet [26] and LaneGCN [4] were the first models, +to use a GNN for motion prediction. VectorNet uses local graph to obtain +polyline-level features for agent trajectories and lanes. Afterwards these +features are used in a global interaction graph, which is fully connected, +undirected and homogeneous. In contrast to Vectornet, LaneGCN uses a + +GNN +agent-node +edge +pos +features +map-node ++ +K modes +graph update +MLP +MLP +MLP +MLP +MLP +Predictor +Embedding +pos +features ++ +MLP +MLP +MLP +MLP +features +L layers +trajectory +score +Temporal +Encoding +2 +3 +2 +2 +3 +128 +128 +128 +128 +128 +128 +128 +128 +for each +for each +for each +30, 2 +1 +Fig. 2: Proposed concept. Inputs are embedded in separate embedding modules. Heterogeneous GNN is used to generate latent representation of all agents +in the scene. Prediction head outputs a future trajectory for each agent. +segment of a polyline as node in their lane graph, therefore capturing the +map at a higher resolution. The map-nodes in the heterogeneous graph +proposed in our model use a similar map representation as LaneGCN. Heat +[2] and HDGT [1] propose a heterogeneous interaction graph, where the +nodes represent higher-level features, such as agent trajectories or lanes. +Heat constructs the street layout with a CNN from BEV images. HDGT +uses a simplified PointNet [27] to encode lane features from a vectorized +format. [28], and [29] introduce a graph-based spatial-temporal convolution. +Our proposed heterogeneous graph differs from above mentioned works by +the differently modelled temporal information. Instead of fusing temporal +information outside of the graph [2], [1], or using a separate graph for +each time-step [28], [29], we combine time variant information, e.g. agent +trajectories, in one graph. Time information is preserved by the usage of a +temporal encoding, see Sec. ??. To the knowledge of the authors, we are +the first to model the whole encoding step in a single graph for the task of +motion prediction. +III. CONCEPT +The general pipeline is depicted in Fig. 2. The model consists of an +embedding part followed by an encoder-decoder structure. As encoder +we propose a spatio temporal static heterogeneous graph, which includes +encoding the past trajectory as well as social context attention and the +encoding of the street layout. The graph yields a latent feature vector per +agent. The decoder is a normal MLP that outputs multi-modal predictions +for each agent in the scene. We use a scene-centric data representation. +A. Embedding +First, Agent-nodes, map-nodes and edge features are embedded to a +higher dimension f using a set of Multilayer Perceptrons (MLPs), each +with a linear layer, followed by ReLU Activation and Layer-Normalization. +A detailed view of the embedding process is depicted in Fig. 2. In order to +represent the timestamp of an agent-node, a temporal encoding τττ, similar +to the positional encoding in Transformers [13], is added to the agent-nodes +in the last step of the embedding. +τ(t, 2i) = sin +� +t/10000 +2i +f +� +(1) +τ(t, 2i + 1) = cos +� +t/10000 +2i+1 +f +� +(2) +aaat +i = W +W +W 1 +� +aaat +i ∥ τττ(t) +� +(3) +where τ(t, 2i) and τ(t, 2i + 1) refer to the even resp. odd index of feature +dimension in τττ(t). +B. Hetero GNN +The heterogeneous graph is defined as G = {N, E}, where N denotes +the set of nodes and E denotes the set of edges with their corresponding +edge features. A scene consists of traffic participants, hereafter referred to +as agents, and the HD-Map. In this work, we used two types of nodes, +N = {A, M}. A refers to the set of agent-nodes, where a single agent- +node aaat +i refers to time-step t of the observed past trajectory of the i-th +agent. aaat +i initially consists of the current position, velocity and orientation, +so that aaai = (xi, yi, vxi, vyi, hi)⊺. M refers to the set of map-modes, +where a single map-mode m +m +mi refers to a segment of a centerline of the +vectorized HD-Map, therefore initially consisting of direction and position, +m +m +mi = (xi, yi, ∆xi, ∆yi)⊺. +The set of the different directed edge types E of the heterogeneous graph +can be seen in Fig. 3. The connections of a specific edge type from node +type j to node type i with relation r are stored in the adjacency matrix +AAAj,r,i and eeej,r,i denote the corresponding edge features. Each edge in the +graph contains embedded edge features, initially consisting of the difference +of the 2d-positions. +To update the node features xxx(l) +i,r ∈ RF of node i in Layer l for a specific +edge type a basic message passing scheme is used. +ˆxˆxˆx(l) +i,r = γ(l) +r +� +�xxx(l−1) +i +, +� +j∈N (i) +φ(l) +r +� +xxx(l−1) +i +,xxx(l−1) +j +,eee(l−1) +j,r,i +� +� +� +(4) +φ(l) is a MLP used to calculate the messages of the neighboring nodes +xxxj while also using the edge features eeej,r,i from the edge connecting the +corresponding node j to node i with relation r. The neighboring nodes are +determined by the associated adjacency matrix. In Eq. 4 the messages are +aggregated using a sum. γ(l) denotes another MLP that is used to calculate +the edge type specific update ˆxˆxˆx(l) +i,r. Each layer of the GNN consists of +message passing followed by ReLU Activation, Residual-Connection and a +Layer Normalization, where we use the sum over all edge types to get the +final message passing output of layer l.: +xxx(l) +i += norm +� +�ReLU +� +� � +r∈E(i) +ˆxˆxˆx(l) +i,r +� +� + xxx(l−1) +i +� +� +(5) +In the proposed heterogeneous graph, not every edge is used simultane- +ously to update the nodes, instead a multi-stage approach is used, which +consists of agent-node and map-node encoding, followed by the fusion of +agents with the HD-Map and finished by the generation of a latent feature +vector per agent. +1) Map Context: +Map-nodes are connected to other map-nodes +using the spatial relations predecessor, successor, left neighbour and right +neighbour. During message passing it is favorable to prefer information +propagation along the road-direction rather than perpendicular to it. That’s +because most road users travel along the road and not across. We accomplish +this by adding new edges along the road connecting a map node with its i-th +predecessor respectively successor, using the i-th power of the corresponding + +{agent, suc, agent} +{agent, attent, agent} +{agent, merge, agent} +{agent, pred, agent} +(a) Edges among agent-nodes. Timestep information is presented +with color shading. Agent belonging to one agent trajectory are +presented on the left. Social context is visualized on the right. +? +{map, suc-3, map} +{map, suc-2, map} +{map, suc-1, map} +{map, pre-1, map} +{map, pre-2, map} +{map, pre-3, map} +{map, pre-4, map} +{map, right, map} +{map, left, map} +{map, suc-4, map} +(b) Edges among map-nodes. For a better view, only the edges +from one map-node are depicted. +? +{agent, drives-on, map} +{map, gives-traffic-info, agent} +(c) Edges between agent-nodes and map-nodes. For a better view +one agent-node is depicted as destination on the left, on the right +one map-node is selected as destination. +Fig. 3: Overview of the edges used by the heterogeneous GNN. +adjacency, eg.g. {map, pre-2, map}. A detailed view of the edges between +map-nodes is given in Fig. 3b and is similar to LaneGCN [4]. For message +generation we propose an extension to the basic GCN-Conv [5]. We include +the usage of edge features in the message generation φ resulting in the node +updates ˆxˆxˆx(l) +i,r +ˆxˆxˆx(l) +i,r = +� +j∈N (i) +1 +� +deg(i) +� +deg(j) +�� +xxx(l−1) +j ++ eee(l−1) +j,r,i +� +W +W +W +� ++ bbb +(6) +where W +W +W and bbb refer to learnable parameters. Edge and node features are +added together, which reduces the number of learnable parameters without +decreasing performance, see Seq. IV-C To gather a good encoding of the +HD-Map Data we use five layers, where each layer is constructed like Eq. 5. +2) Agent Context: An agent-node is connected to its predecessor +and successor belonging to the past trajectory of the agent. The corre- +sponding edges are named {agent, pre, agent} and {agent, suc, agent}. +For social context {agent, social, agent}, every agent-node is connected +to agent-nodes of the previous, same and future timestamp, which belong +to other agents. The respective edges are shown in Fig. 3a. Updating the +agent-nodes is similar to the map-nodes. In order to pass information from +the first to the last agent-node of an agents trajectory, the number of used +layers n corresponds to the number of time-steps of the past trajectory, +e.g. Argoverse: n = 20 INTERACTION: n = 10. Messages are generated +using Eq. 6. +Social context is added during the last two layers with a multi head +graph attention module (GATv2) [8]. Therefore edges of type {agent, attend, +agent} are used. The node updates for relation r are given by +ˆxˆxˆx(l) +i,r = αr +i,ixxx(l−1) +i +W +W +W 1 + +� +j∈N (i) +αr +i,jxxx(l−1) +j +W +W +W 2 +(7) +where the attention coefficients αi,j are calculated as +αi,j = softmax (LeakyReLU ([xxxi ∥ xxxj ∥ eeej,r,i]W +W +W 3)aaa) +(8) +3) Context fusion: +To properly fuse the HD-Map with the past +trajectories of the agents, we use two edge types {agent, drives-on, map} +and {map, gives-traffic-info, agent}. These two edges use a multi head +GATv2 [8] module. The source nodes are selected based on the euclidean +distance dth of the 2d-position to the target nodes. dth is dynamically +calculated using the velocity of the agent-nodes and a threshold time tth. +This compensates for faster moving agents. Furthermore we include the +edges introduced in Seq. ?? and Seq. III-B.1. In total we use two fusion +layers. +Since the latent representation of an agents past trajectory is spread out +between all agent-nodes belonging to this specific agent, for every agent +the last observed agent-node at tobs is selected as final feature vector and +therefore updated by a multi head GATv2 [8] module using edges to the past +agent-nodes of that agent. Fig. 3a shows the edges {agent, merge, agent} +for this purpose. +C. Motion-Prediction Head +We used a combination of regression and scoring in separate MLPs +to generate K possible trajectories per agent. For each mode a new +regression and classification MLP is instantiated. Input is the latent feature +vector for each agent. To calculate the trajectory score we also use the +predicted trajectory. The two MLPs are similar and consist of a linear +layer with a residual connection, ReLU Activation, Layer Normalization +and another linear layer. The model outputs the predicted trajectories τττ of +shape [A, K, Tf, 2], and the scores sss of shape [A, K], where A is the +number of agents and Tf equals the prediction horizon. +D. Loss +The Loss L consists of a regression Loss and a classification Loss. +L = Lreg + λLcls +(9) +As regression Loss Lreg a smooth L1 Loss is used. Lreg is only calculated +for the mode kmin with minimal final displacement error (FDE) to the ground +truth to prevent mode collapse. +Lreg = +1 +AT +A +� +a +Tf +� +t +� +n∈{x,y} +smoothL1 +� +τa,t,kmin,n, ˆτa,t,n +� +(10) +with +smoothL1(x, y) = +� +0.5 ∗ (x − y)2, +if|x − y| < 1 +|x − y| − 0.5, +otherwise +(11) +where ˆt refers to the ground truth. The classification loss Lcls is a max- +margin loss with margin m. +Lcls = +1 +A(K − 1) +A +� +a +K +� +k̸=kmin +max +� +0, sa,k + m − sa,kmin +� +(12) +IV. EVALUATION +In the following we evaluate our model on the INTERACTION dataset +[30] and the Argoverse motion forecast dataset [31]. First we introduce +the datasets, the evaluation metrics and the used hyperparameter settings. +Afterwards we conduct an ablation studies on the architecture and finally +compare our model to the state-of-the-art. + +A. Experimental Settings +The Argoverse Motion Forecast Dataset is a large scale collection of +323557 samples, each with a duration of 5s, resulting in a total of 320h. +The data was collected in Miami and Pittsburgh with 10 Hz. The task is to +predict the future locations of one agent for 3s, given its history of the last +2 seconds. HD-Map data is provided in an argoverse specific format. +The INTERACTION Dataset is a highly interactive dataset, recorded +in 5 different locations, including Roundabouts, Merging Scenarios and +intersections in Germany, USA and China. It consists of around 16,5h of +data including 40054 trajectories, sampled at 10Hz. The task is to predict +the future locations of all agents in the scene for 3s, given their history +for the last second. The HD-Map data is provided using the Lanelet2 [32] +format. +To evaluate the results quantitatively on Argoverse we use Minimum +Average Displacement Error (minADE), Minimum Final Displacement Error +(minFDE) and Minimum Miss Rate (minMR). For multi-modal predictions +minFDE refers to the minimum euclidean distance of the predicted trajectory +and the ground truth at the prediction horizon TF over all modes. minADE +is defined as the euclidean distance between the ground truth and the +predicted positions averaged by time. minMR indicates the ratio of the +predictions where the final position of the trajectory of the best mode +is more than a certain threshold, usually 2m, away from the ground +truth. On INTERACTION we use Minimum Joint Average Displacement +error (minJADE), Minimum Joint Final Displacement Error (minJFDE) and +Minimum Joint Miss Rate (minJMR) as metrics to measure the performance +of joint motion prediction. ”Joint” is hereby referring to the fact, that the +mode is selected for all agents at once and therefore the metrics are also +averaged by all agents in the scene. +Training on each dataset was done on a RTX 3080 GPU for 40 epochs, +starting with an initial learning rate of 1e-3 and a decay of 0.5 every fifth +epoch. We used the Adam [33] optimizer, a batch size of 8 and a weight +decay of 0.5% for all weights not used in a normalization layer. All attention +modules have 4 heads and the result of each head is concatenated. Training +on Argoverse took 33h, and 8h on INTERACTION. +For Argoverse [31] we set the last time-step of the ego-agents as the +origin of the local fixed coordinate system. For INTERACTION [30] the +origin is set to the geometric center point of all the trajectories in the scene. +For both datasets we use a threshold-distance of 80m to the origin of the +local coordinate system to determine the relevant lanes and agents for the +graph. +B. Results +Tab. I shows the results of our model of the Argoverse and INTERAC- +TION dataset. We achieve state of the art results, while having only 2.5 +Mio Parameters. Our approach already reaches the same level as the other +approaches while having significantly less parameter. This means that the +knowledge is represented very efficiently, leaving space for further features +to be included or to be run more efficiently on computing hardware. +TABLE I: Results on argoverse test, regular INTERACTION single and +regular INTERACTION multi test dataset. +argoverse +K=6 +No. of +single +minADE +minFDE +minMR +Parameters +HoliGraph (ours) +0.98 +1.65 +0.172 +2.5 Mio +DenseTNT [34] +0.94 +1.49 +0.105 +1.4 Mio +Scene Trans.[24] +0.80 +1.23 +0.13 +15.3 Mio +LaneGCN [4] +0.87 +1.36 +0.16 +3.6 Mio +interaction +K=6 +No. of +single +minADE +minFDE +minMR +Parameters +HoliGraph (ours) +0.213 +0.529 +0.029 +2.5 Mio +DenseTNT [34] +0.2819 +0.6371 +0.028 +1.4 Mio +HDGT [1] +0.1085 +0.3361 +0.014 +12 Mio +interaction +K=6 +No. of +multi +minJADE +minJFDE +minJMR +Parameters +HoliGraph (ours) +0.362 +1.043 +0.138 +2.5 Mio +ReCoG2 [35] +0.330 +0.932 +0.194 +- +HDGT [1] +0.2162 +0.7309 +0.1384 +12 Mio +C. Ablation Study +To investigate the effect of using different types of context information +on the prediction accuracy, we conducted an ablation study. Tab. II shows +that the performance on the INTERACTION validation dataset is increasing +when providing the model with more context information. It also shows the +importance of edge features to provide the model with additional relational +information. +TABLE II: Results on INTERACTION validation dataset for different types +of context information as input. History means each agent only knows his +past trajectory. Map means the usage of map-nodes. Social means that agents +are also connected to other agents. Relational refers to the usage of edge +features. +context information +K=6 +history +map +social +relational +minJADE +minJFDE +minJMR +✓ +0.607 +1.745 +0.311 +✓ +✓ +0.562 +1.601 +0.272 +✓ +✓ +0.458 +1.254 +0.188 +✓ +✓ +✓ +0.441 +1.212 +0.178 +✓ +✓ +✓ +✓ +0.362 +1.043 +0.138 +In Tab. III we investigated the effect of residual connections during node +update, the temporal encoding of agent-nodes and the way of including +edge features. The residual connections improve the model performance up +to 17 %. Adding timestamp information directly to the agent-nodes with +the temporal encoding from Eq. 3 further improves performance. The third +row in Tab. III refers to the architecture used in the final model, since the +concatenation of edge features with node features during graph update only +results in a small performance gain, but significantly increases the number +of learnable parameters from 2.5 Mio to 4.2 Mio. +TABLE III: Results on INTERACTION validation dataset for different +architectures. Residual means residual connections during node update. +Temporal refers to the temporal encoding of agent-nodes and concat refers +to the concatenation of edge and node features. +architecture +K=6 +residual +temp +concat +minJADE +minJFDE +minJMR +0.426 +1.214 +0.177 +✓ +0.383 +1.090 +0.151 +✓ +✓ +0.362 +1.043 +0.138 +✓ +✓ +✓ +0.361 +1.039 +0.137 +Lastly we investigated the influence of different attention mechanisms. +While all three modules resulting in roughly the same number of learnable +parameters, gatv2 module outperforms the other attention-modules. +TABLE IV: Results on INTERACTION validation dataset for different +attention mechanism. +attention modules +K=6 +gat +gatv2 +transformer +minJADE +minJFDE +minJMR +[7] +[8] +[36] +✓ +0.434 +1.207 +0.178 +✓ +0.362 +1.043 +0.138 +✓ +0.416 +1.149 +0.163 +D. Qualitative Results +Some qualitative results are depicted in Fig. 4. Fig. 4a shows the +performance of the model in a complex intersection with a lot of interactions +between the traffic participants. In the scene, vehicles as well as pedestrians +are present. Our model is able to predict all agents well. For most agents, +the lateral prediction is almost perfect, while their longitudinal prediction +shows small deviations. On the right side of Fig. 4b a pedestrian is crossing +the road. Nearly all modes indicate a light left turn. This is a result of the +attention to the map-nodes, as the driving direction of the road is to the +right. We will use this showcase as a motivation to distinguish in our future +work between road bound and non-road bound users. +V. CONCLUSION +In this paper we have proposed a new way to represent temporal +information in heterogeneous graphs for motion prediction. Instead of +compressing the temporal information, we embed the whole past trajectories +of all agents into the GNN. We achieve state of the art results, while having +considerably less learnable parameters. We did an extensive ablation study +to verify the effectiveness of each design decision. The evaluation was based + +(a) Dense traffic scene +(b) Pedestrian crossing +Fig. 4: Qualitative results on INTERACTION validation dataset. History is depicted in dark green, ground truth in light green, predictions in light red. The +prediction with the highest score is depicted in red. The map-nodes are depicted as light red dots. +on the prediction of vehicles along a road network and conducted on two +different state-of-the-art datasets. As the holistic graph representation allows +to include arbitrary information, we are going to further distinguish between +road bound agents, e.g. cars, trucks and motorcycles, and non-road bound +agents, e.g. pedestrians. +ACKNOWLEDGMENT +The research leading to these results was conducted within the project +KIsSME (Artificial Intelligence for selective near-real-time recordings of +scenario and maneuver data in testing highly automated vehicles) and +was funded by the German Federal Ministry for Economic Affairs and +Climate Action. Responsibility for the information and views set out in +this publication lies entirely with the authors. +REFERENCES +[1] +X. Jia, P. Wu, L. Chen, et al., Hdgt: Heterogeneous +driving graph transformer for multi-agent trajectory +prediction via scene encoding, 2022. +[2] +X. Mo, Y. Xing, and C. Lv, Heterogeneous edge- +enhanced graph attention network for multi-agent +trajectory prediction, 2021. +[3] +Y. Yuan, X. Weng, Y. 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Feng, et al., Masked label +prediction: Unified message passing model for semi- +supervised classification, 2020. + diff --git a/j9FRT4oBgHgl3EQfWzfr/content/tmp_files/load_file.txt b/j9FRT4oBgHgl3EQfWzfr/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6144b56b8f65b20ffe171401efb1269a247848e3 --- /dev/null +++ b/j9FRT4oBgHgl3EQfWzfr/content/tmp_files/load_file.txt @@ -0,0 +1,571 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf,len=570 +page_content='©2023 IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Personal use of this material is permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Accepted to be published in: 2023 IEEE International Conference on Robotics and Automation (ICRA), May 29 - June 2, 2023, London arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='13545v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='RO] 31 Jan 2023 Holistic Graph-based Motion Prediction Daniel Grimm1, Philip Sch¨orner1, Moritz Dreßler2 and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='-Marius Z¨ollner1,2 Abstract— Motion prediction for automated vehicles in com- plex environments is a difficult task that is to be mastered when automated vehicles are to be used in arbitrary situ- ations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Many factors influence the future motion of traffic participants starting with traffic rules and reaching from the interaction between each other to personal habits of human drivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Therefore we present a novel approach for a graph- based prediction based on a heterogeneous holistic graph representation that combines temporal information, properties and relations between traffic participants as well as relations with static elements like the road network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' The information are encoded through different types of nodes and edges that both are enriched with arbitrary features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' We evaluated the approach on the INTERACTION and the Argoverse dataset and conducted an informative ablation study to demonstrate the benefit of different types of information for the motion prediction quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' INTRODUCTION Machine learning has improved in recent years and excels in domains, where it is hard to find an explicit mathematical description of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' In autonomous driving machine learning led to great improvements in perception tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' However, driving in crowded scenes remains challenging for autonomous vehicles (AVs), mainly because the motion prediction becomes harder due to the increasing number of possible interactions among the traffic participants while paying attention to the road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' This problem is not restricted to autonomous driving and can easily be transferred to other use cases, where autonomous systems interact and share their space with humans, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' a logistic robot in a warehouse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' In this work we focus on motion prediction for AVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Recent works [1], [2] solve the spatio-temporal characteristic of the problem in a two staged fusion approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Firstly dynamic information, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' the past trajectory of the traffic participants, is fused over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Secondly information is shared between the traffic participants and the road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' [3] propose a simultaneous temporal and spacial fusion of the past trajectories, using a masked transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' This allows to capture the dynamic context at a higher resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' However, map information is modelled as a birds eye view image and processed via CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' This is not optimal, because each agent should process only surrounding road elements, that really influence their future motion and not the complete map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Works like [1] and [4] model the map as a homogeneous graph, therefore allowing traffic participants to attend to areas of the map they are currently driving on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' But those approaches use the aforementioned staged fusion approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' In contrast, we propose a heterogeneous graph for simultaneous attention to past history, other agents’ time-discrete trajectory and map information without using pre-fused data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' In summary, the contribution of this paper includes: Holistic heterogeneous graph: Formulating the problem as a graph without pre-fused data makes it possible to capture interactions at a higher resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Modularity: More opportunities to encode expert-knowledge in the graph via edges and their features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' The modular construction of the graph also allows for further extensions in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Benchmarking: INTERACTION and Argoverse II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' RELATED WORK Motion prediction is an ongoing research topic in the field of autonomous driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' In this section we provide an overview regarding graph neural 1 FZI Research Center for Information Technology, 76131 Karlsruhe, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' daniel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='grimm, schoerner, zoellner@fzi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='de 2 Karlsruhe Institute of Technology (KIT), Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' 1: Nodes in the heterogeneous graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Map-nodes are depicted in yellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Different colored nodes represent agent-nodes, where nodes of the same color belong to the same trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Time context is visualized with color fading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' The high definition map (HD-Map) is depicted in light gray for better understanding of the traffic scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' networks (GNN) and motion prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' As we are persuing a learned prediction approach we are focussing on learning based approaches for motion prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Graph Neural Networks Graph neural networks are used to extract information from data which can be structured in graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' For homogeneous graphs exist a wide variety of operations to exchange information between nodes, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' GCN [5], GraphSAGE [6], GAT [7] and Gatv2 [8], each following the message passing scheme [9] to update the nodes in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Most previous works, like [5], [7] focus on homogeneous graphs, which is not sufficient in the field of motion prediction, where different entities interact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Heterogeneous graphs consist of different node and edge types [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' [11] propose to model attention in a heterogeneous graph in a two stage approach, called Node-Level Attention and Semantic-Level Attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' [12] introduces ideas of a Transformer [13] in a heterogeneous graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' The attention matrix is calculated dependent on the edge type and node type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' However edge features are not considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Motion Prediction The task of motion prediction is mostly formulated as a seq2seq problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Therefore early motion prediction models, such as [14], [15], [16] rely on Recurrent Neural Network (RNN) structures like LSTM [17] or GRU [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' With the success of Convolutional Neural Networks (CNNs) in the domain of image classification [19], [20], it became possible to use a 2D birds eye view (BEV) image of the street layout in motion prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' [21], [22] and [23] encode a rich representation of the environment including road elements, dynamic context and other traffic participants in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Due to the success of Transformer [13] in Natural Language Processing, which is also a seq2seq problem, works such as [24] and [25] adopted the attention mechanism for motion prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' [3] combines attention over time and other agents in one Transformer called Agentformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Attention is done in a fully connected fashion, not regarding spatial distance between agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' To the authors knowledge VectorNet [26] and LaneGCN [4] were the first models, to use a GNN for motion prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' VectorNet uses local graph to obtain polyline-level features for agent trajectories and lanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Afterwards these features are used in a global interaction graph, which is fully connected, undirected and homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' In contrast to Vectornet, LaneGCN uses a GNN agent-node edge pos features map-node + K modes graph update MLP MLP MLP MLP MLP Predictor Embedding pos features + MLP MLP MLP MLP features L layers trajectory score Temporal Encoding 2 3 2 2 3 128 128 128 128 128 128 128 128 for each for each for each 30, 2 1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' 2: Proposed concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Inputs are embedded in separate embedding modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Heterogeneous GNN is used to generate latent representation of all agents in the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Prediction head outputs a future trajectory for each agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' segment of a polyline as node in their lane graph, therefore capturing the map at a higher resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' The map-nodes in the heterogeneous graph proposed in our model use a similar map representation as LaneGCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Heat [2] and HDGT [1] propose a heterogeneous interaction graph, where the nodes represent higher-level features, such as agent trajectories or lanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Heat constructs the street layout with a CNN from BEV images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' HDGT uses a simplified PointNet [27] to encode lane features from a vectorized format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' [28], and [29] introduce a graph-based spatial-temporal convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Our proposed heterogeneous graph differs from above mentioned works by the differently modelled temporal information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Instead of fusing temporal information outside of the graph [2], [1], or using a separate graph for each time-step [28], [29], we combine time variant information, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' agent trajectories, in one graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Time information is preserved by the usage of a temporal encoding, see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='. To the knowledge of the authors, we are the first to model the whole encoding step in a single graph for the task of motion prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' CONCEPT The general pipeline is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' The model consists of an embedding part followed by an encoder-decoder structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' As encoder we propose a spatio temporal static heterogeneous graph, which includes encoding the past trajectory as well as social context attention and the encoding of the street layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' The graph yields a latent feature vector per agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' The decoder is a normal MLP that outputs multi-modal predictions for each agent in the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' We use a scene-centric data representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Embedding First, Agent-nodes, map-nodes and edge features are embedded to a higher dimension f using a set of Multilayer Perceptrons (MLPs), each with a linear layer, followed by ReLU Activation and Layer-Normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' A detailed view of the embedding process is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' In order to represent the timestamp of an agent-node, a temporal encoding τττ, similar to the positional encoding in Transformers [13], is added to the agent-nodes in the last step of the embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' τ(t, 2i) = sin � t/10000 2i f � (1) τ(t, 2i + 1) = cos � t/10000 2i+1 f � (2) aaat i = W W W 1 � aaat i ∥ τττ(t) � (3) where τ(t, 2i) and τ(t, 2i + 1) refer to the even resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' odd index of feature dimension in τττ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Hetero GNN The heterogeneous graph is defined as G = {N, E}, where N denotes the set of nodes and E denotes the set of edges with their corresponding edge features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' A scene consists of traffic participants, hereafter referred to as agents, and the HD-Map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' In this work, we used two types of nodes, N = {A, M}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' A refers to the set of agent-nodes, where a single agent- node aaat i refers to time-step t of the observed past trajectory of the i-th agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' aaat i initially consists of the current position, velocity and orientation, so that aaai = (xi, yi, vxi, vyi, hi)⊺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' M refers to the set of map-modes, where a single map-mode m m mi refers to a segment of a centerline of the vectorized HD-Map, therefore initially consisting of direction and position, m m mi = (xi, yi, ∆xi, ∆yi)⊺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' The set of the different directed edge types E of the heterogeneous graph can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' The connections of a specific edge type from node type j to node type i with relation r are stored in the adjacency matrix AAAj,r,i and eeej,r,i denote the corresponding edge features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Each edge in the graph contains embedded edge features, initially consisting of the difference of the 2d-positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' To update the node features xxx(l) i,r ∈ RF of node i in Layer l for a specific edge type a basic message passing scheme is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' ˆxˆxˆx(l) i,r = γ(l) r � �xxx(l−1) i , � j∈N (i) φ(l) r � xxx(l−1) i ,xxx(l−1) j ,eee(l−1) j,r,i � � � (4) φ(l) is a MLP used to calculate the messages of the neighboring nodes xxxj while also using the edge features eeej,r,i from the edge connecting the corresponding node j to node i with relation r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' The neighboring nodes are determined by the associated adjacency matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' 4 the messages are aggregated using a sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' γ(l) denotes another MLP that is used to calculate the edge type specific update ˆxˆxˆx(l) i,r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Each layer of the GNN consists of message passing followed by ReLU Activation, Residual-Connection and a Layer Normalization, where we use the sum over all edge types to get the final message passing output of layer l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=': xxx(l) i = norm � �ReLU � � � r∈E(i) ˆxˆxˆx(l) i,r � � + xxx(l−1) i � � (5) In the proposed heterogeneous graph, not every edge is used simultane- ously to update the nodes, instead a multi-stage approach is used, which consists of agent-node and map-node encoding, followed by the fusion of agents with the HD-Map and finished by the generation of a latent feature vector per agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' 1) Map Context: Map-nodes are connected to other map-nodes using the spatial relations predecessor, successor, left neighbour and right neighbour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' During message passing it is favorable to prefer information propagation along the road-direction rather than perpendicular to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' That’s because most road users travel along the road and not across.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' We accomplish this by adding new edges along the road connecting a map node with its i-th predecessor respectively successor, using the i-th power of the corresponding {agent, suc, agent} {agent, attent, agent} {agent, merge, agent} {agent, pred, agent} (a) Edges among agent-nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Timestep information is presented with color shading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Agent belonging to one agent trajectory are presented on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Social context is visualized on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' {map, suc-3, map} {map, suc-2, map} {map, suc-1, map} {map, pre-1, map} {map, pre-2, map} {map, pre-3, map} {map, pre-4, map} {map, right, map} {map, left, map} {map, suc-4, map} (b) Edges among map-nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' For a better view, only the edges from one map-node are depicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' {agent, drives-on, map} {map, gives-traffic-info, agent} (c) Edges between agent-nodes and map-nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' For a better view one agent-node is depicted as destination on the left, on the right one map-node is selected as destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' 3: Overview of the edges used by the heterogeneous GNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' adjacency, eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' {map, pre-2, map}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' A detailed view of the edges between map-nodes is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' 3b and is similar to LaneGCN [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' For message generation we propose an extension to the basic GCN-Conv [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' We include the usage of edge features in the message generation φ resulting in the node updates ˆxˆxˆx(l) i,r ˆxˆxˆx(l) i,r = � j∈N (i) 1 � deg(i) � deg(j) �� xxx(l−1) j + eee(l−1) j,r,i � W W W � + bbb (6) where W W W and bbb refer to learnable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Edge and node features are added together, which reduces the number of learnable parameters without decreasing performance, see Seq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' IV-C To gather a good encoding of the HD-Map Data we use five layers, where each layer is constructed like Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' 2) Agent Context: An agent-node is connected to its predecessor and successor belonging to the past trajectory of the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' The corre- sponding edges are named {agent, pre, agent} and {agent, suc, agent}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' For social context {agent, social, agent}, every agent-node is connected to agent-nodes of the previous, same and future timestamp, which belong to other agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' The respective edges are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Updating the agent-nodes is similar to the map-nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' In order to pass information from the first to the last agent-node of an agents trajectory, the number of used layers n corresponds to the number of time-steps of the past trajectory, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Argoverse: n = 20 INTERACTION: n = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Messages are generated using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Social context is added during the last two layers with a multi head graph attention module (GATv2) [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Therefore edges of type {agent, attend, agent} are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' The node updates for relation r are given by ˆxˆxˆx(l) i,r = αr i,ixxx(l−1) i W W W 1 + � j∈N (i) αr i,jxxx(l−1) j W W W 2 (7) where the attention coefficients αi,j are calculated as αi,j = softmax (LeakyReLU ([xxxi ∥ xxxj ∥ eeej,r,i]W W W 3)aaa) (8) 3) Context fusion: To properly fuse the HD-Map with the past trajectories of the agents, we use two edge types {agent, drives-on, map} and {map, gives-traffic-info, agent}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' These two edges use a multi head GATv2 [8] module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' The source nodes are selected based on the euclidean distance dth of the 2d-position to the target nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' dth is dynamically calculated using the velocity of the agent-nodes and a threshold time tth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' This compensates for faster moving agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Furthermore we include the edges introduced in Seq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' and Seq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' III-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' In total we use two fusion layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Since the latent representation of an agents past trajectory is spread out between all agent-nodes belonging to this specific agent, for every agent the last observed agent-node at tobs is selected as final feature vector and therefore updated by a multi head GATv2 [8] module using edges to the past agent-nodes of that agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' 3a shows the edges {agent, merge, agent} for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Motion-Prediction Head We used a combination of regression and scoring in separate MLPs to generate K possible trajectories per agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' For each mode a new regression and classification MLP is instantiated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Input is the latent feature vector for each agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' To calculate the trajectory score we also use the predicted trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' The two MLPs are similar and consist of a linear layer with a residual connection, ReLU Activation, Layer Normalization and another linear layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' The model outputs the predicted trajectories τττ of shape [A, K, Tf, 2], and the scores sss of shape [A, K], where A is the number of agents and Tf equals the prediction horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Loss The Loss L consists of a regression Loss and a classification Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' L = Lreg + λLcls (9) As regression Loss Lreg a smooth L1 Loss is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Lreg is only calculated for the mode kmin with minimal final displacement error (FDE) to the ground truth to prevent mode collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Lreg = 1 AT A � a Tf � t � n∈{x,y} smoothL1 � τa,t,kmin,n, ˆτa,t,n � (10) with smoothL1(x, y) = � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='5 ∗ (x − y)2, if|x − y| < 1 |x − y| − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='5, otherwise (11) where ˆt refers to the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' The classification loss Lcls is a max- margin loss with margin m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Lcls = 1 A(K − 1) A � a K � k̸=kmin max � 0, sa,k + m − sa,kmin � (12) IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' EVALUATION In the following we evaluate our model on the INTERACTION dataset [30] and the Argoverse motion forecast dataset [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' First we introduce the datasets, the evaluation metrics and the used hyperparameter settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Afterwards we conduct an ablation studies on the architecture and finally compare our model to the state-of-the-art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Experimental Settings The Argoverse Motion Forecast Dataset is a large scale collection of 323557 samples, each with a duration of 5s, resulting in a total of 320h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' The data was collected in Miami and Pittsburgh with 10 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' The task is to predict the future locations of one agent for 3s, given its history of the last 2 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' HD-Map data is provided in an argoverse specific format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' The INTERACTION Dataset is a highly interactive dataset, recorded in 5 different locations, including Roundabouts, Merging Scenarios and intersections in Germany, USA and China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' It consists of around 16,5h of data including 40054 trajectories, sampled at 10Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' The task is to predict the future locations of all agents in the scene for 3s, given their history for the last second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' The HD-Map data is provided using the Lanelet2 [32] format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' To evaluate the results quantitatively on Argoverse we use Minimum Average Displacement Error (minADE), Minimum Final Displacement Error (minFDE) and Minimum Miss Rate (minMR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' For multi-modal predictions minFDE refers to the minimum euclidean distance of the predicted trajectory and the ground truth at the prediction horizon TF over all modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' minADE is defined as the euclidean distance between the ground truth and the predicted positions averaged by time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' minMR indicates the ratio of the predictions where the final position of the trajectory of the best mode is more than a certain threshold, usually 2m, away from the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' On INTERACTION we use Minimum Joint Average Displacement error (minJADE), Minimum Joint Final Displacement Error (minJFDE) and Minimum Joint Miss Rate (minJMR) as metrics to measure the performance of joint motion prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' ”Joint” is hereby referring to the fact, that the mode is selected for all agents at once and therefore the metrics are also averaged by all agents in the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Training on each dataset was done on a RTX 3080 GPU for 40 epochs, starting with an initial learning rate of 1e-3 and a decay of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='5 every fifth epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' We used the Adam [33] optimizer, a batch size of 8 and a weight decay of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='5% for all weights not used in a normalization layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' All attention modules have 4 heads and the result of each head is concatenated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Training on Argoverse took 33h, and 8h on INTERACTION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' For Argoverse [31] we set the last time-step of the ego-agents as the origin of the local fixed coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' For INTERACTION [30] the origin is set to the geometric center point of all the trajectories in the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' For both datasets we use a threshold-distance of 80m to the origin of the local coordinate system to determine the relevant lanes and agents for the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Results Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' I shows the results of our model of the Argoverse and INTERAC- TION dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' We achieve state of the art results, while having only 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='5 Mio Parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Our approach already reaches the same level as the other approaches while having significantly less parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' This means that the knowledge is represented very efficiently, leaving space for further features to be included or to be run more efficiently on computing hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' TABLE I: Results on argoverse test, regular INTERACTION single and regular INTERACTION multi test dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' argoverse K=6 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' of single minADE minFDE minMR Parameters HoliGraph (ours) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='98 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='172 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='5 Mio DenseTNT [34] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='94 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='105 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='4 Mio Scene Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' [24] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='13 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='3 Mio LaneGCN [4] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='87 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='16 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='6 Mio interaction K=6 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' of single minADE minFDE minMR Parameters HoliGraph (ours) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='213 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='529 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='029 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='5 Mio DenseTNT [34] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='2819 0.' metadata={'source': 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minJADE minJFDE minJMR Parameters HoliGraph (ours) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='362 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='043 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='138 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='5 Mio ReCoG2 [35] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='330 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='932 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='194 HDGT [1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='2162 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='7309 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='1384 12 Mio C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Ablation Study To investigate the effect of using different types of context information on the prediction accuracy, we conducted an ablation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' II shows that the performance on the INTERACTION validation dataset is increasing when providing the model with more context information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' It also shows the importance of edge features to provide the model with additional relational information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' TABLE II: Results on INTERACTION validation dataset for different types of context information as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' History means each agent only knows his past trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Map means the usage of map-nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Social means that agents are also connected to other agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Relational refers to the usage of edge features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' context information K=6 history map social relational minJADE minJFDE minJMR ✓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='607 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} 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+page_content=' III we investigated the effect of residual connections during node update, the temporal encoding of agent-nodes and the way of including edge features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' The residual connections improve the model performance up to 17 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Adding timestamp information directly to the agent-nodes with the temporal encoding from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' 3 further improves performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' The third row in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' III refers to the architecture used in the final model, since the concatenation of edge features with node features during graph update only results in a small performance gain, but significantly increases the number of learnable parameters from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='5 Mio to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='2 Mio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' TABLE III: Results on INTERACTION validation dataset for different architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Residual means residual connections during node update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Temporal refers to the temporal encoding of agent-nodes and concat refers to the concatenation of edge and node features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' architecture K=6 residual temp concat minJADE minJFDE minJMR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='426 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='214 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='177 ✓ 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='039 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='137 Lastly we investigated the influence of different attention mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' While all three modules resulting in roughly the same number of learnable parameters, gatv2 module outperforms the other attention-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' TABLE IV: Results on INTERACTION validation dataset for different attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' attention modules K=6 gat gatv2 transformer minJADE minJFDE minJMR [7] [8] [36] ✓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='434 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='207 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='178 ✓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='362 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='043 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='138 ✓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='416 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='149 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='163 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Qualitative Results Some qualitative results are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' 4a shows the performance of the model in a complex intersection with a lot of interactions between the traffic participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' In the scene, vehicles as well as pedestrians are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Our model is able to predict all agents well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' For most agents, the lateral prediction is almost perfect, while their longitudinal prediction shows small deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' On the right side of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' 4b a pedestrian is crossing the road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Nearly all modes indicate a light left turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' This is a result of the attention to the map-nodes, as the driving direction of the road is to the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' We will use this showcase as a motivation to distinguish in our future work between road bound and non-road bound users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' CONCLUSION In this paper we have proposed a new way to represent temporal information in heterogeneous graphs for motion prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Instead of compressing the temporal information, we embed the whole past trajectories of all agents into the GNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' We achieve state of the art results, while having considerably less learnable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' We did an extensive ablation study to verify the effectiveness of each design decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' The evaluation was based (a) Dense traffic scene (b) Pedestrian crossing Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' 4: Qualitative results on INTERACTION validation dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' History is depicted in dark green, ground truth in light green, predictions in light red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' The prediction with the highest score is depicted in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' The map-nodes are depicted as light red dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' on the prediction of vehicles along a road network and conducted on two different state-of-the-art datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' As the holistic graph representation allows to include arbitrary information, we are going to further distinguish between road bound agents, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' cars, trucks and motorcycles, and non-road bound agents, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' pedestrians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' ACKNOWLEDGMENT The research leading to these results was conducted within the project KIsSME (Artificial Intelligence for selective near-real-time recordings of scenario and maneuver data in testing highly automated vehicles) and was funded by the German Federal Ministry for Economic Affairs and Climate Action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' Responsibility for the information and views set out in this publication lies entirely with the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FRT4oBgHgl3EQfWzfr/content/2301.13545v1.pdf'} +page_content=' REFERENCES [1] X.' metadata={'source': 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Corneta,b*, G. Garbarinob, F. Zontoneb, Y. Chushkinb, J. Jacobsb, E. Pinedac, T. Deschampsa, S. Lia, A. +Roncaa,b, J. Shena,b, G. Morardd, N. Neubere, M. Frey e, R. Busche, I. Gallinoe, M. Mezouarb, G. Vaughanb, +and B. Rutaa,b* +aInstitute of Light and Matter, UMR5306 Université Lyon 1-CNRS, Université de Lyon, F-69622 +Villeurbanne, France +bEuropean Synchrotron Radiation Facility, 71 avenue des Martyrs, CS 40220, Grenoble 38043, France +cDepartment of Physics, Institute of Energy Technologies, Universitat Politècnica de Catalunya— +BarcelonaTech, 08019 Barcelona, Spain +d Université Grenoble Alpes, Université Savoie Mont Blanc, CNRS, IRD, Université Gustave Eiffel, ISTerre, +38000 Grenoble, France +e Chair of Metallic Materials, Saarland University, Campus C6.3, 66123 Saarbrücken, Germany +*corresponding authors +antoine.cornet@univ-lyon1.fr +beatrice.ruta@univ-lyon1.fr + +Abstract +A fascinating feature of metallic glasses is their ability to explore different configurations under +mechanical deformations. This effect is usually observed through macroscopic observables, while little +is known on the consequence of the deformation at atomic level. Using the new generation of +synchrotrons, we probe the atomic motion and structure in a metallic glass under hydrostatic +compression, from the onset of the perturbation up to a severely-compressed state. While the +structure indicates reversible densification under compression, the dynamic is dramatically +accelerated and exhibits a hysteresis with two regimes. At low pressures, the atomic motion is +heterogeneous with avalanche-like rearrangements suggesting rejuvenation, while under further +compression, aging leads to a super-diffusive dynamics triggered by internal stresses inherent to the +glass. These results highlight the complexity of the atomic motion in non-ergodic systems and support +a theory recently developed to describe the surprising rejuvenation and strain hardening of metallic +glasses under compression. +Introduction +Every glass has its own story, which is encoded in the evolution of its properties. Once a glass is formed +by rapidly cooling from the melt, its final state depends on the applied temperature protocol and +spontaneously evolves with time1. Fast cooling rates create glasses trapped in more energetically +unstable configurations with larger structural disorder, lower elastic moduli and larger frozen-in free +volume than slow cooling protocols 2. Upon successive annealing, the glass ages and relaxes towards +energetically more stable minima in the potential energy landscape (PEL), exploring continuously +different configurations. This process, called physical aging, is particularly strong in metallic glasses +(MGs), and modifies the mechanical, structural and thermal properties of the material 3,4. In stark +contrast, fast thermal cycling or mechanical deformation can rejuvenate the system, driving the glass +into energetically less favoured configurations with increased plasticity 5–12. In some cases, the +rejuvenated MG would be equivalent to quenched glasses theoretically obtainable with cooling rates + +2 + +much faster than those reachable in a laboratory 6. In the presence of almost hydrostatic compression, +this rejuvenation leads to the suppression of shear banding and the inhibition of catastrophic +mechanical failures, making deformed MGs appealing for technological applications 7. Diffraction +studies suggest a broadening of the interatomic distances in severely deformed MGs, which is in +opposite to the well-known increase of structural order during physical aging 13. Changes in correlation +lengths at medium range order (MRO) have been also reported in pre-deformed Pd-based MGs and +are accompanied by an acceleration of the microscopic relaxation dynamics possibly due to an increase +in free volume 14,15. +The majority of studies deal with ex-situ compressed glasses, while little is known on the microscopic +physical mechanisms occurring during the compression, owing to the experimental difficulty of in-situ +experiments under high pressures. Theoretical works ascribe the pressure-induced rejuvenation and +strain hardening of MGs to the creation of an additional local minimum in the PEL associated to +rearrangements of the energy for cage dynamics 16,17. This process would lead to the occurrence of +two distinguishable dynamical regimes under pressure, whose existence has not been experimentally +observed so far. By combining in-situ high pressure, high energy X-Ray Photon Correlation +Spectroscopy (XPCS) and high energy X-ray Diffraction (XRD) in a 4th generation synchrotron source, +we here provide the experimental evidence of how the atomic dynamics evolve under the application +of pressure in a Pt42.5Cu27Ni9.5P21 MG, unveiling a complex, non-monotonous behaviour which is in +agreement with recent theoretical works. + +Fig. 1 | Dynamical rejuvenation under densification at room temperature. a) Sketch of an XPCS experiment showing the +sample within the diamond anvil cell, the diffracted intensity corresponding to the structure factor, the portion of reciprocal +space probed by the detector and a typical speckle pattern. b) top of FSDP covered during the XPCS experiments (the intensity + +6 +0.4 F +1 atm +0 +1 atm +0.2 +3.3 GPa +0 +3.3 GPa +0 +2.8 +2.9 +3.0 +10-1 +100 +101 +102 +103 +scattering vector q (A-1) +6t (s)0 +1000 +2000 +Diamond Anvil Cell +b) +0.8 +2 +0.65 m +(e +Speckle pattern +2000 +10003 + +integrated across the detector area) measured at atmospheric pressure and at 3.3 GPa. Glitches in the I(Q) comes from the +detector. c) Corresponding Intermediate Scattering Functions showing the acceleration of the dynamics with pressure. +Results +So far, the relatively low flux of high-energy coherent x-rays in 3rd generation synchrotrons, limited the +use of XPCS in bulky sample environments, including diamond anvil cells (DACs) and other high- +pressure apparatus. The current development of 4th generation synchrotron sources, such as the +upgraded ESRF synchrotron (France), provides a monochromatic high flux (1012 photon/s) of coherent +x-rays at energies as high as 21 keV 18 with an unprecedented high quality, allowing for pressure +dependent studies. A schematic view of the XPCS experiments is shown in Fig. 1a) and S1. The scattered +speckle pattern is collected in a wide angle geometry covering the maximum of the first sharp +diffraction peak (FSDP) of the glass which is at about q1=2.87 Å-1 for our as-cast Pt42.5Cu27Ni9.5P21 MG at +ambient temperature and atmospheric pressure (Fig. 1b). By increasing pressure, the maximum of the +FSDP shifts toward high scattering vectors, as shown in Fig. 1b) for 3.3 GPa. As the FSDP originates +from the medium range order, in the absence of important structural rearrangements its position can +be related to the macroscopic density of the glass 19. The agreement between the thermal expansion +coefficients of 3.85x10-5 K-1 obtained from us with high energy XRD (Supplementary Fig. S1), and the +3.95x10-5 K-1 value reported in literature dilatometry data 20 supports the validity of the density-MRO +relation for the Pt42.5Cu27Ni9.5P21. Therefore, the continuous shift toward high scattering vectors, q, in +Fig. 1b) reflects the monotonic rise in the glass density as the pressure increases. +The evolution of the internal dynamics of the glass accompanying the density change can be described +by the pressure dependence of the intermediate scattering function (ISF), F(δt), which monitors the +temporal decay of the electron density fluctuations at the probed q and pressure. As shown in Fig. 1c), +a pressure increase from 1 atm to 3.3 GPa results in a dramatic shift of more than one order of +magnitude in the ISF toward smaller δt, which implies a pressure-induced acceleration of the atomic +dynamic of the same magnitude. Fitting the data with the Kohlrausch-Williams-Watts (KWW) +phenomenological model |�(��)|� = ���(��/�)� with τ the relaxation time and β the shape exponent, +we find a relaxation time τ = 571s and τ = 38s for 1 atm and 3.3 GPa respectively at 300K. This pressure- +induced acceleration of the dynamics by a factor 15 suggests a rejuvenation of the glass under in-situ +hydrostatic compression and is larger than that observed in ex-situ deformed Pd-based MGs (factor +3.2 at 300K) 14 and around a single isolated shear band (factor 3.3 at 300K) in a Zr65Cu25Al10 glass 21. + +4 + + +Fig. 2 | Pressure dependence of structure and dynamics. a) Static structure factor measured with high energy XRD under +compression. b) corresponding maximum of the FSDP during both compression and decompression. c) TTCFs of selected +scans acquired at 0 (1 atm), 3.3 and 6.3 GPa for similar elapsed times after the pressure change. d) Selected ISFs showing the +transition from rejuvenation to relaxation with increasing pressure. Black dotted lines correspond to KWW fits to the data. e) +Averaged relaxation time during compression (full symbols). Data of a second as-cast sample measured during a different +XPCS experiment are reported as well to confirm the reproducibility of the results (empty symbols). +The overall evolution of the structure and dynamics during HP compression is reported in Fig. 2. The +static structure factor, S(Q), varies only slightly with pressure, and exhibits a (4.91x10-3 Å-1/GPa) linear +shift of q1 with pressure up to 7 GPa, which is completely reversible with no hysteresis within the +uncertainty of our measurement (Fig. 2a) and b)). In contrast, the collective atomic dynamics exhibits +a complex evolution during the compression stage as shown by selected two-times correlation +functions (TTCFs, Fig. 2c) and ISFs (Fig. 2d) measured after similar elapsed times from the pressure +change at the different nominal pressures. The TTCF is a time-resolved representation of the ISFs, +where the width of the high correlation contour is proportional to τ, the characteristic time of the +rearrangements at the microscopic length scale. +At atmospheric pressure, high correlation values remain for most of the scan, which indicate almost +arrested dynamics in the 600s total acquisition time. This is the classical picture of a glass well below +the glass transition temperature, where large-scale dynamics are frozen and only slow local atomic +rearrangements occur. The dramatic pressure-induced acceleration at 3.3 GPa corresponds to a sharp +and narrow high-correlation contour and a fast decay of the ISF. As pressure increases from 3.3 GPa to +6.3 GPa, a non-monotonous evolution of the dynamics occurs, as evidenced by the larger width of the +high correlation contours in the TTCFs and the shifts of the ISF to slower dynamics at high pressure. +To better clarify the nature of the atomic motion under hydrostatic compression, Fig. 2e) show +relaxation times averaged over different scans acquired over a period of 3h at each single pressure, +covering thus both the early stage deformation and the severely-compressed state. Two distinct +dynamical regimes can be identified. An acceleration of the particle dynamics up to 3.3 GPa, followed + +[F(6t) +0.6 +4000 +frames +0.6 +0.4 +(1)2 +0 GPa +0.4 +3.3 GPa +200 +O +0.2 +2000 +4.9GPa +100 +0.2 +O +6.3 GPa +0 +O +10-1 +100 +101 +102 +103 +0 +1 +2 +3 +4. +5 +6 +7 +0 +2000 +4000 +6t (s) +Pressure(GPa) +frames3.3 GPa +0.0 +2.875 +口 +4900S +0.8 +0 +2 +4 +6 +8 +10 +1 +2 +3 +4 +5 +6 +4000 +Pressure (GPa) +frames +0.6 +scattering vector q (A-1) +(24.1) +(0)2 +0.4 +2000 +d) +e) +0.2 +1st Sample +outside +2nd Sample +DAC +0 +01 +0.8 +500 +6.3 GPa +4900(e +b) +C) +0.7 GPa +2.910- +3.0 +compression +3.3 GPa +口 +decompression +0.8 +2.905 +2.5 +4.9 GPa +4000 +2.900 +frames +0.6 +6.3 GPa +2.0 +01.5 +A +0.4 +2.890 +2000 +tb +1.0- +0.2 +2.885 +0.5 +2.880 +05 + +by a progressive slow down at larger pressure values, suggesting the existence of a rejuvenation and a +relaxation regime at low and high pressure, respectively. These results have been confirmed by +repeating the experimental protocol in a different XPCS experiment on a second sample (see Methods +for further details). Interestingly, the pressure-induced acceleration of the dynamics is visible even at +the lowest pressure of 0.1 GPa, which corresponds to the preloading of the cell, where hardly any +structural change is visible from XRD, showing the great sensitivity of the dynamics with respect to +pressure. Artefacts related to the cell assembly, including the PTM, have been excluded as they give +rise only to a static background, free of any dynamical contribution (Fig. S8). It is interesting to note +that although this acceleration of a factor 2.5 in a small pressure interval is significant, it remains small +when compared to the pressure-induced shifts of the structural relaxation times reported in softer +molecular liquid glass-former 22,23. +To visualize how the dynamic varies with time during isobars in both the rejuvenation and relaxation +regimes, Fig. 3) reports TTCFs measured at the extremum pressures of 3.3 and 6.3 GPa as a function of +the elapsed time after pressure change. At low pressure, rejuvenation leads to heterogeneous +dynamics with relaxation times fluctuating around an average constant value, as evidenced by the +variation on the thickness of the red contour in the TTCFs at 3.3 GPa. We rule out the presence of +possible artefacts, such as fluctuations in the incident flux and potential sample movement (Fig. S2 and +S3). At about 6800s and 7100s at 3.3 GPa, complete decorrelation happens over one pixel in the TTCF, +which is evidence for massive atomic rearrangements with a time scale lower than our acquisition time +of 0.1s, while a steady acceleration of the dynamics is visible after 11600 s. We note that this +heterogeneous dynamical regime does not stabilize over time during our experiment, as fluctuations +are still visible on the TTCF after 12000s in the severely compressed state. +In sharp contrast to the heterogeneous, rejuvenation regime, the TTCFs show smoothly and +continuously slowing down dynamics at 6.3 GPa, with decorrelation times growing with the time +elapsed since pressure change. The transition from the low-pressure heterogeneous but constant +dynamics regime to the homogeneous, high-pressure aging regime is not sharp but continuous, as +visible from the evolution of τ in Fig. 2e) and the TTCFs at 4.9 GPa which shows this intermediate +regime, where both physical aging and fast massive atomic rearrangements are observed (Fig. S3). The +existence of the two dynamical regimes is confirmed also by the reproducibility of the results in a +second experiment (Fig. S4). The last row of the Fig. 3) corresponds to the TTCFs acquired at 3.3 GPa +during decompression. It is highly similar to the aging regime visible at 6.3 GPa in compression, and +does not match the heterogeneous dynamics observed at the corresponding pressure in compression. +The pressure evolution of the dynamics is therefore not fully reversible, and exhibits a hysteresis +evolution with a slow-down of the atomic motion of even a factor 10 during decompression (Fig. S5), +in contrast with the apparently elastic behaviour of the structure under deformation (Fig. 2b). + +6 + + +Fig. 3 | Temporal evolution of the atomic motion during isobars. TTCFs from scans acquired during compression (3.3 and +6.3 GPa) and during decompression (3.3 GPa), showing heterogeneous dynamics and physical aging at low and high pressure, +respectively, and the hysteresis evolution of the dynamics during decompression. +The ever-slowing dynamics at 6.3 GPa strongly resembles the physical aging usually observed in +thermally activated structural relaxations, associated to the interplay between density changes and +MRO ordering processes 24–26. In this regime, the corresponding ISFs evaluated at successively larger +waiting times, tw, elapsed from the pressure change, shift continuously toward longer decay times (Fig. +4a) and can be rescaled into a master curve when normalizing δt by τ (Fig. 4b). The validity of the +temporal scaling confirms the homogeneous nature of the collective motion. The corresponding +evolution of τ as a function of tw is reported in the inset of Fig. 4b) and echoes the results obtained in +MGs at atmospheric pressure and high temperature 26, that is a first rapid aging regime which obeys a +phenomenological equation �(��) = ��exp (��/�∗) followed by a constant dynamical state (last point +at tw>8000s, excluded from fit), less visible here. The yellow line in the inset corresponds to a fit of the +previous equation to τ(tw). It yields τ*=2300s and τ0=34s, respectively compatible to and ten times +smaller than atmospheric pressure high temperature literature data 25–27. This means that despite the +rejuvenation at early stages after pressure compression (and therefore the slower value of τ0), the rate +of aging is similar in both temperature and pressure studies. + +1.0 +300 +delay time 6t (s) +3.3GPa +0.8 +100 +0.6 +100 +0.4 +200 +0.2 +Compression +0.0 +4600 +4800 +5000 +5200 +6600 +6800 +7000 +7200 +9400 +9600 +9800 +10000 +11600 +11800 +12000 +12200 +Labtime(s) +OOE +1.0 +delay time 6t (s) +200 +6.3GPa +0.8 +100 +0.6 +0.4 +100 +200 +0.2 +300 +0.0 +1200 +1400 +1600 +1800 +3200 +3400 +3200 +4600 +4800 +5000 +5200 +6400 +6600 +6800 +Lab time (s) +Decompression +300 +1.0 +delay time 6t (s) +200 +3.3GPa +0.8 +100 +0.6 +0.4 +100 +200 +0.2 +300 +0.0 +600 +800 +1000 +2000 +2200 +2400 +3200 +3400 +3600 +Labtime (s)7 + + +Fig. 4 | Aging and wave-vector dependence of the dynamics in the homogeneous regime. a) ISFs measured at 6.3 GPa as a +function of the elapsed time from pressure change. Black dotted lines correspond to KWW fits to the data. Aging is visible +through the shift to long delay times with increasing waiting time (from left to right ISFs). b) Scaling of the ISF as a function +of the reduced time δt/τ. Inset shows the evolution of the corresponding τ and the best fit to the equation �(��) = +�����(��/�∗) (line). c) Wave-vector dependence of the dynamics at 6.3 GPa: top) KWW shape parameter; and bottom) +relaxation time (symbols) and integrated intensity in the detector (grey line) measured at 6.3 GPa. +Interestingly, the same physical mechanism seems to control the atomic rearrangements in both the +rejuvenation and relaxation regimes. All data can be described by compressed ISFs with an averaged +compressed shape parameter, β, ranging from 1.6 to 2, depending on the degree of heterogeneity of +the dynamics. Similar compressed values of β have been reported in all MGs under temperature +studies 26,28,29. Thanks to the high signal to noise ratio of the data and the large area detector used +during the XPCS measurements, we can evaluate the dynamics of the glass at different wave-vectors +q even in the nonergodic state, bypassing the problem of aging 27. Although the probed q-range is +limited by the size of the detector (Fig. S6), the relaxation time follows a τ(q)=1/cqα dependence from +the probed wave-vector, with 0<α<1. This is shown in Fig. 4c for 6.3 GPa where the fit yields +α=0.36±0.04. The wave-vector dependence of the dynamics and the constant compressed shape of +the ISFs implies that the ISFs can be described by |�(��)| = ��(��/�(�))� = ���������� +� + with +θ=1/α=2.74 and k=α·β= 0.73 at 6.3 GPa 30. This expression confirms the complex nature of the +dynamics of MGs and contrast with the high temperature diffusive motion of liquid metals which +would instead corresponds to τ(q)=1/Dq2 and thus to |�(��)| = �� �!��, with D the diffusion +coefficient. +To characterize the evolution of the dynamics over the complete compression/decompression cycle, +we have defined a dynamical heterogeneity parameter, in the following way. We first extract the +temporal evolution of the correlations "(�, �� = �) = 〈%(�) ∙ %(� ' �)〉/〈%〉� at a fixed delay time +between frames corresponding to the structural relaxation time τ obtained by the KWW analysis of +the individual ISFs (red curve in Fig. 5a). We then compute distributions of the correlation values +observed at a single pressure (Fig. 5b) by averaging all the different histograms of "(�, �� = �) at this +pressure. Heterogeneous dynamics lead to broad, potentially multimodal dynamics as illustrated by +the distribution obtained at 3.3 GPa, where two main distinct contributions are visible in addition to +the long tail at large values. Overall, the distributions broaden toward both the low and high +correlation values when pressure increases from 0.1 GPa to 3.3 GPa, and shrink afterward. As the width +of the distribution translates directly to the behaviour of the dynamics, we define a heterogeneity +parameter ΔC as the smallest width that contains 90% of the values of the statistics of the distribution +(as shown in Fig. 5b at 0.1GPa). Similar results are observed also for lower percentages of ΔC (Fig. S9). +The evolution of this heterogeneity parameter shows the pathway of the dynamics during the + +0.4 +Ot) +0.4 +5 +I(q) (a. +A +Q +D +P +0.2 +2 +0.2 +101 +2 +4 +8 +100 +0.0 +tw (103s) +0.0 +100 +101 +102 +10-3 +10-2 +10-1 +100 +101 +2.6 +2.7 +2.8 +6t (s) +ot/t +scattering vector q (A-1)a) +b) +2.2 +1.0 +1.0 +B +0.8 +0.8 +1.8 +10 +105 +0.6 +z(3g) +0.6 +8 +1048 + +compression-decompression cycle, and is displayed in Fig. 5c). The compression regime is inversely +related to the evolution of τ, with the bell shaped curve centred around 3.3 GPa, and corresponds to +the dynamical transition between the rejuvenated and relaxed regimes described above. Interestingly, +the decompression pathway shows the hysteresis deduced from the TTCFs at 3.3 GPa for increasing +and decreasing pressures (Fig. 3). The decreasing pressure does not impact significantly the dynamical +behaviour until 0.5 GPa, with a heterogeneity that remain relatively constant, possibly going through +a limited increase. At 0.5 GPa, the heterogeneity rises up to a value similar to the maximum observed +in compression. This pressure step corresponds to a fully deflated membrane in the DAC, and one +could associate the dynamic fluctuations to mechanical instabilities of the cell. The pressure stability +of 0.04 GPa over the course of the measurement dismisses however this possible artefact. + +Fig. 5 | Dynamical pathway during full compression-decompression cycle. a) Typical evolution of "(�), ��) = +〈%(�)) ∙ %(��)〉/〈%〉� at a fixed delay time �� = �) − �� = �. b) Corresponding distributions of "(�, �� = �) during the +compression stage for τ the structural relaxation time obtained from the KWW fits of the ISFs. Distributions are offset +vertically for clarity. c) Dynamical heterogeneity ΔC as a function of the applied pressure. This parameter represents the width +of the distributions in panel b), defined as the smallest interval that contains 90% of the correlations. The first point +corresponds to the loading pressure of 0.1 GPa. The fixed delay time �� = � is not accessible at 1 atm because dynamics is +too slow to observe a full decorrelation in the TTCFs. +Discussion +The existence of two dynamical regimes controlling the atomic motion of MGs under hydrostatic +pressure is consistent with results from recent theoretical works, which suggest that increasing +pressure leads to the formation of a second metastable higher-energy state in the potential energy +landscape 16,17. In this picture, fast dynamics correspond to temperature-assisted transitions within this +two-level system which leads to rejuvenation at low pressures. With further increase of pressure the +second metastable state vanishes, and dynamics reverse then to the slow structural relaxation, +similarly to our data 16,17. It would be interesting to know, whether the model could describe also the +heterogeneous to homogeneous evolution of the particle motion during compression. + +C) +1.0150 +1.0175 +0.011 +0.010 +1.0075 +400 +0.009 +1.0050 +t +0.008 +200 +0.007 +0.006 +compression +decompression +0 +200 +400 +600 +0 +1 +2 +3 +4 +5 +6 +7 +ti (s) +Pressure (GPa)b) +300 +7GPa +6.3 GPa +correlations +250 +4.9GPa +3.3 GPa +200 +1.5 GPa +0.1GPa +number of +150 +100 +a) +50 +0 +△C +1.005 +1.010 +1.015 +1.020 +1.025 +C(t, 6t = t)9 + +The decorrelation events observed in the rejuvenation regime, especially when fast and complete +decorrelation occurs, are the sign of cascade or avalanche-like cooperative relaxation mechanisms, +where local relaxation events trigger neighbouring events in a chain reaction31. While the trigger for +thermally activated relaxation in MGs is highly localized and independent of the stability of the system +32,33, this chain reaction implies a high spatial density of local minima in the PEL of the glass 34, as +isolated minima do not interact with each other. Such avalanche-like dynamics have been reported as +an aging mechanism in the similar Pd43Cu27Ni10P20 metallic glass 29, in a mechanically stressed metallic +glass ribbon 28, and as a mediator of aging and/or crystallization in a hard-sphere glass 33,35. Regardless +of the final structural state (aged glass or crystal), Yanagishima et al. showed that the avalanche events +statistically appear in regions of lower local density and bond orientational order 33, reinforcing the +heterogeneity of the PEL mentioned above at low pressures. Therefore, the avalanches-like dynamics +observed at low pressures witnesses a higher degree of inhomogeneity in the glass structure in this +pressure range, in agreement with the as-cast nature of our glass. As individual avalanches do not +necessarily increase the local order in the glass 33, and longer time is necessary for the aging trend to +emerge at low pressures, the rejuvenation regime persists for several steps in pressure and for many +hours per pressure without any signature of relaxation. The transition from rejuvenation to aging hints +also toward an effect of the excess free volume, which is present in the as-cast glass but seems to +reduce greatly during the relaxation at high pressures, as suggested by the dynamical hysteresis, even +if further measurements would be necessary to investigate more this aspect. XRD studies report the +occurrence of elastic deformations during the compression of MGs, supporting the idea of a +homogeneous fractal network model for the glass 36, as opposed to the heterogeneous structural +model of liquid-like regions of loosely bonded atoms embedded in a solid-like matrix 3,37,38. Our work +shows that the presence of apparently reversible structural changes under hydrostatic pressure +compression (Fig. 2b), is not a sufficient condition to assure a simple elastic structural mechanism +under compression, as they can be accompanied by a dramatic hysteresis evolution of the dynamics +(Fig. 3 and S5). +The τ(q)≈1/qα wave-vector dependence of the relaxation time implies a super-diffusive collective +particle motion in the glass at all pressures, which differs from the well-known structural dependence +of the relaxation time observed in supercooled liquids in the proximity of the FSDP 39,40. Above the +glass transition temperature, the equilibrium dynamics is associated to cage-escape processes, and the +long-time collective motion is sub-diffusive leading to a stretched exponential decay of the ISFs, +described thus by a value of β<1 40,41. In the glassy state, atomic mobility of MGs originates from fast +secondary relaxation processes, such as the β- and γ-processes 3,42. These processes control the stress +response of the material in the non-ergodic state and have been associated to cooperative string-like +particle motions in nanometric liquid-like regions 43,44. Compressed ISFs and super-diffusive dynamics +have been reported in many different complex systems as colloidal gels, clays, concentrated emulsions, +oxides and soft colloids 30,45,46. In these systems, the anomalous dynamic has been associated to the +presence of random local stresses in the materials, which are then released triggering the faster-than- +exponential collective dynamics 30,31,46–48. In MGs this stress propagation could be related to the +kinetics of structural rearrangements induced by the stress field controlled by the β- and γ- relaxation +processes. Further studies will be necessary to clarify the nature of the collective dynamics in MGs and +their evolution paths under annealing and pressure. + + +10 + +Methods: +Glass synthesis: We prepared a PtCuNi precursor by arc-melting the pure metallic components (with +purity >99.95%) under a Ti-gettered Ar-atmosphere (with purity >99.999%). We then alloyed +inductively the elemental P with the PtCuNi precursor in a fused-silica tube under Ar-atmosphere. In +order to obtain as low as possible oxide content, the alloy was subjected to a fluxing treatment in +dehydrated B2O3 for more than 6 hours at 1473 K. The ribbons were produced by melt spinning of the +master alloy on a rotating copper wheel under high-purity Ar atmosphere. The resulting glass ribbons +of Pt42.5Cu27Ni9.5P21 at .% had a thickness of 20 µm and a width of 2 mm. +High Pressure: the sample was cut from the as-cast 20 µm thick ribbon to a rough shape of 50x50x20 +µm3. The sample was subsequently pre-loaded at 0.1 GPa in a membrane driven Diamond Anvil Cell +with a ruby sphere and 4:1 methanol/ethanol mixture as pressure-transmitting medium (PTM), to +ensure a perfectly hydrostatic compression up to 10 GPa 49. The DAC was equipped with 600 µm +diamonds (culet size) and a pre-indented laser drilled stainless steel gasket to make a 60 µm x 300 µm +(height x diameter) experimental volume. The compression cycle up to 7 GPa is shown in Fig. S1: similar +pressures were reached in compression and decompression, and the elapsed time at each pressure +was around three hours in compression and one hour in decompression. The pressure was measured +from the wavelength of the Chromium 2E→4A2 transition in a ruby sphere after and before each +pressure change, and a dedicated pressure protocol on the membrane ensured a pressure variation +lower than 0.12 GPa at all pressures (Fig. S7). +X-Ray Diffraction: The structure of the metallic glass under pressure was monitored by two different +runs of x-ray diffraction. The first run, conducted at beamline ID27 at ESRF synchrotron, France, +reproduced the pressure protocol of the XPCS experiment. Experiment was performed using an +incident energy of 33 keV, an EIGER2 X CdTe 9M (active area = 233.1 x 244.7 mm2, pixel size = 75 µm) +detector and a DAC loaded with 4:1 methanol/ethanol mixture as PTM, a sample and a ruby sphere +for pressure determination. Background was collected at each pressure by measuring the scattering +pattern of a location inside the DAC next to the sample. The maximum scattering vector probed in this +run is q=12 Å-1. Azimuthal integration of the 2D scattered patterns was performed using the pyFAI +python library 50,51 to yield 1d diffraction patterns, and the computation of the (background corrected) +Faber-Zimman structure factor with Krogh-Moe-Norman normalization 52 was performed using the +python-based Amorpheus software 53. +To assess quantitatively the link between the peak position and the sample density, x-ray diffraction +data was collected as a function of temperature at atmospheric pressure to compare the shift of the +first sharp diffraction to the coefficient of thermal expansion measured by dilatometry (Fig. S1). The +XRD data was collected at the beamline ID15a 54 at the ESRF synchrotron, France. Data acquisition +using an incident beam energy of 68.5 keV and the scattered diffraction pattern was collected with a +Pilatus3 X CdTe 2M detector (active area = 253.7 x 288.8 mm2, pixel size = 172 µm). A sample to +detector distance of 1.087m was chosen to maximize the resolution on the first sharp diffraction peak. +The background was acquired in the same condition with an empty sample. Diffraction patterns were +azimuthally integrated using routines from the pyFAI library 50,51, and locally implemented corrections +for outliers rejection, background, polarization of the X-rays and detector geometry, response, and +transparency, to yield 1D diffraction patterns. + +11 + +XPCS: In order to optimize high-energy and high-pressure XPCS studies, we performed three different +XPCS campaigns for a total of 3 weeks of beamtime at beamline ID10 at the ESRF synchrotron, France. +The main data have been collected by using a 20.95 keV partially coherent monochromatic beam with +a photon flux of 4.2x1011 photon/s, focalized by a 2D Be lens transfocator to 50.5x14.2 µm2 (HxV, +FWHM) cut by a pair of slits for an illumination area of 8x8 µm2 on sample. The second sample was +measured in a second run with an incident energy set to 21.67 keV with a flux of 7.3x1011 photon/s +focalized to a beamsize of 5.2x4.2 µm2 (HxV, FWHM) on sample. To record the speckle patterns, we +placed an Eiger2 4M CdTe detector (active area = 155.1 x 162.2 mm2, pixel size = 75 µm) 5 meters +downstream at an angle corresponding to the pressure-dependent position of the FSDP, whose +maximum is at 2.79 Å-1 at atmospheric pressure and 25°C. The top part of this FSDP is reconstructed +by integrating the intensity in the detector, allowing the monitoring of the position of the peak during +the measurements. An additional PILATUS detector has been also employed to control the evolution +of the structure in a broader Q range during the experiment. A constant acquisition time of 0.1s/frame +was kept throughout the whole XPCS experiment, with scans ranging from 6000 frames to 14000 +frames depending on τ. Intensity-Intensity correlation functions, g2(t), and TTCFs are extracted from +the successive speckle patterns using the event correlator method described in 55. The ISFs are then +obtained from the g2(t) through the Siegert relation +�(,, ��) = 1 ' . ∙ |�(,, ��)|�, whose validity in +non-ergodic systems is assured by the use of large area detectors 56,57. In this expression γ is the +experimental contrast related to the degree of coherence of the beam. TTCFs have been evaluated +from the normalized correlation 〈%(�)) ∙ %(��)〉/〈%〉� between all pairs of scattering patterns recorded +during a scan at a given q. The main diagonal corresponds to the elapsed time of the measurement +with t(frame 1) = t(frame 2) = t, while any point off this diagonal express the correlation value at a +certain delay time δt = t(frame 2) – t(frame 1) after the first frame is recorded. 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Ultralow-angle dynamic light scattering with a charge coupled device +camera based multispeckle, multitau correlator. Review of Scientific Instruments 70, 3214–3221 +(1999). + + +Acknowledgements +We acknowledge ESRF (Grenoble, France), for the provision of experimental facilities. Parts of this +research were carried out at ID10 and ID27 beamlines under the LTP project HC4529. We gratefully +thank M. di Michiel for providing in-house experimental time at the ID15a beamline and for his +assistance during the experiment. We would also like to thank T. Poreba, K. Lhoste and D. Duran for +assistance. This project has received funding from the European Research Council (ERC) under the +European Union’s Horizon 2020 research and innovation programme (Grant Agreement No 948780). +All data needed to evaluate the conclusions in the paper are present in the paper and/or the +Supplementary Materials. + +Competing Interests +The authors declare that there are no financial or non-financial competing interests. + +Author Contributions +B. R., A. C., Y. C. and F. Z. conceived the study. N.N. and M.F. prepared the samples. G. G., J. J. and M. +M. provided technical and scientific support for all high pressure experiments. A. C., B. R., F.Z., Y. C., S. +L., T. D., N. N., M. F., E. P., J. S. and A.R. conducted the HP-XPCS experiments at beamline ID10. A.C., +B.R., S. L., G.V. and M. di M. conducted the high temperature XRD measurements at beamline ID15A. +A.C., J.S., B.R. and G.G. performed the HP-XRD experiments at beamline ID27. A.C. analysed all data +with the support of B.R., Y.C., G.V., G.G. and G.M.. A.C. and B.R. wrote the manuscript with inputs from +all authors. + + +17 + +Supplementary Materials to “Denser glasses relax faster: a +competition between rejuvenation and aging during in-situ +high pressure compression at the atomic scale” +A. Cornet et al. + +1. Compression decompression cycle and First Sharp Diffraction Peak evolution: + +Figure S1 – a) Pressure protocol. All XPCS measurements are performed during the different isobars. b) Evolution with +pressure of the top of the diffraction peak during the XPCS measurements. Data are reconstructed by integrating the detector +intensity. For clarity, the curves correspond to the compression stage only. c) Evaluation of thermal expansion coefficient +from the evolution of the FSDP position q1 with temperature measured at atmospheric pressure with synchrotron high energy +XRD. + +Similar pressures were reached upon compression and decompression to allow direct comparison. The +first step in compression at 0.1 GPa corresponds to the preloading of the cell, and the pressure of the +subsequent steps are of 1.5, 3.3, 4.9, 6.3, and 7 GPa. The last decompression step at 0.5 GPa +corresponds to the situation where the diamond anvil cell (DAC) membrane is fully deflated, but the +DAC remains mechanically locked. The top of the First Sharp Diffraction Peak (FSDP) can be +reconstructed by integrating the intensity collected on the detector during a complete XPCS scan. The +quantitative estimate of the peak position q1 is used to verify the consistency of the XPCS experiment +with X-ray diffraction (XRD) experiment. Here, the continuous shift of the FSDP toward the high +scattering vectors confirms the XRD results shown in the main text. q1 being linked to the characteristic +distance ℓ of the medium range order of the glass by q1=2π/ℓ, we can quantitatively assess the validity +of the link between q1 and the macroscopic density of the glass by comparing the coefficient of +thermal expansion (CTE) derived from high energy XRD measurement to the CTE obtained from +dilatometry. The CTE α is inferred from the evolution of (ρ(T)-ρ(25°C))/ρ(25°C) ∝ (q1(25°C)/q1(T))3-1. +We obtain α = 3.85x10-5 K-1 in the glassy state, in good agreement with the reported value obtained by +dilatometry1 α = 3.95x10-5 K-1. This shows that the FSDP is directly linked through the macroscopic +density for this glass. + + +es +0.004 += + 0.002 +1 +1 +g +0 +0.000 +0 +20000 +40000 +60000 +80000 +2.602.652.702.752.802.85 +100 +200 +Time (s) +Scattering vector q (A-1) +Temperature (°C)(e +b) +C +0.012 +atmosphericpressure +6 +0.010 +(GPa) +5 +0.008 +4 +sure +S(q) +Pressure +0.006 +318 + +2. Ruling out artefacts as the source of the heterogeneous dynamics + +Figure 6 - Trace (total intensity on detector) and selected TTCFs in compression at 3.3 GPa (left panels, elapsed time = +9700s), 4.9 GPa (central panels, elapsed time = 9800s) and in decompression at 3.3 GPa (right panels, elapsed time = 860s). +Although the intensity impinging on the sample is in general stable, some fluctuations can occur during +a full week of beamtime, and are usually due to adjustment of the electron beam in the storage ring +and refilling mode. As the magnitude of these fluctuations is usually small compared to the total +intensity, data are not affected. In Figure S2 we report selected TTCFs and the corresponding trace +(total intensity in the detector as a function of time) which show two different situations. In the left +panel and the beginning of the central panel, heterogeneous dynamics appear while trace is stable. +Differently, in the central and right panel trace shows fluctuations related to the re-fill in the storage +ring with no influence on the dynamics. This demonstrates that fluctuations of the incoming beam +intensity are not responsible for the observation of heterogeneous dynamics. These data allow us to +rule out also possible sample movements as sources of induced decorrelations. The position of the +sample was monitored by a microscope before and after each scan. Large movements on the scale of +10 µm (>5 times the Rayleigh Criterion) would then been detected, which is not the case. We exclude +also smaller, micrometres movements as if the decorrelation would be associated only to a change of +the scattering volume, the decorrelation time τ should be identical before and after the event, which +is generally not the case. In both the left and central panel, the ‘decorrelations’ in the TTCFs lead to +different dynamic profiles, which implies that the relaxation of the systems has changed. Another +example is reported also in Fig. S3 as described below. +It is well-known that some Pressure Transmitting Medium (PTM) can alter the property of a glass under +compression. This is the case for instance of gas loading with He that can enter the large open network +of silica glasses2. Due to its large molecular structure this is not the case for the alcohol mixture chosen +here as PTM even in the presence of large open structures2. +3. Additional data on the rejuvenation and relaxation regimes +Fig. S3 shows the TTCFs measured at 4.9 GPa in compression. The data shows aging regimes separated +by a cascade relaxation at 9600s. The mix of cascade relaxation and aging between the two well +defined dynamical regimes at 3.3 and 6.3 GPa shows the transition is not abrupt but continuous. The +third TTCF at 4.9 GPa is also a further confirmation of the absence of sample movement as the source + +6 +6 +time +time +time +500 +300 +300 +00 +00 +500 +300 +time +time +(s) +110 +1000time (s) +time (s) +time (s) +0 +200 +400 +600 +0 +200 +400 +600 +0 +250 +500 +750 1000 +trace +(x104) +15 +1019 + +of the heterogeneous dynamical regime observed at low pressures. If sample movement caused the +decorrelation event at 9600s, decorrelation time τ should be identical before and after the event, +which is obviously not the case. The TTCFs at 0.5 GPa at the end of the decompression stage show the +heterogeneous dynamical regime is eventually recovered but at a lower pressure compared to +compression, in accordance with the evolution of the dynamical heterogeneity introduced in the figure +5. + +Figure S3 - TTCFs obtained at 4.9 GPa during compression (top row) and 0.5 GPa in decompression (bottom row). +4. Repeatability: results from a second experiments +We controlled the repeatability of the results on a new sample measured in a different run. As shown +in Fig. 2e and Fig. S4, results of both runs overlap and show the same heterogenous vs homogeneous +transition with pressure, which confirm the robusteness of the data showed in this study. + +Figure S4: Partial TTCFs measured in a second sample in the low pressure heterogeneous (left) and high +pressure homogeneous (right) regimes. +5. Hysteresis evolution of the dynamics under pressure compression and decompression +The hysteresis shown in the main text from the comparison of the TTCFs at 3.3 GPa and the complete +pathway of the heterogeneity parameter is also visible from the characteristic relaxation time of the +intermediate scattering function. In the figure S5 we represent intermediate scattering functions (ISFs) +from the compression and decompression stages at 1.5 GPa and 3.3 GPa. To provide an accurate +comparison, we chose to plot the curves obtained at the most similar elapsed times tw (defining the +duration of the isobar). Both the shift of the curves and the relaxation times inferred from the KWW + +1000 +1000 +3.4 GPa +570P. +0 +0 +1000 +1000 +0 +200 +400 +600 +800 +10001200 +0 +200 +400 +600 +800 +1000 1200 +time (s) +time (s)1.0 +300 +200 +0.8 +6t( +100 +time +0.6 +0 +lay +100 +0.4 +4.9 GPa +del +200 +0.2 +Compression +300 +0.0 +1000 +1200 +1400 +1600 +4000 +4200 +4400 +4600 +9400 +9600 +9800 +10000 +10000 +11000 +11200 +11400 +Labtime(s) +1.0 +400 +(s) 39 +0.8 +200 +delay time +0.6 +0 +0.4 +200 +0.5 GPa +0.2 +400 +Decompression +0.0 +500 +750 +1000 +1250 +1900 +2150 +2400 +2650 +2900 +Lab time (s)20 + +model fitted to the data show slower dynamics during the decompression stage, confirming the +hysteretic behaviour. + +Figure S5: ISFs measured at 1.5 GPa and 3.3 GPa in compression and decompression at similar elapsed times tw, +showing the hysteresis within a pressure cycle. Dashed lines correspond to fits of the KKW model to the data. +The corresponding relaxation times τ are reported. +6. Wave-vector dependent XPCS study +To determine the dependence of the ISF with respect to the scattering vector q, we added a binning +on the raw data. In the figure S6 we plot a colour representation the total scattered intensity in the +detector within a single scan of 7000 frames with an acquisition time of 0.1s. The distribution of the +intensity clearly shows the maximum of the diffraction peak. The grey area corresponds to the raw +mask applied during the pre-processing of the data, which covers the shadow of the vacuum tube +between the sample and the detector and Kossel lines from the diamonds. On the right panel, the +segmentation of the unmasked area of the detector into several bands at different is visible. The TTCFs +and ISFs were then extracted for the data corresponding to each of these bins. + + +500 +500 +0 +0 +0 +500 +1000 +1500 +2000 +0 +500 +1000 +1500 +20002000 +2000 +1500 +1500 +1000 +10000.4 +Tcompression = 135s +口 +Tcompression = 38s +000 +吃 +O +TDecompression = 244s +TDecompression = 244s +9 +0.2 +O +Compression,tw=2800s +Compression,tw=4900s +00 +Decompression,tw=2400s +Decompression,tw=3400s +0.0 +10-1 +100 +101 +102 +103 +10-1 +100 +101 +102 +ot (s) +at (s)1.0 +0.8 +脚 +2 +0.6 +1.5 +1 +GPa: +3.3 +GPa +021 + +Figure S6: Left panel: Integrated intensity in the detector during an XPCS scan. A portion of the reddish ring +associated to the FSDP is well visible. Right panel: Q-binning of the data to extract the q-dependence evolution +of the ISFs. + + +7. Pressure protocol and stability + +Figure S7 - Loading protocol for pressure stability. Simultaneous recording of the membrane and ruby pressures (left panel) +showing the stabilisation effect of the loading protocol. Pressure overshot without the adapted pressure protocol (upper +right panel) and pressure drifts recorded during the experiment using the adapted protocol. +As XPCS is extremely sensitive to any structural change, it is essential to minimize any potential +pressure drift during the measurements. The typical pressure increase after reaching the desired set- +point for our DAC configuration is shown in the upper right panel, and can be higher than 1 GPa over +one hour. To mitigate this issue, we have determined how much we can reverse the membrane +pressure to stabilize the pressure on the sample. More precisely, we measured how much one can +decrease the membrane pressure after an initial increase before we can see any change in the sample +pressure (down to our precision of 0.01 GPa), as shown in the left panels. We have applied this loading +protocol during the measurements, and we report all pressure drifts, taken as the pressure difference +between the beginning and the end of a pressure steps. We can see the pressure drifts are now limited +to about 0.1 GPa. Importantly, this drift is random and does not depend on the nominal pressure, so +it is not responsible for the two-steps pressure effect on the dynamics reported in this study. +8. Diamond Anvil Cell XPCS background +Fig. S8 contains two ISFs obtained under pressure with beam focused on the sample or in the +experimental volume next to the sample. The absence of a decorrelation in the second case +demonstrates that the contribution of the Diamond Anvil Cell, which comprises contributions from the +diamonds and the pressure-transmitting medium, are only static contributions. The sample dynamics +probed with XPCS is therefore not affected by the high pressure cell. + + +1.0 +56 +7 +60 +8.5 +(bar) +compression +sample pressure (GPa) +54 +6.5 +0.8 +decompression +pressure +58- +(GPa) +8 +52 +6 +0.6 +56- +50 +5.5 +7.5 +0.4 +54- +△PlMAX = 0.12 GPa +48 +5 +0.2 +7 +0.0 +O +46 +4.5 +52 +0 +12 +24 +48 +60 +72 +0 +12 +24 +36 +48 +60 +0 +1 +2 +3 +4 +5 +6 +7 +8 +time (s) +time (s) +pressure(GPa)time (s) +time (s) +time (s) +0 +500 +1000 +1500 +2000 +2500 +3000 +3500 +0 +6 +12 +18 +24 +30 +36 +0 +12 +24 +36 +48 +60 +(bar) +40 +50- +1.25 +1.8 +sample pressure (GPa) +1.00 +1 GPa +membranepressure +4.5 +(GPa) +39 +2 GPa +0.75 +3.9 GPa +48- +1.6 +0.50 +5.9 GPa +38 +8.2 GPa +0.25 +0.00 +37 +1.4 +46- +3.522 + + + +Figure S8 - Correlation function g2 from beam targeting the sample (orange) and beam out of the sample, showing only the +contribution of the diamond and pressure transmitting medium. +9. Dynamical heterogeneity parameter +The dynamical heterogeneity parameter ΔC characterize the level of inhomogeneity of the glass atomic +scale dynamics. This parameter corresponds to the smallest width that encompasses 90% of the +distribution of the correlation values at a fixed delay time τ for each pressure. This 90% threshold was +chosen to reflect the extremes values taken by the correlation values. However, we show that the +evolution of with respect to pressure is not threshold strictly dependent on the value of this threshold. +In the figure S9 we reproduce the figure 5c) of the main text for different threshold values: 90%, 80%, +70%, and 60% (upper left, upper right, lower left and lower right panels respectively). + +Figure S9 – Dynamical heterogeneity parameter as a function of pressure for different threshold value: 90% (top left), 80% +(top right), 70% (bottom left) and 60% (bottom right). + +Pressure (GPa) +Pressure(GPa) +0 +2 +4 +6 +0 +N +4 +6 +0.009 +0.011 +0.008 +0.010 +(%06) +(%08) +0.009 +0.007 +AC +0.008 +AC +0.006 +0.007 +0.005 +0.006 +0.007 +0.0050 +0.006 +0.0045 +(70%) +0.0040 +AC +0.005 +△C +0.0035 +0.004 +compression +0.0030 +decompression +0 +2 +4 +6 +0 +2 +4 +6 +Pressure(GPa) +Pressure(GPa)1.025 +1.020 +diamond +sample +1.015 +1.010 +口 +10-1 +100 +101 +102 +6t (s)23 + + +The result obtained for a width of 90% is reproducible quantatively down to a width of 70%, and +qualitatively down to a width of 60%. Overall, this confirms the robustness of the heterogeneity +parameter ΔC. + +References: +1. Stolpe, M. Synchrotron x-ray diffraction studies of bulk metallic glass forming liquids and glasses. +(Saarländische Universitäts- und Landesbibliothek, 2019). doi:10.22028/D291-32093. +2. Weigel, C. et al. Vitreous Silica Distends in Helium Gas: Acoustic Versus Static Compressibilities. +Phys. Rev. Lett. 109, 245504 (2012). + + + + diff --git a/odE0T4oBgHgl3EQfqQG1/content/tmp_files/load_file.txt b/odE0T4oBgHgl3EQfqQG1/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..44f2a0d4728a1a8c6c73ab6656ba9e6d54efe650 --- /dev/null +++ b/odE0T4oBgHgl3EQfqQG1/content/tmp_files/load_file.txt @@ -0,0 +1,1198 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf,len=1197 +page_content='1 Denser glasses relax faster: a competition between rejuvenation and aging during in-situ high pressure compression at the atomic scale A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Corneta,b*, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Garbarinob, F.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Université Gustave Eiffel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' ISTerre,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 38000 Grenoble,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' France e Chair of Metallic Materials,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Saarland University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Campus C6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3, 66123 Saarbrücken, Germany corresponding authors antoine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='cornet@univ-lyon1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='fr beatrice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='ruta@univ-lyon1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='fr Abstract A fascinating feature of metallic glasses is their ability to explore different configurations under mechanical deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' This effect is usually observed through macroscopic observables, while little is known on the consequence of the deformation at atomic level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Using the new generation of synchrotrons, we probe the atomic motion and structure in a metallic glass under hydrostatic compression, from the onset of the perturbation up to a severely-compressed state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' While the structure indicates reversible densification under compression, the dynamic is dramatically accelerated and exhibits a hysteresis with two regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' At low pressures, the atomic motion is heterogeneous with avalanche-like rearrangements suggesting rejuvenation, while under further compression, aging leads to a super-diffusive dynamics triggered by internal stresses inherent to the glass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' These results highlight the complexity of the atomic motion in non-ergodic systems and support a theory recently developed to describe the surprising rejuvenation and strain hardening of metallic glasses under compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Introduction Every glass has its own story, which is encoded in the evolution of its properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Once a glass is formed by rapidly cooling from the melt, its final state depends on the applied temperature protocol and spontaneously evolves with time1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Fast cooling rates create glasses trapped in more energetically unstable configurations with larger structural disorder, lower elastic moduli and larger frozen-in free volume than slow cooling protocols 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Upon successive annealing, the glass ages and relaxes towards energetically more stable minima in the potential energy landscape (PEL), exploring continuously different configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' This process, called physical aging, is particularly strong in metallic glasses (MGs), and modifies the mechanical, structural and thermal properties of the material 3,4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' In stark contrast, fast thermal cycling or mechanical deformation can rejuvenate the system, driving the glass into energetically less favoured configurations with increased plasticity 5–12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' In some cases, the rejuvenated MG would be equivalent to quenched glasses theoretically obtainable with cooling rates 2 much faster than those reachable in a laboratory 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' In the presence of almost hydrostatic compression, this rejuvenation leads to the suppression of shear banding and the inhibition of catastrophic mechanical failures, making deformed MGs appealing for technological applications 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Diffraction studies suggest a broadening of the interatomic distances in severely deformed MGs, which is in opposite to the well-known increase of structural order during physical aging 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Changes in correlation lengths at medium range order (MRO) have been also reported in pre-deformed Pd-based MGs and are accompanied by an acceleration of the microscopic relaxation dynamics possibly due to an increase in free volume 14,15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The majority of studies deal with ex-situ compressed glasses, while little is known on the microscopic physical mechanisms occurring during the compression, owing to the experimental difficulty of in-situ experiments under high pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Theoretical works ascribe the pressure-induced rejuvenation and strain hardening of MGs to the creation of an additional local minimum in the PEL associated to rearrangements of the energy for cage dynamics 16,17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' This process would lead to the occurrence of two distinguishable dynamical regimes under pressure, whose existence has not been experimentally observed so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' By combining in-situ high pressure, high energy X-Ray Photon Correlation Spectroscopy (XPCS) and high energy X-ray Diffraction (XRD) in a 4th generation synchrotron source, we here provide the experimental evidence of how the atomic dynamics evolve under the application of pressure in a Pt42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='5Cu27Ni9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='5P21 MG, unveiling a complex, non-monotonous behaviour which is in agreement with recent theoretical works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 1 | Dynamical rejuvenation under densification at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' a) Sketch of an XPCS experiment showing the sample within the diamond anvil cell, the diffracted intensity corresponding to the structure factor, the portion of reciprocal space probed by the detector and a typical speckle pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' b) top of FSDP covered during the XPCS experiments (the intensity 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='4 F 1 atm 0 1 atm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3 GPa 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3 GPa 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='0 10-1 100 101 102 103 scattering vector q (A-1) 6t (s)0 1000 2000 Diamond Anvil Cell b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='8 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='65 m (e Speckle pattern 2000 10003 integrated across the detector area) measured at atmospheric pressure and at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Glitches in the I(Q) comes from the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' c) Corresponding Intermediate Scattering Functions showing the acceleration of the dynamics with pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Results So far, the relatively low flux of high-energy coherent x-rays in 3rd generation synchrotrons, limited the use of XPCS in bulky sample environments, including diamond anvil cells (DACs) and other high- pressure apparatus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The current development of 4th generation synchrotron sources, such as the upgraded ESRF synchrotron (France), provides a monochromatic high flux (1012 photon/s) of coherent x-rays at energies as high as 21 keV 18 with an unprecedented high quality, allowing for pressure dependent studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' A schematic view of the XPCS experiments is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 1a) and S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The scattered speckle pattern is collected in a wide angle geometry covering the maximum of the first sharp diffraction peak (FSDP) of the glass which is at about q1=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='87 Å-1 for our as-cast Pt42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='5Cu27Ni9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='5P21 MG at ambient temperature and atmospheric pressure (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' By increasing pressure, the maximum of the FSDP shifts toward high scattering vectors, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 1b) for 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' As the FSDP originates from the medium range order, in the absence of important structural rearrangements its position can be related to the macroscopic density of the glass 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The agreement between the thermal expansion coefficients of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='85x10-5 K-1 obtained from us with high energy XRD (Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' S1), and the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='95x10-5 K-1 value reported in literature dilatometry data 20 supports the validity of the density-MRO relation for the Pt42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='5Cu27Ni9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='5P21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Therefore, the continuous shift toward high scattering vectors, q, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 1b) reflects the monotonic rise in the glass density as the pressure increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The evolution of the internal dynamics of the glass accompanying the density change can be described by the pressure dependence of the intermediate scattering function (ISF), F(δt), which monitors the temporal decay of the electron density fluctuations at the probed q and pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 1c), a pressure increase from 1 atm to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3 GPa results in a dramatic shift of more than one order of magnitude in the ISF toward smaller δt, which implies a pressure-induced acceleration of the atomic dynamic of the same magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Fitting the data with the Kohlrausch-Williams-Watts (KWW) phenomenological model |�(��)|� = ���(��/�)� with τ the relaxation time and β the shape exponent, we find a relaxation time τ = 571s and τ = 38s for 1 atm and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3 GPa respectively at 300K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' This pressure- induced acceleration of the dynamics by a factor 15 suggests a rejuvenation of the glass under in-situ hydrostatic compression and is larger than that observed in ex-situ deformed Pd-based MGs (factor 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='2 at 300K) 14 and around a single isolated shear band (factor 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3 at 300K) in a Zr65Cu25Al10 glass 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 2 | Pressure dependence of structure and dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' a) Static structure factor measured with high energy XRD under compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' b) corresponding maximum of the FSDP during both compression and decompression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' c) TTCFs of selected scans acquired at 0 (1 atm), 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3 GPa for similar elapsed times after the pressure change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' d) Selected ISFs showing the transition from rejuvenation to relaxation with increasing pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Black dotted lines correspond to KWW fits to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' e) Averaged relaxation time during compression (full symbols).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Data of a second as-cast sample measured during a different XPCS experiment are reported as well to confirm the reproducibility of the results (empty symbols).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The overall evolution of the structure and dynamics during HP compression is reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The static structure factor, S(Q), varies only slightly with pressure, and exhibits a (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='91x10-3 Å-1/GPa) linear shift of q1 with pressure up to 7 GPa, which is completely reversible with no hysteresis within the uncertainty of our measurement (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 2a) and b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' In contrast, the collective atomic dynamics exhibits a complex evolution during the compression stage as shown by selected two-times correlation functions (TTCFs, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 2c) and ISFs (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 2d) measured after similar elapsed times from the pressure change at the different nominal pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The TTCF is a time-resolved representation of the ISFs, where the width of the high correlation contour is proportional to τ, the characteristic time of the rearrangements at the microscopic length scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' At atmospheric pressure, high correlation values remain for most of the scan, which indicate almost arrested dynamics in the 600s total acquisition time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' This is the classical picture of a glass well below the glass transition temperature, where large-scale dynamics are frozen and only slow local atomic rearrangements occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The dramatic pressure-induced acceleration at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3 GPa corresponds to a sharp and narrow high-correlation contour and a fast decay of the ISF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' As pressure increases from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3 GPa to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3 GPa, a non-monotonous evolution of the dynamics occurs, as evidenced by the larger width of the high correlation contours in the TTCFs and the shifts of the ISF to slower dynamics at high pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' To better clarify the nature of the atomic motion under hydrostatic compression, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 2e) show relaxation times averaged over different scans acquired over a period of 3h at each single pressure, covering thus both the early stage deformation and the severely-compressed state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Two distinct dynamical regimes can be identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' An acceleration of the particle dynamics up to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3 GPa, followed [F(6t) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='6 4000 frames 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='4 (1)2 0 GPa 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3 GPa 200 O 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='2 2000 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='9GPa 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='2 O 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3 GPa 0 O 10-1 100 101 102 103 0 1 2 3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 5 6 7 0 2000 4000 6t (s) Pressure(GPa) frames3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3 GPa 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='875 口 4900S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='8 0 2 4 6 8 10 1 2 3 4 5 6 4000 Pressure (GPa) frames 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='6 scattering vector q (A-1) (24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='1) (0)2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='4 2000 d) e) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='2 1st Sample outside 2nd Sample DAC 0 01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='8 500 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3 GPa 4900(e b) C) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='7 GPa 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='910- 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='0 compression 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3 GPa 口 decompression 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='905 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='9 GPa 4000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='900 frames 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3 GPa 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='0 01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='5 A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='890 2000 tb 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='0- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='885 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='880 05 by a progressive slow down at larger pressure values, suggesting the existence of a rejuvenation and a relaxation regime at low and high pressure, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' These results have been confirmed by repeating the experimental protocol in a different XPCS experiment on a second sample (see Methods for further details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Interestingly, the pressure-induced acceleration of the dynamics is visible even at the lowest pressure of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='1 GPa, which corresponds to the preloading of the cell, where hardly any structural change is visible from XRD, showing the great sensitivity of the dynamics with respect to pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Artefacts related to the cell assembly, including the PTM, have been excluded as they give rise only to a static background, free of any dynamical contribution (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' S8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' It is interesting to note that although this acceleration of a factor 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='5 in a small pressure interval is significant, it remains small when compared to the pressure-induced shifts of the structural relaxation times reported in softer molecular liquid glass-former 22,23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' To visualize how the dynamic varies with time during isobars in both the rejuvenation and relaxation regimes, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 3) reports TTCFs measured at the extremum pressures of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3 GPa as a function of the elapsed time after pressure change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' At low pressure, rejuvenation leads to heterogeneous dynamics with relaxation times fluctuating around an average constant value, as evidenced by the variation on the thickness of the red contour in the TTCFs at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' We rule out the presence of possible artefacts, such as fluctuations in the incident flux and potential sample movement (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' S2 and S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' At about 6800s and 7100s at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3 GPa, complete decorrelation happens over one pixel in the TTCF, which is evidence for massive atomic rearrangements with a time scale lower than our acquisition time of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='1s, while a steady acceleration of the dynamics is visible after 11600 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' We note that this heterogeneous dynamical regime does not stabilize over time during our experiment, as fluctuations are still visible on the TTCF after 12000s in the severely compressed state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' In sharp contrast to the heterogeneous, rejuvenation regime, the TTCFs show smoothly and continuously slowing down dynamics at 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3 GPa, with decorrelation times growing with the time elapsed since pressure change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The transition from the low-pressure heterogeneous but constant dynamics regime to the homogeneous, high-pressure aging regime is not sharp but continuous, as visible from the evolution of \uf0e1τ\uf0f1 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 2e) and the TTCFs at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='9 GPa which shows this intermediate regime, where both physical aging and fast massive atomic rearrangements are observed (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The existence of the two dynamical regimes is confirmed also by the reproducibility of the results in a second experiment (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The last row of the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 3) corresponds to the TTCFs acquired at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3 GPa during decompression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' It is highly similar to the aging regime visible at 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3 GPa in compression, and does not match the heterogeneous dynamics observed at the corresponding pressure in compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The pressure evolution of the dynamics is therefore not fully reversible, and exhibits a hysteresis evolution with a slow-down of the atomic motion of even a factor 10 during decompression (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' S5), in contrast with the apparently elastic behaviour of the structure under deformation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 6 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 3 | Temporal evolution of the atomic motion during isobars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' TTCFs from scans acquired during compression (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3 GPa) and during decompression (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3 GPa), showing heterogeneous dynamics and physical aging at low and high pressure, respectively, and the hysteresis evolution of the dynamics during decompression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The ever-slowing dynamics at 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3 GPa strongly resembles the physical aging usually observed in thermally activated structural relaxations, associated to the interplay between density changes and MRO ordering processes 24–26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' In this regime, the corresponding ISFs evaluated at successively larger waiting times, tw, elapsed from the pressure change, shift continuously toward longer decay times (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 4a) and can be rescaled into a master curve when normalizing δt by τ (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The validity of the temporal scaling confirms the homogeneous nature of the collective motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The corresponding evolution of τ as a function of tw is reported in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 4b) and echoes the results obtained in MGs at atmospheric pressure and high temperature 26, that is a first rapid aging regime which obeys a phenomenological equation �(��) = ��exp (��/�∗) followed by a constant dynamical state (last point at tw>8000s, excluded from fit), less visible here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The yellow line in the inset corresponds to a fit of the previous equation to τ(tw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' It yields τ*=2300s and τ0=34s, respectively compatible to and ten times smaller than atmospheric pressure high temperature literature data 25–27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' This means that despite the rejuvenation at early stages after pressure compression (and therefore the slower value of τ0), the rate of aging is similar in both temperature and pressure studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='0 300 delay time 6t (s) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3GPa 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='8 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='6 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='4 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='2 Compression 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='0 4600 4800 5000 5200 6600 6800 7000 7200 9400 9600 9800 10000 11600 11800 12000 12200 Labtime(s) OOE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='0 delay time 6t (s) 200 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3GPa 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='8 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='4 100 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='2 300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='0 1200 1400 1600 1800 3200 3400 3200 4600 4800 5000 5200 6400 6600 6800 Lab time (s) Decompression 300 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='0 delay time 6t (s) 200 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3GPa 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='8 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='4 100 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='2 300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='0 600 800 1000 2000 2200 2400 3200 3400 3600 Labtime (s)7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 4 | Aging and wave-vector dependence of the dynamics in the homogeneous regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' a) ISFs measured at 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3 GPa as a function of the elapsed time from pressure change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Black dotted lines correspond to KWW fits to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Aging is visible through the shift to long delay times with increasing waiting time (from left to right ISFs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' b) Scaling of the ISF as a function of the reduced time δt/τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Inset shows the evolution of the corresponding τ and the best fit to the equation �(��) = �����(��/�∗) (line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' c) Wave-vector dependence of the dynamics at 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3 GPa: top) KWW shape parameter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' and bottom) relaxation time (symbols) and integrated intensity in the detector (grey line) measured at 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Interestingly, the same physical mechanism seems to control the atomic rearrangements in both the rejuvenation and relaxation regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' All data can be described by compressed ISFs with an averaged compressed shape parameter, β, ranging from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='6 to 2, depending on the degree of heterogeneity of the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Similar compressed values of β have been reported in all MGs under temperature studies 26,28,29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Thanks to the high signal to noise ratio of the data and the large area detector used during the XPCS measurements, we can evaluate the dynamics of the glass at different wave-vectors q even in the nonergodic state, bypassing the problem of aging 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Although the probed q-range is limited by the size of the detector (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' S6), the relaxation time follows a τ(q)=1/cqα dependence from the probed wave-vector, with 0<α<1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' This is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 4c for 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3 GPa where the fit yields α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='36±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The wave-vector dependence of the dynamics and the constant compressed shape of the ISFs implies that the ISFs can be described by |�(��)| = ��(��/�(�))� = ���������� � with θ=1/α=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='74 and k=α·β= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='73 at 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3 GPa 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' This expression confirms the complex nature of the dynamics of MGs and contrast with the high temperature diffusive motion of liquid metals which would instead corresponds to τ(q)=1/Dq2 and thus to |�(��)| = �� �!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='��, with D the diffusion coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' To characterize the evolution of the dynamics over the complete compression/decompression cycle, we have defined a dynamical heterogeneity parameter, in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' We first extract the temporal evolution of the correlations "(�, �� = �) = 〈%(�) ∙ %(� \' �)〉/〈%〉� at a fixed delay time between frames corresponding to the structural relaxation time τ obtained by the KWW analysis of the individual ISFs (red curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 5a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' We then compute distributions of the correlation values observed at a single pressure (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 5b) by averaging all the different histograms of "(�, �� = �) at this pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Heterogeneous dynamics lead to broad, potentially multimodal dynamics as illustrated by the distribution obtained at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3 GPa, where two main distinct contributions are visible in addition to the long tail at large values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Overall, the distributions broaden toward both the low and high correlation values when pressure increases from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='1 GPa to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3 GPa, and shrink afterward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' As the width of the distribution translates directly to the behaviour of the dynamics, we define a heterogeneity parameter ΔC as the smallest width that contains 90% of the values of the statistics of the distribution (as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 5b at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='1GPa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Similar results are observed also for lower percentages of ΔC (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' S9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The evolution of this heterogeneity parameter shows the pathway of the dynamics during the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='4 Ot) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='4 5 I(q) (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' A Q D P 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='2 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='2 101 2 4 8 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='0 tw (103s) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='0 100 101 102 10-3 10-2 10-1 100 101 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='8 6t (s) ot/t scattering vector q (A-1)a) b) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='0 B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='8 10 105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='6 z(3g) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='6 8 1048 compression-decompression cycle, and is displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 5c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The compression regime is inversely related to the evolution of \uf0e1τ\uf0f1, with the bell shaped curve centred around 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3 GPa, and corresponds to the dynamical transition between the rejuvenated and relaxed regimes described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Interestingly, the decompression pathway shows the hysteresis deduced from the TTCFs at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3 GPa for increasing and decreasing pressures (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The decreasing pressure does not impact significantly the dynamical behaviour until 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='5 GPa, with a heterogeneity that remain relatively constant, possibly going through a limited increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' At 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='5 GPa, the heterogeneity rises up to a value similar to the maximum observed in compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' This pressure step corresponds to a fully deflated membrane in the DAC, and one could associate the dynamic fluctuations to mechanical instabilities of the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The pressure stability of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='04 GPa over the course of the measurement dismisses however this possible artefact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 5 | Dynamical pathway during full compression-decompression cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' a) Typical evolution of "(�), ��) = 〈%(�)) ∙ %(��)〉/〈%〉� at a fixed delay time �� = �) − �� = �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' b) Corresponding distributions of "(�, �� = �) during the compression stage for τ the structural relaxation time obtained from the KWW fits of the ISFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Distributions are offset vertically for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' c) Dynamical heterogeneity ΔC as a function of the applied pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' This parameter represents the width of the distributions in panel b), defined as the smallest interval that contains 90% of the correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The first point corresponds to the loading pressure of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='1 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The fixed delay time �� = � is not accessible at 1 atm because dynamics is too slow to observe a full decorrelation in the TTCFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Discussion The existence of two dynamical regimes controlling the atomic motion of MGs under hydrostatic pressure is consistent with results from recent theoretical works, which suggest that increasing pressure leads to the formation of a second metastable higher-energy state in the potential energy landscape 16,17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' In this picture, fast dynamics correspond to temperature-assisted transitions within this two-level system which leads to rejuvenation at low pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' With further increase of pressure the second metastable state vanishes, and dynamics reverse then to the slow structural relaxation, similarly to our data 16,17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' It would be interesting to know, whether the model could describe also the heterogeneous to homogeneous evolution of the particle motion during compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' C) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='0150 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='0175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='010 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='0075 400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='009 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='0050 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='008 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='006 compression decompression 0 200 400 600 0 1 2 3 4 5 6 7 ti (s) Pressure (GPa)b) 300 7GPa 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3 GPa correlations 250 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='9GPa 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3 GPa 200 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='5 GPa 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='1GPa number of 150 100 a) 50 0 △C 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='005 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='010 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='015 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='020 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='025 C(t, 6t = t)9 The decorrelation events observed in the rejuvenation regime, especially when fast and complete decorrelation occurs, are the sign of cascade or avalanche-like cooperative relaxation mechanisms, where local relaxation events trigger neighbouring events in a chain reaction31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' While the trigger for thermally activated relaxation in MGs is highly localized and independent of the stability of the system 32,33, this chain reaction implies a high spatial density of local minima in the PEL of the glass 34, as isolated minima do not interact with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Such avalanche-like dynamics have been reported as an aging mechanism in the similar Pd43Cu27Ni10P20 metallic glass 29, in a mechanically stressed metallic glass ribbon 28, and as a mediator of aging and/or crystallization in a hard-sphere glass 33,35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Regardless of the final structural state (aged glass or crystal), Yanagishima et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' showed that the avalanche events statistically appear in regions of lower local density and bond orientational order 33, reinforcing the heterogeneity of the PEL mentioned above at low pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Therefore, the avalanches-like dynamics observed at low pressures witnesses a higher degree of inhomogeneity in the glass structure in this pressure range, in agreement with the as-cast nature of our glass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' As individual avalanches do not necessarily increase the local order in the glass 33, and longer time is necessary for the aging trend to emerge at low pressures, the rejuvenation regime persists for several steps in pressure and for many hours per pressure without any signature of relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The transition from rejuvenation to aging hints also toward an effect of the excess free volume, which is present in the as-cast glass but seems to reduce greatly during the relaxation at high pressures, as suggested by the dynamical hysteresis, even if further measurements would be necessary to investigate more this aspect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' XRD studies report the occurrence of elastic deformations during the compression of MGs, supporting the idea of a homogeneous fractal network model for the glass 36, as opposed to the heterogeneous structural model of liquid-like regions of loosely bonded atoms embedded in a solid-like matrix 3,37,38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Our work shows that the presence of apparently reversible structural changes under hydrostatic pressure compression (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 2b), is not a sufficient condition to assure a simple elastic structural mechanism under compression, as they can be accompanied by a dramatic hysteresis evolution of the dynamics (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 3 and S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The τ(q)≈1/qα wave-vector dependence of the relaxation time implies a super-diffusive collective particle motion in the glass at all pressures, which differs from the well-known structural dependence of the relaxation time observed in supercooled liquids in the proximity of the FSDP 39,40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Above the glass transition temperature, the equilibrium dynamics is associated to cage-escape processes, and the long-time collective motion is sub-diffusive leading to a stretched exponential decay of the ISFs, described thus by a value of β<1 40,41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' In the glassy state, atomic mobility of MGs originates from fast secondary relaxation processes, such as the β- and γ-processes 3,42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' These processes control the stress response of the material in the non-ergodic state and have been associated to cooperative string-like particle motions in nanometric liquid-like regions 43,44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Compressed ISFs and super-diffusive dynamics have been reported in many different complex systems as colloidal gels, clays, concentrated emulsions, oxides and soft colloids 30,45,46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' In these systems, the anomalous dynamic has been associated to the presence of random local stresses in the materials, which are then released triggering the faster-than- exponential collective dynamics 30,31,46–48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' In MGs this stress propagation could be related to the kinetics of structural rearrangements induced by the stress field controlled by the β- and γ- relaxation processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Further studies will be necessary to clarify the nature of the collective dynamics in MGs and their evolution paths under annealing and pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 10 Methods: Glass synthesis: We prepared a PtCuNi precursor by arc-melting the pure metallic components (with purity >99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='95%) under a Ti-gettered Ar-atmosphere (with purity >99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='999%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' We then alloyed inductively the elemental P with the PtCuNi precursor in a fused-silica tube under Ar-atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' In order to obtain as low as possible oxide content, the alloy was subjected to a fluxing treatment in dehydrated B2O3 for more than 6 hours at 1473 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The ribbons were produced by melt spinning of the master alloy on a rotating copper wheel under high-purity Ar atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The resulting glass ribbons of Pt42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='5Cu27Ni9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='5P21 at .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='% had a thickness of 20 µm and a width of 2 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' High Pressure: the sample was cut from the as-cast 20 µm thick ribbon to a rough shape of 50x50x20 µm3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The sample was subsequently pre-loaded at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='1 GPa in a membrane driven Diamond Anvil Cell with a ruby sphere and 4:1 methanol/ethanol mixture as pressure-transmitting medium (PTM), to ensure a perfectly hydrostatic compression up to 10 GPa 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The DAC was equipped with 600 µm diamonds (culet size) and a pre-indented laser drilled stainless steel gasket to make a 60 µm x 300 µm (height x diameter) experimental volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The compression cycle up to 7 GPa is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' S1: similar pressures were reached in compression and decompression, and the elapsed time at each pressure was around three hours in compression and one hour in decompression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The pressure was measured from the wavelength of the Chromium 2E→4A2 transition in a ruby sphere after and before each pressure change, and a dedicated pressure protocol on the membrane ensured a pressure variation lower than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='12 GPa at all pressures (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' S7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' X-Ray Diffraction: The structure of the metallic glass under pressure was monitored by two different runs of x-ray diffraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The first run, conducted at beamline ID27 at ESRF synchrotron, France, reproduced the pressure protocol of the XPCS experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Experiment was performed using an incident energy of 33 keV, an EIGER2 X CdTe 9M (active area = 233.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='1 x 244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='7 mm2, pixel size = 75 µm) detector and a DAC loaded with 4:1 methanol/ethanol mixture as PTM, a sample and a ruby sphere for pressure determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Background was collected at each pressure by measuring the scattering pattern of a location inside the DAC next to the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The maximum scattering vector probed in this run is q=12 Å-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Azimuthal integration of the 2D scattered patterns was performed using the pyFAI python library 50,51 to yield 1d diffraction patterns, and the computation of the (background corrected) Faber-Zimman structure factor with Krogh-Moe-Norman normalization 52 was performed using the python-based Amorpheus software 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' To assess quantitatively the link between the peak position and the sample density, x-ray diffraction data was collected as a function of temperature at atmospheric pressure to compare the shift of the first sharp diffraction to the coefficient of thermal expansion measured by dilatometry (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The XRD data was collected at the beamline ID15a 54 at the ESRF synchrotron, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Data acquisition using an incident beam energy of 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='5 keV and the scattered diffraction pattern was collected with a Pilatus3 X CdTe 2M detector (active area = 253.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='7 x 288.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='8 mm2, pixel size = 172 µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' A sample to detector distance of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='087m was chosen to maximize the resolution on the first sharp diffraction peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The background was acquired in the same condition with an empty sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Diffraction patterns were azimuthally integrated using routines from the pyFAI library 50,51, and locally implemented corrections for outliers rejection, background, polarization of the X-rays and detector geometry, response, and transparency, to yield 1D diffraction patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 11 XPCS: In order to optimize high-energy and high-pressure XPCS studies, we performed three different XPCS campaigns for a total of 3 weeks of beamtime at beamline ID10 at the ESRF synchrotron, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The main data have been collected by using a 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='95 keV partially coherent monochromatic beam with a photon flux of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='2x1011 photon/s, focalized by a 2D Be lens transfocator to 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='5x14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='2 µm2 (HxV, FWHM) cut by a pair of slits for an illumination area of 8x8 µm2 on sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The second sample was measured in a second run with an incident energy set to 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='67 keV with a flux of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3x1011 photon/s focalized to a beamsize of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='2x4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='2 µm2 (HxV, FWHM) on sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' To record the speckle patterns, we placed an Eiger2 4M CdTe detector (active area = 155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='1 x 162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='2 mm2, pixel size = 75 µm) 5 meters downstream at an angle corresponding to the pressure-dependent position of the FSDP, whose maximum is at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='79 Å-1 at atmospheric pressure and 25°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The top part of this FSDP is reconstructed by integrating the intensity in the detector, allowing the monitoring of the position of the peak during the measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' An additional PILATUS detector has been also employed to control the evolution of the structure in a broader Q range during the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' A constant acquisition time of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='1s/frame was kept throughout the whole XPCS experiment, with scans ranging from 6000 frames to 14000 frames depending on τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Intensity-Intensity correlation functions, g2(t), and TTCFs are extracted from the successive speckle patterns using the event correlator method described in 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=" The ISFs are then obtained from the g2(t) through the Siegert relation +�(,, ��) = 1 ' ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' ∙ |�(,, ��)|�, whose validity in non-ergodic systems is assured by the use of large area detectors 56,57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' In this expression γ is the experimental contrast related to the degree of coherence of the beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' TTCFs have been evaluated from the normalized correlation 〈%(�)) ∙ %(��)〉/〈%〉� between all pairs of scattering patterns recorded during a scan at a given q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The main diagonal corresponds to the elapsed time of the measurement with t(frame 1) = t(frame 2) = t, while any point off this diagonal express the correlation value at a certain delay time δt = t(frame 2) – t(frame 1) after the first frame is recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' To quantify the evolution of the dynamics with pressure, we extracted the characteristic times τ of all scans acquired during the compression by fitting KWW functions to the F(q,δt) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' We further averaged the different values of τ at a single pressure, to get an average \uf0e1τ\uf0f1 for each isobars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' No contribution to the dynamics has been observed from the background (diamonds and pressure transmitting medium) as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' For the analysis of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=" 5, the computation of 〈%(�) ∙ %(� ' �)〉/〈%〉� have been done on the raw data, i." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' taking into account also the aging within each scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Although this potentially leads to an overestimation of the heterogeneity parameter in the aging regime, we found this effect to be very limited leading to a well-defined transition between the rejuvenation and aging regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Ediger, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' D.' metadata={'source': 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+page_content=' & Sillescu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The glass transition dynamics of polymer micronetwork colloids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' A mode coupling analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 106, 3743–3756 (1997).' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Acknowledgements We acknowledge ESRF (Grenoble, France), for the provision of experimental facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Parts of this research were carried out at ID10 and ID27 beamlines under the LTP project HC4529.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' We gratefully thank M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' di Michiel for providing in-house experimental time at the ID15a beamline and for his assistance during the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' We would also like to thank T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Poreba, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Lhoste and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Duran for assistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No 948780).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Competing Interests The authors declare that there are no financial or non-financial competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Author Contributions B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=', Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' conceived the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' prepared the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' provided technical and scientific support for all high pressure experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=', B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' R.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' conducted the HP-XPCS experiments at beamline ID10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='C.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' di M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' conducted the high temperature XRD measurements at beamline ID15A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=', B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' performed the HP-XRD experiments at beamline ID27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' analysed all data with the support of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=', Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=', G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=', G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='. A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' wrote the manuscript with inputs from all authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 17 Supplementary Materials to “Denser glasses relax faster: a competition between rejuvenation and aging during in-situ high pressure compression at the atomic scale” A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Cornet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Compression decompression cycle and First Sharp Diffraction Peak evolution: Figure S1 – a) Pressure protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' All XPCS measurements are performed during the different isobars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' b) Evolution with pressure of the top of the diffraction peak during the XPCS measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Data are reconstructed by integrating the detector intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' For clarity, the curves correspond to the compression stage only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' c) Evaluation of thermal expansion coefficient from the evolution of the FSDP position q1 with temperature measured at atmospheric pressure with synchrotron high energy XRD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Similar pressures were reached upon compression and decompression to allow direct comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The first step in compression at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='1 GPa corresponds to the preloading of the cell, and the pressure of the subsequent steps are of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='5, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='9, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3, and 7 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The last decompression step at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='5 GPa corresponds to the situation where the diamond anvil cell (DAC) membrane is fully deflated, but the DAC remains mechanically locked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The top of the First Sharp Diffraction Peak (FSDP) can be reconstructed by integrating the intensity collected on the detector during a complete XPCS scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The quantitative estimate of the peak position q1 is used to verify the consistency of the XPCS experiment with X-ray diffraction (XRD) experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Here, the continuous shift of the FSDP toward the high scattering vectors confirms the XRD results shown in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' q1 being linked to the characteristic distance ℓ of the medium range order of the glass by q1=2π/ℓ, we can quantitatively assess the validity of the link between q1 and the macroscopic density of the glass by comparing the coefficient of thermal expansion (CTE) derived from high energy XRD measurement to the CTE obtained from dilatometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The CTE α is inferred from the evolution of (ρ(T)-ρ(25°C))/ρ(25°C) ∝ (q1(25°C)/q1(T))3-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' We obtain α = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='85x10-5 K-1 in the glassy state, in good agreement with the reported value obtained by dilatometry1 α = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='95x10-5 K-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' This shows that the FSDP is directly linked through the macroscopic density for this glass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' es 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='004 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='002 1 1 g 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='000 0 20000 40000 60000 80000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='602.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='652.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='702.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='752.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='85 100 200 Time (s) Scattering vector q (A-1) Temperature (°C)(e b) C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='012 atmosphericpressure 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='010 (GPa) 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='008 4 sure S(q) Pressure 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='006 318 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Ruling out artefacts as the source of the heterogeneous dynamics Figure 6 - Trace (total intensity on detector) and selected TTCFs in compression at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3 GPa (left panels, elapsed time = 9700s), 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='9 GPa (central panels, elapsed time = 9800s) and in decompression at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3 GPa (right panels, elapsed time = 860s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Although the intensity impinging on the sample is in general stable, some fluctuations can occur during a full week of beamtime, and are usually due to adjustment of the electron beam in the storage ring and refilling mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' As the magnitude of these fluctuations is usually small compared to the total intensity, data are not affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' In Figure S2 we report selected TTCFs and the corresponding trace (total intensity in the detector as a function of time) which show two different situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' In the left panel and the beginning of the central panel, heterogeneous dynamics appear while trace is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Differently, in the central and right panel trace shows fluctuations related to the re-fill in the storage ring with no influence on the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' This demonstrates that fluctuations of the incoming beam intensity are not responsible for the observation of heterogeneous dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' These data allow us to rule out also possible sample movements as sources of induced decorrelations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The position of the sample was monitored by a microscope before and after each scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Large movements on the scale of 10 µm (>5 times the Rayleigh Criterion) would then been detected, which is not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' We exclude also smaller, micrometres movements as if the decorrelation would be associated only to a change of the scattering volume, the decorrelation time τ should be identical before and after the event, which is generally not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' In both the left and central panel, the ‘decorrelations’ in the TTCFs lead to different dynamic profiles, which implies that the relaxation of the systems has changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Another example is reported also in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' S3 as described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' It is well-known that some Pressure Transmitting Medium (PTM) can alter the property of a glass under compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' This is the case for instance of gas loading with He that can enter the large open network of silica glasses2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Due to its large molecular structure this is not the case for the alcohol mixture chosen here as PTM even in the presence of large open structures2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Additional data on the rejuvenation and relaxation regimes Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' S3 shows the TTCFs measured at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='9 GPa in compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The data shows aging regimes separated by a cascade relaxation at 9600s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The mix of cascade relaxation and aging between the two well defined dynamical regimes at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3 GPa shows the transition is not abrupt but continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The third TTCF at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='9 GPa is also a further confirmation of the absence of sample movement as the source 6 6 time time time 500 300 300 00 00 500 300 time time (s) 110 1000time (s) time (s) time (s) 0 200 400 600 0 200 400 600 0 250 500 750 1000 trace (x104) 15 1019 of the heterogeneous dynamical regime observed at low pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' If sample movement caused the decorrelation event at 9600s, decorrelation time τ should be identical before and after the event, which is obviously not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The TTCFs at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='5 GPa at the end of the decompression stage show the heterogeneous dynamical regime is eventually recovered but at a lower pressure compared to compression, in accordance with the evolution of the dynamical heterogeneity introduced in the figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Figure S3 - TTCFs obtained at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='9 GPa during compression (top row) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='5 GPa in decompression (bottom row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Repeatability: results from a second experiments We controlled the repeatability of the results on a new sample measured in a different run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 2e and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' S4, results of both runs overlap and show the same heterogenous vs homogeneous transition with pressure, which confirm the robusteness of the data showed in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Figure S4: Partial TTCFs measured in a second sample in the low pressure heterogeneous (left) and high pressure homogeneous (right) regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Hysteresis evolution of the dynamics under pressure compression and decompression The hysteresis shown in the main text from the comparison of the TTCFs at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3 GPa and the complete pathway of the heterogeneity parameter is also visible from the characteristic relaxation time of the intermediate scattering function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' In the figure S5 we represent intermediate scattering functions (ISFs) from the compression and decompression stages at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='5 GPa and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' To provide an accurate comparison, we chose to plot the curves obtained at the most similar elapsed times tw (defining the duration of the isobar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Both the shift of the curves and the relaxation times inferred from the KWW 1000 1000 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='4 GPa 570P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 0 0 1000 1000 0 200 400 600 800 10001200 0 200 400 600 800 1000 1200 time (s) time (s)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='0 300 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='8 6t( 100 time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='6 0 lay 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='9 GPa del 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='2 Compression 300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='0 1000 1200 1400 1600 4000 4200 4400 4600 9400 9600 9800 10000 10000 11000 11200 11400 Labtime(s) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='0 400 (s) 39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='8 200 delay time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='4 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='5 GPa 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='2 400 Decompression 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='0 500 750 1000 1250 1900 2150 2400 2650 2900 Lab time (s)20 model fitted to the data show slower dynamics during the decompression stage, confirming the hysteretic behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Figure S5: ISFs measured at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='5 GPa and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3 GPa in compression and decompression at similar elapsed times tw, showing the hysteresis within a pressure cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Dashed lines correspond to fits of the KKW model to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The corresponding relaxation times τ are reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Wave-vector dependent XPCS study To determine the dependence of the ISF with respect to the scattering vector q, we added a binning on the raw data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' In the figure S6 we plot a colour representation the total scattered intensity in the detector within a single scan of 7000 frames with an acquisition time of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The distribution of the intensity clearly shows the maximum of the diffraction peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The grey area corresponds to the raw mask applied during the pre-processing of the data, which covers the shadow of the vacuum tube between the sample and the detector and Kossel lines from the diamonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' On the right panel, the segmentation of the unmasked area of the detector into several bands at different is visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The TTCFs and ISFs were then extracted for the data corresponding to each of these bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 500 500 0 0 0 500 1000 1500 2000 0 500 1000 1500 20002000 2000 1500 1500 1000 10000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='4 Tcompression = 135s 口 Tcompression = 38s 000 吃 O TDecompression = 244s TDecompression = 244s 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='2 O Compression,tw=2800s Compression,tw=4900s 00 Decompression,tw=2400s Decompression,tw=3400s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='0 10-1 100 101 102 103 10-1 100 101 102 ot (s) at (s)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='8 脚 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='5 1 GPa: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='3 GPa 021 Figure S6: Left panel: Integrated intensity in the detector during an XPCS scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' A portion of the reddish ring associated to the FSDP is well visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Right panel: Q-binning of the data to extract the q-dependence evolution of the ISFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Pressure protocol and stability Figure S7 - Loading protocol for pressure stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Simultaneous recording of the membrane and ruby pressures (left panel) showing the stabilisation effect of the loading protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Pressure overshot without the adapted pressure protocol (upper right panel) and pressure drifts recorded during the experiment using the adapted protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' As XPCS is extremely sensitive to any structural change, it is essential to minimize any potential pressure drift during the measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The typical pressure increase after reaching the desired set- point for our DAC configuration is shown in the upper right panel, and can be higher than 1 GPa over one hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' To mitigate this issue, we have determined how much we can reverse the membrane pressure to stabilize the pressure on the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' More precisely, we measured how much one can decrease the membrane pressure after an initial increase before we can see any change in the sample pressure (down to our precision of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='01 GPa), as shown in the left panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' We have applied this loading protocol during the measurements, and we report all pressure drifts, taken as the pressure difference between the beginning and the end of a pressure steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' We can see the pressure drifts are now limited to about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='1 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Importantly, this drift is random and does not depend on the nominal pressure, so it is not responsible for the two-steps pressure effect on the dynamics reported in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Diamond Anvil Cell XPCS background Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' S8 contains two ISFs obtained under pressure with beam focused on the sample or in the experimental volume next to the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The absence of a decorrelation in the second case demonstrates that the contribution of the Diamond Anvil Cell, which comprises contributions from the diamonds and the pressure-transmitting medium, are only static contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' The sample dynamics probed with XPCS is therefore not affected by the high pressure cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='0 56 7 60 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='5 (bar) compression sample pressure (GPa) 54 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='8 decompression pressure 58- (GPa) 8 52 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='6 56- 50 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='4 54- △PlMAX = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='12 GPa 48 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='2 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='0 O 46 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='5 52 0 12 24 48 60 72 0 12 24 36 48 60 0 1 2 3 4 5 6 7 8 time (s) time (s) pressure(GPa)time (s) time (s) time (s) 0 500 1000 1500 2000 2500 3000 3500 0 6 12 18 24 30 36 0 12 24 36 48 60 (bar) 40 50- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='8 sample pressure (GPa) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='00 1 GPa membranepressure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='5 (GPa) 39 2 GPa 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='75 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='9 GPa 48- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='50 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='9 GPa 38 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='2 GPa 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='00 37 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='4 46- 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='522 Figure S8 - Correlation function g2 from beam targeting the sample (orange) and beam out of the sample, showing only the contribution of the diamond and pressure transmitting medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Dynamical heterogeneity parameter The dynamical heterogeneity parameter ΔC characterize the level of inhomogeneity of the glass atomic scale dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' This parameter corresponds to the smallest width that encompasses 90% of the distribution of the correlation values at a fixed delay time τ for each pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' This 90% threshold was chosen to reflect the extremes values taken by the correlation values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' However, we show that the evolution of with respect to pressure is not threshold strictly dependent on the value of this threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' In the figure S9 we reproduce the figure 5c) of the main text for different threshold values: 90%, 80%, 70%, and 60% (upper left, upper right, lower left and lower right panels respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Figure S9 – Dynamical heterogeneity parameter as a function of pressure for different threshold value: 90% (top left), 80% (top right), 70% (bottom left) and 60% (bottom right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Pressure (GPa) Pressure(GPa) 0 2 4 6 0 N 4 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='010 (%06) (%08) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='007 AC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='008 AC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='0050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='0045 (70%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='0040 AC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='005 △C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='0035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='004 compression 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='0030 decompression 0 2 4 6 0 2 4 6 Pressure(GPa) Pressure(GPa)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='025 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='020 diamond sample 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='015 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='010 口 10-1 100 101 102 6t (s)23 The result obtained for a width of 90% is reproducible quantatively down to a width of 70%, and qualitatively down to a width of 60%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Overall, this confirms the robustness of the heterogeneity parameter ΔC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' References: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Stolpe, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Synchrotron x-ray diffraction studies of bulk metallic glass forming liquids and glasses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' (Saarländische Universitäts- und Landesbibliothek, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content='22028/D291-32093.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Weigel, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Vitreous Silica Distends in Helium Gas: Acoustic Versus Static Compressibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} +page_content=' 109, 245504 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE0T4oBgHgl3EQfqQG1/content/2301.02551v1.pdf'} diff --git a/q9FAT4oBgHgl3EQffR3S/content/tmp_files/2301.08581v1.pdf.txt b/q9FAT4oBgHgl3EQffR3S/content/tmp_files/2301.08581v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..4f440fc969a948eca0382d14ea080b9783b32412 --- /dev/null +++ b/q9FAT4oBgHgl3EQffR3S/content/tmp_files/2301.08581v1.pdf.txt @@ -0,0 +1,2045 @@ +MNRAS 000, 1–12 (2022) +Preprint 23 January 2023 +Compiled using MNRAS LATEX style file v3.0 +Tilted discs in six poorly studied cataclysmic variables +Stefan Y. Stefanov,1,2★ Atanas K. Stefanov3 +1Institute of Astronomy and National Astronomical Observatory, Bulgarian Academy of Sciences, 72 Tsarigradsko Shose Boulevard, 1784 Sofia, Bulgaria +2Department of Astronomy, Sofia University "St. Kliment Ohridski", 5 James Bourchier Boulevard, 1164 Sofia, Bulgaria +3Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK +Accepted XXX. Received YYY; in original form ZZZ +ABSTRACT +In this work, we search for negative superhumps (nSHs) in poorly studied cataclysmic variables using TESS data. We find three +eclipsing binaries with nSH signatures: HBHA 4204-09, Gaia DR3 5931071148325476992, and SDSS J090113.51+144704.6. +The last one exhibits IW And-like behaviour in archival ZTF data, and appears to have shallow, grazing eclipses. In addition, +we detect nSH signatures in two non-eclipsing systems: KQ Mon and Gaia DR3 4684361817175293440, by identifying the +orbital period from the superorbital-dependent irradiation of the secondary. We discover nSH signatures in one more system, +[PK2008] HalphaJ103959, by using an orbital period from another work. An improved mass ratio – nSH deficit relation 𝑞(𝜀−) +is suggested by us, which agrees with independent measurements on nova-like variables. With this relation, we estimate the +mass ratios of all systems in our sample, and determine the orbital inclinations for the three that are eclipsing. All systems with +discovered nSHs in this work are excellent targets for follow-up spectroscopic studies. +Key words: +stars: activity – binaries: close – novae, cataclysmic variables – stars: individual: HBHA 4204-09, +Gaia DR3 4684361817175293440, +KQ Mon, +SDSS J090113.51+144704.6, +Gaia DR3 5931071148325476992, +[PK2008] HalphaJ103959 +1 INTRODUCTION +Cataclysmic variables (CVs) are binary systems that consist of a +white-dwarf (WD) primary and a Roche-lobe filling secondary. Mat- +ter from the secondary flows through the first Lagrangian point and +accretes on to the primary. In the case of a non-magnetic or a very +weakly magnetic primary, this mass transfer happens through an +accretion disc (Hellier 2001). In systems with mass-transfer rates of +�M ≃ 1 – 5 ×10−9 M⊙yr−1, thermal instabilities arise in the accretion +disc and cause repeating quasi-periodic outbursts. These outbursts +usually occur once about every few months, last several days, and +can increase the system brightness with up to ∼ 5 mag. CVs with +recorded outbursts are termed dwarf novae (DNe), whereas CVs +with no recorded outbursts are termed nova-likes (NLs). In NLs, +most of the flux originates from the accretion disc, which is in a hot +steady state and is much brighter than the two system components. +The orbital periods of this type of variables can range from ∼ 1 h +to more than 10 h. Not many CVs, however, are observed in the +period range of 2–3 h. This phenomenon is called the ”period gap” +and is explained by transitions in evolutionary stages of this type of +variables (see Warner 1995 for an encyclopedic description of CVs). +NLs can change their brightness on time-scales from seconds to +millennia. Some systems have drops in brightness of several magni- +tudes, which can last from months to years. This behaviour is most +commonly observed in systems with orbital periods (𝑃orb) near the +upper edge of the period gap. Such drops in brightness are cate- +★ E-mail: sstefanov@nao-rozhen.org +gorised as a low state of type VY Scl (King & Cannizzo 1998) and +can also be displayed by magnetic CVs. VY Scl low states are likely +caused by the reduction or the complete cessation of mass transfer in +the system, which significantly decreases the flux coming from the +disc. They are believed to be associated with the magnetic activity of +the secondary. Star spots emerging on the first Lagrangian point may +suppress mass transfer in the system (Livio & Pringle 1994), and the +radius of the secondary itself can be affected by magnetic activity +(Howell 2004). Yet, the exact mechanism of mass-transfer cessation +during VY Scl episodes remains unknown. +Apart from low states and other long-term trends in brightness, +CVs display an abundance of photometric variability on shorter time- +scales. Roche-lobe geometry requires that the secondary takes a char- +acteristic teardrop-like shape. As it orbits the barycentre, it presents +different projections of itself to the observer, which introduces a +photometric variability of period 𝑃orb/2. A similar effect can oc- +cur when the secondary is strongly irradiated by the accretion disc. +In that case, the visibility of the irradiated side of the secondary +is dependent on the orbital phase of the system, and a light-curve +modulation of period 𝑃orb takes place. +Some CVs exhibit variations in brightness that have periods +slightly shorter or slightly longer than 𝑃orb. These variations are +called ”superhumps” and are believed to be caused by a precessing +accretion disc. Superhumps can be of either positive (pSH) or neg- +ative (nSH) type, depending on the sign of 𝑃SH − 𝑃orb. They are +well-studied and commonly seen in SU UMa stars, a DN subclass +(e.g. Kato et al. 2009, 2017); as well as in NLs (Bruch 2023). For +NLs in particular, Bruch gave a sample of 46 systems, 13 have had +© 2022 The Authors +arXiv:2301.08581v1 [astro-ph.SR] 20 Jan 2023 + +2 +Stefanov & Stefanov +pSHs, 16 have had nSHs and 17 have had superhumps of both types +at some point in the past (but not necessarily at the same time). +Each superhump type is associated with processes of different +nature. pSHs are believed to be caused by an apsidally precessing +accretion disc. In this case, the 3:1 resonance induces tidal deforma- +tions, the heat from which causing periodic changes in disc brightness +(Whitehurst 1988; Hirose & Osaki 1990; Lubow 1991). On the other +hand, nSHs can be explained with a retrograde nodal precession of +a tilted accretion disc. The tilt allows for the mass-transfer stream +to go deeper in the gravitational well of the primary, and thus to +release more energy upon impact. The point of impact on the disc +is commonly referred to as the ”bright spot”. The sweeping of the +bright spot across the disc faces introduces an additional photometric +variability that has a period equal to the beating of 𝑃orb and the disc +precession period 𝑃prec (Wood et al. 2009; Montgomery 2009a), i.e. +1 +𝑃nSH += +1 +𝑃orb ++ +1 +𝑃prec +. +(1) +Superhumps of both types can be used to estimate some physical +properties of these systems. The nSH deficit 𝜀− is defined as +𝜀− = 𝑃nSH − 𝑃orb +𝑃orb +(2) +and has been shown to correlate with the mass ratio of the system +𝑞 = 𝑀1/𝑀2 in several works (e.g. Wood et al. 2009; Montgomery +2009a). A detailed study of nSHs can be found in Kimura et al. +(2020b); Kimura & Osaki (2021), where Kepler photometry of the +NL system KIC 9406652 was analysed. The light curve of this partic- +ular object has identifiable 𝑃orb, 𝑃nSH signals as well as superorbital +ones (i.e. 𝑃prec signatures). +In this work, we present our results from a search for nSHs in poorly +studied CVs that are similar to KIC 9406652. Section 2 presents our +methods for searching and data reduction, and gives a list of objects +with discovered nSH signatures. Section 3 contains a literature review +and discussion of each system we found to have nSH behaviour. In +Section 4, we attempt to estimate some physical parameters in said +systems, and in Section 5, we summarise the findings of this work. +2 ANALYSIS +2.1 Data from TESS +The Transiting Exoplanet Survey Satellite (TESS; Ricker et al. 2015) +mission is an all-sky survey in the red-infrared that continues to +provide with long-term measurements of remarkable photometric +precision. The TESS Science Processing Operations Center pipeline +(SPOC; Jenkins et al. 2016) offers light curves from two different +reduction techniques: Simple Aperture Photometry (SAP) and Pre- +Search Data Conditioning Simple Aperture Photometry (PDCSAP). +A comprehensive comparison between the two is given in Kine- +muchi et al. (2012). PDCSAP tries to reduce effects of instrumental +nature, but can sometimes introduce systematics in periodograms, +and analysis should proceed with care. Bruch (2022) found in par- +ticular that the additional conditioning in PDCSAP may distort DNe +light curves, and chose to use the simpler SAP technique in order to +search for periodic variations in CVs. We use SAP light curves too +in all analysis to follow. +2.2 Photometric features of tilted accretion discs +Negative superhumps are direct evidence for a titled accretion disc, +but finding their signatures is only possible in systems of known +Figure 1. A model CV system at an orbital phase 𝜑orb = 0.5, as it would be +seen by an observer. Four precession phases 𝜑prec of a disc with tilt 𝜃 = 6◦ +are illustrated. The orbital plane of the system is defined by dotted lines. It +divides space into two half-spaces: the near (above the plane) and the far +one (below the plane), with respect to the observer. The precession phase is +defined such that the system is the brightest at 𝜑prec = 0. In this orientation, +the disc has the largest projected area at 𝜑prec = 0. Conversely, at 𝜑prec = 0.5, +it has the smallest projection, but faces towards the secondary and irradiates +it the most. Kimura & Osaki (2021), Figure 9 gives a full description of CV +configurations in titled-disc regimes. +𝑃orb. This is a strong restriction, since not many CVs have had their +orbital periods measured. To expand the population of stars with +known 𝑃orb, we searched for systems with several significant peaks +in the power spectrum, in a frequency region above the period gap. +In the case of two neighbouring prominent peaks, it could be that +those are signatures of 𝑃pSH, 𝑃orb and not 𝑃nSH, 𝑃orb. Nevertheless, +this degeneracy can be lifted with the following rationale. +In systems with a precessing tilted accretion disc, the disc orienta- +tion changes with respect to the secondary for different orbital phases +𝜑orb and different disc precession phases 𝜑prec. The former is defined +such that the secondary is at inferior conjunction at 𝜑orb = 0.0; the +latter is defined1 such that the light maximum in the disc precession +cycle is at 𝜑prec = 0.0. The observed irradiation of the secondary +by the bright disc varies with both 𝜑prec and 𝜑orb. Consider a non- +eclipsing system at 𝜑orb = 0.5 (Figure 1). The orbital plane of the +system divides space into two half-spaces, one of which the observer +finds themselves in. One part of the system resides in the same half- +space as the observer, and the other part is in the opposite half-space +(i.e. on the other side of the orbital plane with respect to the ob- +server). We shall refer to those as the ”near half-space” and the ”far +half-space”. As an example, in Figure 1, the part of the accretion disc +that lies in the near half-space is: its rear side at 𝜑prec = 0, its right +side at 𝜑prec = 0.25 and so on. +For an observer, the near half-space of a system is more photomet- +rically accessible than the far half-space.2 At 𝜑prec = 0, the luminous +disc reveals the most of itself to the observer, and the average sys- +tem brightness across 𝜑orb is the greatest. However, the irradiated +region of the secondary is in the far half-space, and thus the 𝜑orb +variation in brightness is minimal in amplitude. In the opposite case +of 𝜑prec = 0.5, the observer sees the smallest possible projection +of the disc, and the average system brightness across 𝜑orb is the +smallest – but the irradiated region of the secondary is now in the +near half-space, and the 𝜑orb variation in brightness is maximal in +amplitude. +1 These definitions are consistent with Kimura et al. (2020b). +2 This is only untrue in the special case of 𝑖 = 90◦, when the observer lies in +the orbital plane, and both half-spaces are thus equally accessible. +MNRAS 000, 1–12 (2022) + +Ppree +Ppree +Ppree +PpreeTilted discs in six poorly studied CVs +3 +At the same time, nSHs introduce additional complexities in vari- +ability that need to be accounted for. Kimura & Osaki (2021) discuss +this issue and carry out the following procedure. A given light curve +is initially split into subsets of different time intervals. Then, for each +subset, they: (1) fold by 𝑃nSH and construct an average light-curve +profile of the nSH, (2) subtract said profile from each subset, (3) +split the subset into different 𝜑prec windows, (4) fold each window +by 𝑃orb. This technique results in multiple orbital phase curves, each +corresponding to a different 𝜑prec window. If these phase curves show +a 𝜑prec-dependent irradiation of the secondary, the system has a pre- +cessing tilted accretion disc and the observed superhump is negative. +It is this consideration that could lift the pSH-nSH degeneracy in the +power spectrum. +In order to address the nSH contamination, we use a variant of the +nSH-subtraction technique by Kimura & Osaki (2021) with the fol- +lowing adjustments: all data is smoothed by a fourth-order Savitzky- +Golay filter (Savitzky & Golay 1964) of window size 10 d, and no sep- +arate subsets are considered; in (1), nSH light-curve profiles are con- +structed with median filters of window size 1101;3 in (3), four 𝜑prec +intervals are considered with centres at 𝜑prec = 0.00, 0.25, 0.50, 0.75 +and of width 0.1.4 +2.3 Target selection +The International Variable Star Index (VSX; Watson et al. 2006, ac- +cessed 2022 June) is perhaps the most extensive catalogue of known +variable stars. We took all objects from the VSX labelled as CV or +as NL (𝑛 = 1249), and then sought all for which there were avail- +able TESS SPOC light curves of 120-second cadence (𝑛 = 180). +Lomb-Scargle periodograms (LS periodogram; Lomb 1976; Scar- +gle 1982) of range between 0.125 – 16.000 d−1 were constructed for +those systems. Periodograms were then manually searched for the si- +multaneous presence of at least two neighbouring periodicities in the +region above the period gap, as well as for one periodicity near their +expected beat period. This was done to select NLs with signatures +of all 𝑃prec, 𝑃orb, 𝑃nSH. For most stars, long-term photometry from +the All-Sky Automated Survey for Supernovae (ASAS-SN; Shappee +et al. 2014; Kochanek et al. 2017) and from the Catalina Sky Survey +(CSS, Drake et al. 2009) was available. We attempted to construct +LS periodograms using photometry from said surveys, but data was +found to be sparse and of too long cadence to be usable. +We report on the discovery of nSH behaviour in six poorly studied +CVs. Three of them are eclipsing systems, which enabled us to +directly determine 𝑃orb. For two other systems, 𝑃orb was identified +with the use of 𝜑prec-dependent irradiation of the secondary. The last +CV was found to have 𝑃prec and 𝑃nSH signatures, but not a 𝑃orb one. +Our derived value of 𝑃orb by Equation (1), however, agrees well with +the spectroscopic measurement of Pretorius & Knigge (2008). All six +objects are discussed individually in Section 3. During inspection, +we also found five new eclipsing CVs with no superhump behaviour. +Their measured orbital periods are provided in Table A3, and their +orbital phase curves are shown in Figure A1. +3 REVIEW AND RESULTS +The following sections provide literature review, discussion and in- +terpretation of data for all CVs with discovered nSH behaviour. Each +3 We found that this window size worked generally well for all systems. +4 That is, the intervals 0.95 – 0.05, 0.20 – 0.30, 0.45 – 0.55, 0.70 – 0.80. +CV system has an associated figure containing: (1) available sky- +survey data, (2) TESS photometry from sectors with prominent nSH +behaviour together with corresponding LS periodograms, (3) orbital +phase plots of data in the four 𝜑prec regions discussed in Section +2.2. Measured periodicities of each system are given in Table 1. All +measurements agree well with Equation (1) within uncertainty. +3.1 HBHA 4204-09 +HBHA 4204-095 (Figure 2) is discovered by ASAS-SN. It was clas- +sified as a CV by Jayasinghe et al. (2018) and by ALeRCE (Förster +et al. 2021) in data from the Zwicky Transient Facility (ZTF; Bellm +et al. 2019). This object is part of the ”Catalogue of Stars in the +Northern Milky Way Having H-alpha in Emission” (Kohoutek & +Wehmeyer 1999). The Gaia DR3 distance estimate is 478 ± 3 pc and +ASAS-SN photometry gives a mean brightness of 𝑚𝑉 = 16.19 mag. +We report the presence of previously unknown V-shaped eclipses +in HBHA 4204-09. Using them, we identify the periodogram +peaks corresponding to 𝑃orb, 𝑃nSH, 𝑃prec (Table 1). Aside from +these periodicities, the power spectrum contains a strong signal +at 0.d070655(58), which matches 𝑃orb/2. A collection of peaks at +around 0.d155 is observed, which may be indicative of a pSH sig- +nature. Additional photometry of HBHA 4204-09 can be found in +TESS Sectors 55 and 56, but no superhumps are present in those +data sources. Due to its high orbital inclination, the near and the far +half-spaces defined by the orbital plane are comparably accessible +to the observer. The portion of the secondary in the far half-space +is most irradiated at 𝜑prec = 0.00, while the portion in the near +half-space is most irradiated at 𝜑prec = 0.50. The orbital profiles in +panels (d) and (f) of Figure 2 show stronger secondary irradiation at +aforementioned 𝜑prec, which is expected. +3.2 Gaia DR3 4684361817175293440 +Gaia DR3 4684361817175293440, hereinafter Gaia-4684366 (Fig- +ure 3) was discovered and classified as a NL type CV by Bajer +(2019). The Gaia DR3 distance estimate is 1062+29 +−30 pc. On the long- +term ASAS-SN curve, a 1-mag fall in brightness can be observed +around BTJD 800 – BTJD 1700. A panel with ASAS-SN photom- +etry in this time period is shown in Figure 4. The observed drop in +brightness has a smaller amplitude from what is expected in classic +VY Scl low states. Quasi-cyclic variations of 𝑃 ∼ 20 d resembling +Z Cam outbursts appear after the start of the low state. The system +later returns to normal brightness and outbursts are replaced with a +standstill lasting for ∼ 300 days. This standstill is followed by another +Z Cam outburst episode, after which no more outbursts of this type +are observed. +Our LS periodogram of Gaia-468436 shows three peaks with peri- +ods matching Equation (1). We interpret them as 𝑃orb, 𝑃nSH,𝑃prec in +a system with a tilted precessing disc (Section 2.2). We find two ad- +ditional peaks at 0.d07372(14) and 0.d07702(15) that match 𝑃nSH/2 +and 𝑃orb/2 respectively. In Figure 3(d)–(g), it can be seen that the +light maximum of orbital-phase curves gradually shifts to earlier 𝜑orb +as the disc precession cycle advances. This is direct evidence for a +retrogradely precessing tilted disc (Kimura et al. 2020b). +5 The VSX identifier of this source is ASASSN-V J210752.24+440542.0. +6 The VSX identifier of this source is BMAM-V424. +MNRAS 000, 1–12 (2022) + +4 +Stefanov & Stefanov +Table 1. List of CVs with discovered nSHs using the methods described in Section 2. All periodicities in this table were measured on a Lomb-Scargle periodogram +of range 0.125–16 d−1 and of ten-fold oversampling. All measured 𝑃prec in this table agree within uncertainty with the expected values by Equation (1) using +measured 𝑃orb and 𝑃nSH. Equatorial coordinates come from Gaia DR3 and are in the J2000 epoch. +Name +RA +Dec +TESS Sector +𝑃orb +𝑃nSH +𝑃prec +|𝜀− | +HBHA 4204-09 +21h07m52.s24 ++44◦05′42.′′0 +15, 16 +0.d14128(22) +0.d13657(22) +4.d11(18) +0.0333(22) +Gaia DR3 4684361817175293440 +00h49m59.s93 +−76◦08′27.′′5 +28 +0.d15401(53) +0.d14750(52) +3.d40(27) +0.0423(48) +KQ Mon +07h31m21.s13 +−10◦21′49.′′4 +34 +0.d13456(40) +0.d12894(38) +3.d12(24) +0.0418(41) +SDSS J090113.51+144704.6 +09h01m13.s51 ++14◦47′04.′′7 +44 – 46 +0.d14631(17) +0.d13991(17) +3.d198(70) +0.0437(16) +Gaia DR3 5931071148325476992 +16h36m03.s63 +−52◦33′32.′′6 +39 +0.d14827(46) +0.d14248(43) +3.d57(30) +0.0391(42) +[PK2008] HalphaJ103959 +10h39m59.s98 +−47◦01′26.′′3 +36, 37 +0.d1577(2)† +0.d15285(29) +4.d94(26) +0.0308(22) +†Orbital period measured spectroscopically by Pretorius & Knigge (2008). +0 +500 +1000 +1500 +2000 +2500 +BTJD (d) +14 +15 +Mag itude (mag) +(a) +1720 +1740 +1760 +BTJD (d) +−200 +0 +200 +Res. flu) (e +− +s +−1 +) +(b) +0 +5 +10 +15 +Freque cy (d +−1 +) +0.0 +0.5 +1.0 +Norm. po(er +(c) +0.0 +0.5 +1.0 +1.5 +2.0 +Orbital phase +−50 +0 +50 +Res. flu) (e +− + s +−1 +) +(d) +φ +prec += +0.00 +0.0 +0.5 +1.0 +1.5 +2.0 +Orbital phase +−50 +0 +50 +Res. flu) (e +− + s +−1 +) +(e) +φ +prec += +0.25 +0.0 +0.5 +1.0 +1.5 +2.0 +Orbital phase +−50 +0 +50 +Res. flu) (e +− + s +−1 +) +(f) +φ +prec += +0.50 +0.0 +0.5 +1.0 +1.5 +2.0 +Orbital phase +−50 +0 +50 +Res. flu) (e +− + s +−1 +) +(g) +φ +prec += +0.75 +Figure 2. Photometry and analysis of HBHA 4204-09. (a) Long-term photometry from: ASAS-SN 𝑔 (purple downward triangles), ASAS-SN 𝑉 (green upward +triangles). Temporal coverage of TESS: Sector 15 (light blue), Sector 16 (yellow). (b) Residual (mean-subtracted) SAP flux from TESS data. (c) Associated +LS periodogram of (b), with indications of 𝑃prec, 𝑃orb, 𝑃nSH signals from Table 1 (blue dashes). (d)–(g) Binned orbital profiles around disc precession phases +𝜑prec = 0.00, 0.25, 0.50, 0.75 with nSH subtraction (blue squares) and without one (black circles). The typical standard deviation of data in bins is 25 e− s−1. +This system has a high inclination, and the parts of the secondary in both half-spaces are accessible to the observer. The secondary is expected to be the most +irradiated in panels (d) and (f), where the observed out-of-eclipse profile is non-flat. In panels (e) and (g), where no irradiation of the secondary is expected, the +out-of-eclipse profile is mostly flat. +3.3 KQ Mon +KQ Mon (Figure 5) was classified as a NL-type CV by Bond (1979) +using low-resolution spectra in the optical. Its orbital period was +measured in Schmidtobreick et al. (2005) to be 𝑃orb = 0.d1283(17) +by analysing two nights of time-resolved spectroscopy. Later, Wolfe +et al. (2013) examined far-ultraviolet spectra of KQ Mon from the +International Ultraviolet Explorer. The mass of the primary was esti- +mated to be 𝑀1 ∼ 0.6 𝑀⊙ with the use of synthetic spectra. The same +work argued that the primary contributes little to the total system flux, +and is overwhelmed by the flux of a steady-state accretion disc. It +was concluded that the system is located at a distance of 144–159 pc, +with an inclination of 𝑖 ≤ 60◦ and an accretion rate in the order of +�𝑀 ∼ 10−9 𝑀⊙ yr−1. The Gaia DR3 distance is 628±8 pc, which +disagrees with their estimates. +Our measured value for 𝑃nSH matches the 𝑃orb given in Schmidto- +breick et al. (2005). What we measure as 𝑃orb = 0.d13456(40) would +have corresponded to a pSH signal in their interpretation. But then, +no other signals in the periodogram would have been expected. We, +however, measure a strong third signal at 3.d12(24), which is self- +consistent with the other two by Equation (1). In addition, we observe +a 𝜑prec-dependent amplitude of the orbital phase curve, which could +be explained by a varying irradiation of the secondary. In Figure 3 of +Schmidtobreick et al. (2005), a strong aliasing pattern can be seen. +The authors chose an orbital period 𝑃orb among four possible signals, +two of which agree with our measurements of 𝑃orb, 𝑃nSH. With all +this in mind, we think that the correct value of 𝑃orb is 0.d13456(40), +and that there is presence of a tilted accretion disc in this system. +MNRAS 000, 1–12 (2022) + +Tilted discs in six poorly studied CVs +5 +0 +500 +1000 +1500 +2000 +2500 +BTJD (d) +14 +15 +16 +Magnit(de ( ag) +(a) +2065 +2070 +2075 +2080 +2085 +BTJD (d) +−25 +0 +25 +Res. fl(x (e +− +s +−1 +) +(b) +0 +5 +10 +15 +Freq(ency (d +−1 +) +0.0 +0.5 +1.0 +Nor . po)er +(c) +0.0 +0.5 +1.0 +1.5 +2.0 +Orbital phase +−10 +0 +10 +Res. fl(x (e +− + s +−1 +) +(d) +φ +prec += +0.00 +0.0 +0.5 +1.0 +1.5 +2.0 +Orbital phase +−10 +0 +10 +Res. fl(x (e +− + s +−1 +) +(e) +φ +prec += +0.25 +0.0 +0.5 +1.0 +1.5 +2.0 +Orbital phase +−10 +0 +10 +Res. fl(x (e +− + s +−1 +) +(f) +φ +prec += +0.50 +0.0 +0.5 +1.0 +1.5 +2.0 +Orbital phase +−10 +0 +10 +Res. fl(x (e +− + s +−1 +) +(g) +φ +prec += +0.75 +Figure 3. Photometry and analysis of Gaia-468436. (a) Long-term photometry from: ASAS-SN 𝑔 (purple downward triangles), ASAS-SN 𝑉 (green upward tri- +angles). Temporal coverage of TESS: Sector 28 (light blue). (b) Residual (mean-subtracted) SAP flux from TESS data. (c) Associated LS periodogram of (b), with +indications of 𝑃prec, 𝑃orb, 𝑃nSH signals from Table 1 (blue dashes). (d)–(g) Binned orbital profiles around disc precession phases 𝜑prec = 0.00, 0.25, 0.50, 0.75 +with nSH subtraction (blue squares) and without one (black circles). The typical standard deviation of data in bins is 4 e− s−1. The effect of variable irradiation +in panels (d)–(g) is similar to the one observed in KIC 9406652 (Kimura & Osaki 2021). +1000 +1200 +1400 +1600 +1800 +BTJD (d) +14 +15 +16 +Magnitude (mag) +Figure 4. Z Cam-like episodes of Gaia-468436 in ASAS-SN 𝑔 (blue downward triangles) and ASAS-SN 𝑉 (green upward triangles). The Z Cam behaviour +begins after a ∼ 0.5 mag fall in brightness. Emerging oscillations have a variable amplitude of ∼ 0.8 mag and are quasi-periodic with 𝑃 ∼ 20 days. At about +BTJD 1500, a brightening takes place, which is followed by a standstill at a level of 15.0 mag. At about BTJD 1760, the standstill is replaced with another +oscillatory episode that has outbursts of similar period and amplitude as the former ones. No more Z Cam episodes were observed in Gaia-468436 in this data. +3.4 SDSS J090113.51+144704.6 +SDSS J090113.51+144704.6, hereinafter SDSS-090113 (Figure 6) +first appeared in Szkody et al. (2009) where it was classified as a +CV due to accretion disc features in its spectrum. This system was +included in the catalogue of bright WDs of Raddi et al. (2017). Gaia +DR3 estimated the distance to SDSS-090113 to be 1482+100 +−116 pc. Later, +Mösenlechner et al. (2022) included this system in their time-series +analysis study of subdwarf A-type stars using Kepler K2 data. They +discovered a periodicity of 0.d146, which was suggested to be the +orbital period 𝑃orb. +SDSS-090113 has no recorded low states and its brightness varies +around 𝑚𝑉 = 16.2 mag. Between BTJD 1600 and BTJD 2300, we +recognise an episode of anomalous Z Cam-type outbursts repeating +once about every 25 days (Figure 7). These outbursts begin after a +brightening, which is one of the defining features of the IW And- +phenomenon systems (Kato 2019). This can be explained by a tilted +disc that causes the accretion stream to enter inner disc regions, and +thus to disrupt the accretion cycle. In this new type of accretion, the +inner disc is in a hot state, while the outer disc repeats outbursts +(Kimura et al. 2020a). +Figure 6(d)–(g) shows what seems to be the presence of graz- +ing eclipses in the orbital curve of SDSS-090113. They vary in +depth and width, and for some phases of 𝑃prec they disappear com- +pletely, similar to ES Dra (Kato 2022). Through them, we iden- +tify 𝑃orb, 𝑃nSH, 𝑃prec (Table 1). Two additional peaks are found at +0.d071531(52) and 0.d073161(40) that match 𝑃nSH/2 and 𝑃orb/2 +respectively. +MNRAS 000, 1–12 (2022) + +6 +Stefanov & Stefanov +0 +500 +1000 +1500 +2000 +2500 +BTJD (d) +13.0 +13.5 +Mag itude (mag) +(a) +2230 +2240 +2250 +BTJD (d) +−100 +0 +100 +Res. flu) (e +− +s +−1 +) +(b) +0 +5 +10 +15 +Freque cy (d +−1 +) +0.0 +0.5 +1.0 +Norm. po(er +(c) +0.0 +0.5 +1.0 +1.5 +2.0 +Orbital phase +0 +50 +Res. flu) (e +− + s +−1 +) +(d) +φ +prec += +0.00 +0.0 +0.5 +1.0 +1.5 +2.0 +Orbital phase +0 +50 +Res. flu) (e +− + s +−1 +) +(e) +φ +prec += +0.25 +0.0 +0.5 +1.0 +1.5 +2.0 +Orbital phase +0 +50 +Res. flu) (e +− + s +−1 +) +(f) +φ +prec += +0.50 +0.0 +0.5 +1.0 +1.5 +2.0 +Orbital phase +0 +50 +Res. flu) (e +− + s +−1 +) +(g) +φ +prec += +0.75 +Figure 5. Photometry and analysis of KQ Mon. (a) Long-term photometry from: ASAS-SN 𝑔 (purple downward triangles), ASAS-SN 𝑉 (green upward triangles). +Temporal coverage of TESS: Sector 34 (light blue). (b) Residual (mean-subtracted) SAP flux from TESS data. (c) Associated LS periodogram of (b), with +indications of 𝑃prec, 𝑃orb, 𝑃nSH signals from Table 1 (blue dashes). (d)–(g) Binned orbital profiles around disc precession phases 𝜑prec = 0.00, 0.25, 0.50, 0.75 +with nSH subtraction (blue squares) and without one (black circles). The typical standard deviation of data in bins is 16 e− s−1. The blue and the black curves +differ due to the significant nSH contribution to the observed system flux. In blue curves of different 𝜑prec, there seems to be a change of shape and amplitude +near 𝜑orb = 0.5, which is expected, but could be also due to noise. Kimura & Osaki (2021) provide models for such orbital curves that could explain these +observations. +3.5 Gaia DR3 5931071148325476992 +Gaia DR3 5931071148325476992, hereinafter Gaia-5931077 (Fig- +ure 8) is a poorly studied CV that was discovered in plates by +Prestgard (2020) from the Digitized Sky Survey8 and the Super- +COSMOS H𝛼 survey (Parker et al. 2005). The NOMAD catalogue +(Zacharias et al. 2004) gives an apparent magnitude of 𝑚𝑉 = 16 mag. +No ASAS-SN photometry is available for this system. Its TESS +brightness reads 𝑚TESS = 16.02 mag. There is an X-ray source +(1RXS J163605.9-523335) at a distance of 20.8 arcsec, which is +likely associated with Gaia-593107. In addition, we find two bright +sources of brightness 𝑚TESS = 13.44 and 𝑚TESS = 15.16 mag in +the aperture mask, that are expected to severely contaminate the light +curve. This issue, however, is resolved by the apparent variability of +Gaia-593107 in the discovery images9 of Prestgard, on the basis of +which we attribute the tilted-disc behaviour to this specific system. +Our analysis of TESS light curves shows Gaia-593107 to be an +eclipsing variable with an orbital period of 𝑃orb = 0.d14248(43). +A peak at 𝑃orb/2 is present as well. This allows us to locate +𝑃orb, 𝑃nSH, 𝑃prec (Table 1). A change in eclipse depth is observed in +different phases of the determined 𝜑prec. +7 The VSX identifier of this source is USNO-A2.0 0300-28957281. +8 ESO Online Digitized Sky Survey: http://archive.eso.org/dss/dss +(accessed 2022 October). +9 https://www.aavso.org/vsx_docs/1544030/3344/USNO-A2.0% +200300-28957281.png +3.6 [PK2008] HalphaJ103959 +[PK2008] HalphaJ103959, hereinafter PK-103959 (Figure 9) was +classified as a CV in Pretorius & Knigge (2008), where spectro- +scopic and photometric analyses of the system were carried out. The +orbital period of PK-103959 was measured to be 𝑃orb = 0.d1577(2) +in the same work. Catalina and ASAS-SN photometry has a mean +brightness of 𝑚𝑉 =15.7 mag, with no low states. A gradual in- +crease in brightness can be seen in the period between -1000 and +2000 BTJD. +We find signatures of 𝑃nSH and 𝑃prec (Table 1), but no peaks at +the 𝑃orb by Pretorius & Knigge. However, there are two other visible +peaks at 0.d07885(14) and 0.d07639(13), which correspond to 𝑃orb/2 +by Pretorius & Knigge and 𝑃nSH/2 respectively. +4 DISCUSSION +4.1 Mass-ratio estimates +By using smoothed particle hydrodynamic (SPH) simulations of +tilted accretion discs, Wood et al. (2009) found that the relation +between the mass ratio and the nSH deficit is well-represented by +𝑞(𝜀−) = −0.192|𝜀−|0.5 +10.37|𝜀−| −99.83|𝜀−|1.5 +451.1|𝜀−|2. (3) +This result has been supported by other works using related SPH +simulations (Montgomery 2009a; Thomas & Wood 2015). To com- +pare Equation (3) with observations, we searched for NL objects in +MNRAS 000, 1–12 (2022) + +Tilted discs in six poorly studied CVs +7 +−3000 +−2000 +−1000 +0 +1000 +2000 +3000 +BTJD (d) +15 +16 +17 +Magnit(de ( ag) +(a) +2520 +2540 +2560 +BTJD (d) +−25 +0 +25 +Res. fl(x (e +− +s +−1 +) +(b) +0 +5 +10 +15 +Freq(ency (d +−1 +) +0.0 +0.5 +1.0 +Nor . po)er +(c) +0.0 +0.5 +1.0 +1.5 +2.0 +Orbital phase +−5 +0 +5 +Res. fl(x (e +− + s +−1 +) +(d) +φ +prec += +0.00 +0.0 +0.5 +1.0 +1.5 +2.0 +Orbital phase +−5 +0 +5 +Res. fl(x (e +− + s +−1 +) +(e) +φ +prec += +0.25 +0.0 +0.5 +1.0 +1.5 +2.0 +Orbital phase +−5 +0 +5 +Res. fl(x (e +− + s +−1 +) +(f) +φ +prec += +0.50 +0.0 +0.5 +1.0 +1.5 +2.0 +Orbital phase +−5 +0 +5 +Res. fl(x (e +− + s +−1 +) +(g) +φ +prec += +0.75 +Figure 6. Photometry and analysis of SDSS-090113. (a) Long-term photometry from: ASAS-SN 𝑔 (purple downward triangles), ASAS-SN 𝑉 (green upward +triangles), Catalina 𝑉 (pink squares). Temporal coverage of TESS: Sector 44 (light blue), Sector 45 (yellow), Sector 46 (light blue). (b) Residual (mean-subtracted) +SAP flux from TESS data. (c) Associated LS periodogram of (b), with indications of 𝑃prec, 𝑃orb, 𝑃nSH signals from Table 1 (blue dashes). Data from (b) was +smoothed by a fourth-order Savitzky-Golay filter of window size 10 d before constructing the periodogram. This was done solely for the sake of clear identification +of 𝑃prec by the reader, and not for periodicity measurements. (d)–(g) Binned orbital profiles around disc precession phases 𝜑prec = 0.00, 0.25, 0.50, 0.75 with +nSH subtraction (blue squares) and without one (black circles). The typical standard deviation of data in bins is 4 e− s−1. SDSS-090113 appears to have shallow +grazing eclipses that are barely detectable for some 𝜑prec. Observed eclipses vary in depth and width for different 𝜑prec. This could be explained by a secondary +that partially covers the tilted disc only when the projected area of the disc is large. +1400 +1600 +1800 +2000 +2200 +BTJD (d) +16.0 +16.5 +17.0 +Magnitude (mag) +Figure 7. IW And episodes of SDSS-090113 in ZTF 𝑔 (teal circles) and ZTF 𝑟 (magenta diamonds). Three seasons of photometry are shown. The first season +starts with oscillations that are terminated by brightening. This is one of the defining features of the IW And phenomenon (Kato 2019; Kato et al. 2022). The +second season shows the beginning of a new oscillatory episode that is variable in amplitude. The episode continues in the third season and abruptly ends at +BTJD 2315. +literature for which superhump deficits and mass ratios were mea- +sured independently from one another. Our reasoning is that NLs +share three main similarities with our discovered CVs: (1) their 𝑃orb +are of the same order, (2) they have steady, hot and luminous discs, +(3) samples of both populations exhibit VY-Scl behaviour. Using the +sample of NLs with nSH signatures, given in Bruch (2023), we were +able to find twelve such objects, which we list in Table 2. Figure 10(a) +compares their measurements with the 𝑞(𝜀−) relations provided by +Montgomery (2009a) and Wood et al. (2009). We share the concern +that both works underestimate 𝑞(𝜀−) with respect to past measure- +ments in literature. +There exists a different approach that could estimate mass ratios +using nSHs. By making some assumptions, a 𝑞(𝜀−) relation can be +derived in the following manner. Through linear perturbation theory, +Papaloizou & Terquem (1995) derived the precession rate 𝜔prec of a +differentially rotating fluid disc with a mass profile Σ(𝑟): +𝜔prec = −3 +4 +𝐺𝑀2 +𝑎3 +∫ +Σ𝑟3d𝑟 +∫ +ΣΩ𝑟3d𝑟 +cos 𝜃, +(4) +where 𝑎 is the orbital separation, Ω(𝑟) = +√︁ +𝐺𝑀1/𝑟3 is the Keplerian +angular velocity profile of the disc, and 𝜃 is the disc tilt with respect +to the orbital plane. For a power-law mass profile Σ(𝑟) ∝ 𝑟𝑛, Osaki +MNRAS 000, 1–12 (2022) + +8 +Stefanov & Stefanov +2370 +2380 +BTJD (d) +−50 +0 +50 +Res. flux (e +− +s +−1 +) +(a) +0 +5 +10 +15 +Frequenc( (d +−1 +) +0.0 +0.5 +1.0 +Norm. ower +(b) +0.0 +0.5 +1.0 +1.5 +2.0 +Orbital hase +−20 +0 +20 +Res. flux (e +− + s +−1 +) +(c) +φ +prec += +0.00 +0.0 +0.5 +1.0 +1.5 +2.0 +Orbital hase +−20 +0 +20 +Res. flux (e +− + s +−1 +) +(d) +φ +prec += +0.25 +0.0 +0.5 +1.0 +1.5 +2.0 +Orbital hase +−20 +0 +20 +Res. flux (e +− + s +−1 +) +(e) +φ +prec += +0.50 +0.0 +0.5 +1.0 +1.5 +2.0 +Orbital hase +−20 +0 +20 +Res. flux (e +− + s +−1 +) +(f) +φ +prec += +0.75 +Figure 8. Photometry and analysis of Gaia-593107. (a) Residual (mean-subtracted) SAP flux from TESS data. (b) Associated LS periodogram of (a), with +indications of 𝑃prec, 𝑃orb, 𝑃nSH signals from Table 1 (blue dashes). (c)–(f) Binned orbital profiles around disc precession phases 𝜑prec = 0.00, 0.25, 0.50, 0.75 +with nSH subtraction (blue squares) and without one (black circles). The typical standard deviation of data in bins is 8 e− s−1. Similar to the other eclipsing +binaries in our sample, the secondary is expected to be the most irradiated in panels (c) and (e), where the observed out-of-eclipse profile is non-flat. In panels +(d) and (f), these profiles are mostly flat, and the system brightness does not seem to increase near 𝜑orb = 0.5. +−3000 +−2000 +−1000 +0 +1000 +2000 +3000 +BTJD (d) +15 +16 +Magnitude (mag) +(a) +2290 +2300 +2310 +2320 +2330 +BTJD (d) +0 +50 +Res. flu( (e +− +s +−1 +) +(b) +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +Frequenc) (d +−1 +) +0.0 +0.5 +1.0 +N rm. p wer +(c) +Figure 9. Photometry and analysis of PK-103959. (a) Long-term photometry +from: ASAS-SN 𝑔 (purple downward triangles), ASAS-SN 𝑉 (green upward +triangles), Catalina 𝑉 (pink squares). Temporal coverage of TESS: Sector +36 (light blue), Sector 37 (yellow). (b) Residual (mean-subtracted) SAP flux +from TESS data. (c) Associated LS periodogram of (b), with indications of +𝑃prec, 𝑃orb, 𝑃nSH signals from Table 1 (blue dashes). +& Kato (2013) derived that +𝜈prec +𝜈orb += −3 +4 +2.5 + 𝑛 +4 + 𝑛 +𝑞 +√︁ +1 + 𝑞 +� 𝑅𝑑 +𝑎 +�1.5 +cos 𝜃, +(5) +where 𝜈 = 𝜔/2𝜋 and 𝑅𝑑 is the disc radius.10 Using Equations (1), (2) +and a mass profile of a steady-state disc given by 𝑛 = −0.75 (Shakura +& Sunyaev 1973), Equation (5) can be reduced to +𝜀− +1 + 𝜀− += −21 +52 +𝑞 +√︁ +1 + 𝑞 +� 𝑅𝑑 +𝑎 +�1.5 +cos 𝜃. +(6) +A similar derivation can be found in Montgomery (2009b). This +shows that 𝜀− depends on three parameters: the mass ratio 𝑞, the disc +tilt 𝜃 and the fractional disc radius 𝑅𝑑/𝑎. The third can be reasoned +to be a function of 𝑞 as follows. Suppose that in our systems with +discovered nSHs, accretion discs are in steady state most of the time. +Then, 𝑅𝑑 approaches the tidal truncation radius 𝑟tidal. Paczynski +(1977, Table 1) provided a functional dependence 𝑟tidal(𝑞). Later, +Warner (1995) proposed the approximation +𝑟tidal = 0.60𝑎 +1 + 𝑞 +(7) +for 0.03 < 𝑞 < 1. This is a good approximation in all regions but near +𝑞 = 0.7, where 𝑟tidal(𝑞) is underestimated. Using it would reduce +Equation (6) to +𝜀− +1 + 𝜀− += − 0.188𝑞 +(1 + 𝑞)2 cos 𝜃, +(8) +which does not describe well observational data for 𝑞 > 0.4 (see dot- +ted line in Figure 10(b)). Because of this, we do not use Equation (8). +Instead, we linearly interpolate between data given in Paczynski +(1977, Table 1) in order to evaluate 𝑅𝑑/𝑎 in Equation (6). +The other independent variable in Equation (6) is the disc tilt 𝜃. +Smak (2009) predicts that disc tilts should not exceed 𝜃max = 7◦ for +CVs. In their photometric analysis of KIC 9406652, Kimura et al. +(2020b) concluded that 𝜃 varies between 0–3◦ over the course of +1500 days. Such range of 𝜃 allows the assumption cos 𝜃 ≃ 1, which +is accurate to within one per cent. This motivates us to compute a +𝑞(𝜀−) curve for cos 𝜃 = 1 and compare it against measurements of +Table 2 objects (Figure 10(b); Table A2). The 𝑞(𝜀−) relation becomes +two-fold degenerate in 𝑞 from about |𝜀−| > 0.048. +10 We note that precession is retrograde, which implies 𝜔prec, 𝜈prec < 0. +MNRAS 000, 1–12 (2022) + +Tilted discs in six poorly studied CVs +9 +0.03 +0.04 +0.05 +0.06 +|ε +− +| +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +q +(a) +0.03 +0.04 +0.05 +0.06 +|ε +− +| +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +q +0.9r +tidal +1.1r +tidal +(b) +Figure 10. Different 𝑞(𝜀−) relations against NL variables with 𝜀−, 𝑞 measurements from Table 2. (a) Solid line: Montgomery (2009a), Dashed line: Wood +et al. (2009). Both relations underestimate 𝑞(𝜀−) for given measurements. (b) Dotted line: Equation (8), derived from the approximation by Warner (1995) used +in Equation (6). This fails to accurately describe Paczynski (1977) in 𝑞 regimes near 0.7. Blue curve: a computed 𝑞(𝜀−) relation from our treatment in Section +4.1 that makes use of Paczynski (1977). Shaded region: solutions between 𝑅𝑑 =0.9–1.1𝑟tidal. All measurements belong to this region within uncertainty. +Table 2. A list of NLs with independently measured superhump deficit 𝜀− and mass ratio 𝑞. Equatorial coordinates come from Gaia DR3 and are in the J2000 +epoch. Other references: (𝑎) Gies et al. (2013); (𝑏) Africano et al. (1978); (𝑐) Smak (2019); (𝑑) Subebekova et al. (2020); (𝑒) Skillman et al. (1995); ( 𝑓 ) Bruch +(2022); (𝑔) Rodríguez-Gil et al. (2020); (ℎ) Kozhevnikov (2007); (𝑖) Araujo-Betancor et al. (2003); ( 𝑗) Boyd et al. (2017); (𝑘) Wu et al. (2002); (𝑙) Huber et al. +(1998); (𝑚) Taylor et al. (1998); (𝑛) Hoard & Szkody (1997); (𝑜) Patterson (1999); (𝑝) Arenas et al. (2000); (𝑞) Peters & Thorstensen (2006); (𝑟) Patterson +et al. (1997); (𝑠) Neustroev et al. (2011); (𝑡) de Miguel et al. (2016); (𝑢) Szkody & Howell (1993); (𝑣) Bruch (2023); (𝑤) Gülsecen et al. (2009); (𝑥) Hellier +(1993); (𝑦) Bruch (2022). +Name +RA +Dec +𝑞 +𝑃orb +𝑃nSH +|𝜀− | +KIC 9406652 +19h31m29.s15 ++45◦59′06.′′1 +𝑎0.83 ± 0.07 +𝑎0.25450(2) +𝑎0.23971(2) +0.0581(1) +RW Tri +02h25m36.s16 ++28◦05′50.′′9 +𝑑0.60 ± 0.03 +𝑏0.23188324(4) +𝑐0.2203(14) +0.050(6) +MV Lyr +19h07m16.s29 ++44◦01′07.′′9 +𝑒0.43+0.19 +−0.13 +𝑒0.1329(4) +𝑓 0.12816(1) +0.036(3) +KR Aur +06h15m43.s92 ++28◦35′08.′′6 +𝑔0.39+0.03 +−0.04 +𝑔0.16277164(5) +ℎ0.1571(2) +0.035(1) +DW UMa +10h33m52.s88 ++58◦46′54.′′7 +𝑖0.39 ± 0.12 +𝑖0.136606499(3) +𝑗0.132626(9) +0.02914(7) +TT Ari +02h06m53.s08 ++15◦17′41.′′9 +𝑘0.19 ± 0.04 +𝑘0.1375504(17) +𝑓 0.132921(2) +0.03366(2) +V592 Cas +00h20m52.s22 ++55◦42′16.′′2 +𝑙0.19+0.10 +−0.09 +𝑚0.115063(1) +𝑚0.11193(5) +0.0272(4) +BH Lyn +08h22m36.s05 ++51◦05′24.′′6 +𝑛0.45+0.15 +−0.10 +𝑛0.15587520(5) +𝑜0.1490(11) +0.044(7) +V603 Aql +18h48m54.s64 ++00◦35′02.′′9 +𝑝0.24 ± 0.05 +𝑞0.13820103(8) +𝑟0.1341(3) +0.030(2) +UX UMa +13h36m40.s95 ++51◦54′49.′′4 +𝑠0.43 ± 0.07 +𝑡0.19667118(19) +𝑡0.186700(11) +0.05070(6) +AY Psc +01h36m55.s46 ++07◦16′29.′′3 +𝑢0.45 ± 0.2 +𝑣0.217320654(4) +𝑤0.20645(75) +0.050(3) +TV Col +05h29m25.s53 +−32◦49′03.′′9 +𝑥0.75 ± 0.15 +𝑦0.22860010(2) +𝑦0.215995(1) +0.055140(4) +There are several systems that lie far from our computed curve. +However, they would all agree with a 𝑞(𝜀−) relation where 𝑅𝑑 is +between 0.9–1.1𝑟tidal (also in Figure 10(b)). It becomes apparent +that the curve is much more sensitive to changes in 𝑅𝑑 than to +changes in 𝜃. On this basis, we compute three curves for different +𝑅𝑑 = 0.9𝑟tidal, 1.0𝑟tidal, 1.1𝑟tidal and then we graphically solve 𝑞 for +our measured 𝜀−. Our mass-ratio estimates are listed in Table 3. +Systems with smaller |𝜀−| tend to be better constrained in 𝑞. For +example, PK-103959 has the lowest |𝜀−| = 0.0308(22) in our sam- +ple, and its mass ratio shows little variation for different values of 𝑅d. +Conversely, SDSS-090113 has the largest measured |𝜀−| in our sam- +ple. For different 𝑅d, its 𝑞 estimates differ significantly with respect +to statistical uncertainty. +Table 3. Estimates of the mass ratio 𝑞 of Table 1 systems for disc radii +𝑅𝑑 = 0.9𝑟tidal, 1.0𝑟tidal, 1.1𝑟tidal from Paczynski (1977), using the methods +described in Section 4.1. Upper bounds of 𝑞 entering the region of two-fold +degeneracy are labeled with an asterisk. +Name +𝑞(𝜀−) +0.9𝑟tidal +1.0𝑟tidal +1.1𝑟tidal +HBHA 4204-09 +0.38+0.05 +−0.05 +0.28+0.03 +−0.03 +0.22+0.02 +−0.02 +Gaia-468436 +0.60+0.09∗ +−0.13 +0.43+0.10 +−0.08 +0.33+0.07 +−0.06 +KQ Mon +0.59+0.11∗ +−0.11 +0.42+0.09 +−0.07 +0.32+0.06 +−0.05 +SDSS-090113 +0.64+0.04∗ +−0.04 +0.46+0.03 +−0.03 +0.35+0.02 +−0.02 +Gaia-593107 +0.52+0.11 +−0.10 +0.38+0.08 +−0.07 +0.29+0.06 +−0.05 +PK-103959 +0.33+0.04 +−0.04 +0.25+0.03 +−0.03 +0.20+0.02 +−0.02 +MNRAS 000, 1–12 (2022) + +10 +Stefanov & Stefanov +4.2 Orbital inclination estimates +The eclipse width at half depth Δ𝜑orb is a reasonable measure of the +primary-eclipse duration in the assumption of an axisymmetric disc +(Warner 1995, Section 2.6.2). This duration can be used to determine +the orbital inclination 𝑖 by the relationship +sin2 𝑖 ≈ 1 − 𝑅𝐿(2)2 +cos2(2𝜋𝜑𝑝) , +(9) +where 𝑅𝐿(2) is the radius of the secondary star, and ±𝜑𝑝 are the +phases of mid-immersion and mid-emergence of the primary star +(Horne 1985; Warner 1995).11 The Roche-lobe radius approximation +by Eggleton (1983) +𝑅𝐿(2) = +0.49𝑞2/3 +0.6𝑞2/3 + ln�1 + 𝑞1/3� +(10) +can be used to substitute 𝑅𝐿(2) in Equation (9), such that the relation +can retrieve 𝑖 using only Δ𝜑orb and the superhump-derived 𝑞. We use +this relation to constrain 𝑖 for HBHA 4204-09, SDSS-090113 and +Gaia-593107, which are eclipsing systems. We measured Δ𝜑orb on +nSH-subtracted data. +HBHA 4204-09 has a deep eclipse, with Δ𝜑orb = 0.049(4). Us- +ing our value of 𝑞 ≈ 0.29(3), we compute 𝑖 = 77(1)◦. On the other +hand, SDSS-090113 shows grazing eclipses of variable depth and +width that disappear completely for some 𝜑prec. For this reason, +we can assume Δ𝜑orb ≈ 0. With our value of 𝑞 ≈ 0.47(4), we +compute 𝑖 = 71.◦6(4). For the last eclipsing system in our sample, +Gaia-593107, we measure Δ𝜑orb = 0.06(2). This eclipse width, +combined with 𝑞 ≈ 0.38(8), gives an orbital inclination 𝑖 = 76(3)◦. +5 CONCLUSIONS +In this work we present results from a search for nSHs in poorly +studied CVs. We initially cross-matched TESS light-curve data with +the VSX catalogue for objects labelled as CV or NL. We manually +inspected LS periodograms of objects from this query (𝑛 = 180), +and then selected targets with at least two neighbouring periodicities +above the period gap, including one large-period signal that matches +the beating of the former two. This resulted in six systems with nSHs, +which we list in Table 1. +Spectroscopic measurements of 𝑃orb were available for only one +of the six aforementioned systems. For the rest, we used a couple +of methods to recognise 𝑃orb signatures in their LS periodograms. +Three systems had their orbital period determined by the pres- +ence of eclipses. For the last two, 𝑃orb was identified using the +𝜑prec-dependent irradiation of the secondary caused by the precess- +ing tilted disc (see Section 2.2). For all systems, the light maximum in +the orbital phase profile was found to shift to earlier 𝜑orb, as 𝜑prec ad- +vances. This is strong evidence of nSHs (Kimura et al. 2020b; Kimura +& Osaki 2021) and supports our findings. For SDSS-090113, a pe- +culiar behaviour was observed in ZTF photometry (Figure 7), which +is similar to what is observed in IW And stars (Kato 2019). For +Gaia-46843, two Z Cam-like episodes in ASAS-SN data were found, +which take place after drops in brightness of about 0.5 mag (Figure +4). +Determined periodicities from TESS photometry can constrain +some physical parameters of CV systems. The dependence between +the nSH deficit 𝜀− and the system mass ratio 𝑞 has been explored in +several works already (e.g. Wood et al. 2009; Montgomery 2009a). +11 From here, 𝜑𝑝 ≡ Δ𝜑orb/2. +Referenced 𝑞(𝜀−) relations, however, underestimate independent +measurements of 𝑞 and 𝜀−, which gives some ground for concern +(Figure 10(a)). We tried to come to a better 𝑞(𝜀−) relation by us- +ing the precession rate of a differentially rotating steady-state disc +that extends to the maximum tidal truncation radius 𝑟tidal. This, in +combination with the 𝑟tidal(𝑞) relation given in Paczynski (1977, +Table 1), resulted in better agreement with independent observations +(Figure 10(b)). Mass-ratio estimates using this method are given in +Table 3 for different disk radii 𝑅𝑑 = 0.9𝑟tidal, 1.0𝑟tidal, 1.1𝑟tidal. For +the three systems that are eclipsing, we used the eclipse-mapping +techniques described by Horne (1985); Warner (1995) to compute +the orbital inclination 𝑖 with our values of 𝑞. +There are several subtle points that bear discussion. SAP light +curves from TESS-SPOC may contain instrumental effects that could +affect LS periodogram measurements; and photometry itself may +be contaminated by nearby sources. To address the former, we re- +peated our methods on PDCSAP data, and found small differences in +comparison to Table 1 measurements. Regarding the latter, we used +mean-subtracted fluxes in all analysis, which mitigates effects by +non-variable contaminating objects. None of our systems have bright +sources in their vicinity, except the case of Gaia-593107, which is +discussed in Section 3.5. +Our variant of the nSH subtraction method in Kimura & Osaki +(2021) shares the same shortcomings. If the nSH profile is time- +dependent, it cannot be fully subtracted. In addition, the mass-transfer +stream could happen to obstruct some parts of the disc, causing +the light maximum to occur at earlier orbital phases 𝜑orb (Kimura +et al. 2020b). Inhomogeneities in the stellar surface brightness of the +secondary can produce similar effects, shifting the light maximum +to earlier or to later 𝜑orb. +The technique of using irradiation of the secondary in order to +determine 𝑃orb is entirely based on photometry, and spectroscopic +measurements of 𝑃orb could support its feasibility. In addition, radial- +velocity analysis would put constraints on mass ratios, and would test +the 𝑞(𝜀−) relation we consider in Section 4.1. The newly discovered +systems in this work are therefore strongly encouraged for follow-up +spectroscopic observations. +ACKNOWLEDGEMENTS +This work includes data collected by the TESS mission and made use +of lightkurve, a Python package for Kepler and TESS data analysis +(Lightkurve Collaboration et al. 2018). Funding for the TESS mission +is provided by the NASA’s Science Mission Directorate. This work +has made use of the NASA/IPAC Infrared Science Archive, which is +funded by the National Aeronautics and Space Administration and +operated by the California Institute of Technology. +The CSS survey is funded by the National Aeronautics and Space +Administration under Grant No. NNG05GF22G issued through the +Science Mission Directorate Near-Earth Objects Observations Pro- +gram. The CRTS survey is supported by the U.S. National Science +Foundation under grants AST-0909182 and AST-1313422. +We used the following Python packages for data analysis and visu- +alisation: NumPy (Harris et al. 2020), SciPy (Virtanen et al. 2020), +pandas (McKinney 2010; The Pandas Development Team 2022), +Matplotlib (Hunter 2007) and uncertainties (Lebigot 2010). +We are grateful to R. K. Zamanov and to A. A. Kurtenkov for their +advice during the preparation of this work. We thank the anonymous +referee for their time and their effort. We acknowledge the grants +KΠ-06-H28/2 and KΠ-06-M58/2 from the Bulgarian National Sci- +ence Fund. Both authors contributed equally to this work. It is appre- +MNRAS 000, 1–12 (2022) + +Tilted discs in six poorly studied CVs +11 +ciated that our last names considerably simplified the issue of author +ordering. +DATA AVAILABILITY +This work contains publicly available data from the sky surveys +TESS, ASAS-SN, CSS, CRTS, and ZTF, all of which can be found +in their corresponding databases. +REFERENCES +Africano J. L., Nather R. E., Patterson J., Robinson E. L., Warner B., 1978, +PASP, 90, 568 +Araujo-Betancor S., et al., 2003, ApJ, 583, 437 +Arenas J., Catalán M. S., Augusteijn T., Retter A., 2000, MNRAS, 311, 135 +Bajer M., 2019, VSX Discovery of BMAM-V424, https://www.aavso. +org/vsx/index.php?view=detail.top&oid=1260092 +Bellm E. C., et al., 2019, PASP, 131, 018002 +Bond H. E., 1979, in van Horn H. 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M., Simpson J. C., 2009, MNRAS, 398, 2110 +Wu X., Li Z., Ding Y., Zhang Z., Li Z., 2002, ApJ, 569, 418 +Zacharias N., Monet D. G., Levine S. E., Urban S. E., Gaume R., Wycoff +G. L., 2004, in American Astronomical Society Meeting Abstracts. p. +48.15 +de Miguel E., et al., 2016, MNRAS, 457, 1447 +APPENDIX A: EXTRA MATERIAL +This appendix contains orbital phase curves (Figure A1) and a list +of 𝑃orb measurements (Table A3) of discovered eclipsing binaries +with no nSH behaviour. Our calculated 𝑞(𝜀−) curves using the tidal +truncation disc radii relation 𝑟tidal(𝑞) in Paczynski (1977) are shown +in Table A2. Computed values can be approximated by +|𝜀−| = +𝑎𝑞 +(1 + 𝑞)𝑏 , +(A1) +MNRAS 000, 1–12 (2022) + +12 +Stefanov & Stefanov +Table A1. Equation (A1) fit coefficients 𝑎 and 𝑏 that approximate Table A2 +data for different values of 𝑅𝑑. +𝑅𝑑 +𝑎 +𝑏 +|𝜀− | interval +0.9𝑟tidal +0.155 +1.706 +[0.006, 0.041] +1.0𝑟tidal +0.182 +1.694 +[0.006, 0.048] +1.1𝑟tidal +0.210 +1.680 +[0.006, 0.057] +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +Orbital phas +−5 +0 +R s. flux (e +− +s +−1 +) +(a) TYC 8920-22-1 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +Orbital phase +−50 +0 +Res. flux (e +− +s +−1 +) +(b) Gaia DR3 5755874037751559424 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +Orbi(al phas +−25 +0 +R s. fl)x ( +− +s +−1 +) +(c) A +TO J213.4592-29.8897 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +Orbital phase +−5 +0 +5 +Res. flux (e +− +s +−1 +) +(d) 3XLSS J231521.7-541842 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +Orbital phase +−10 +0 +Res. flux (e +− +s +−1 +) +(e) Gaia DR3 5499736649371768192 +Figure A1. Orbital phase curves of Table A3 systems. Light curves were +smoothed by a fourth-order Savitzky-Golay filter of window size 10 d before +folding. Orbital phases are offset with +0.50 for the sake of clarity. +where 𝑎 and 𝑏 are fit coefficients. Values of best fits are given in +Table A1 for 𝑅𝑑 = 0.9𝑟tidal, 1.0𝑟tidal, 1.1𝑟tidal. +This paper has been typeset from a TEX/LATEX file prepared by the author. +Table A2. Tabular form of our derived 𝑞(𝜀−) relation for 𝑅𝑑 = 0.9−1.1𝑟tidal +using 𝑟tidal from Paczynski (1977, Table 1). +|𝜀− | +𝑞(𝜀−) +0.9𝑟tidal +1.0𝑟tidal +1.1𝑟tidal +0.006 +0.039 +0.032 +0.031 +0.007 +0.046 +0.038 +0.033 +0.008 +0.053 +0.044 +0.038 +0.009 +0.061 +0.05 +0.043 +0.010 +0.069 +0.057 +0.048 +0.011 +0.077 +0.064 +0.054 +0.012 +0.086 +0.071 +0.060 +0.013 +0.095 +0.078 +0.066 +0.014 +0.104 +0.085 +0.071 +0.015 +0.113 +0.093 +0.078 +0.016 +0.124 +0.101 +0.084 +0.017 +0.135 +0.108 +0.091 +0.018 +0.145 +0.117 +0.097 +0.019 +0.156 +0.126 +0.104 +0.020 +0.167 +0.135 +0.110 +0.021 +0.178 +0.144 +0.118 +0.022 +0.191 +0.153 +0.126 +0.023 +0.205 +0.162 +0.134 +0.024 +0.218 +0.171 +0.142 +0.025 +0.232 +0.181 +0.149 +0.026 +0.246 +0.193 +0.157 +0.027 +0.262 +0.204 +0.165 +0.028 +0.279 +0.216 +0.173 +0.029 +0.297 +0.227 +0.182 +0.030 +0.314 +0.239 +0.192 +0.031 +0.332 +0.251 +0.201 +0.032 +0.352 +0.265 +0.211 +0.033 +0.373 +0.280 +0.221 +0.034 +0.393 +0.295 +0.231 +0.035 +0.414 +0.310 +0.241 +0.036 +0.436 +0.324 +0.251 +0.037 +0.461 +0.340 +0.264 +0.038 +0.487 +0.358 +0.277 +0.039 +0.513 +0.375 +0.289 +0.040 +0.539 +0.392 +0.302 +0.041 +0.566 +0.410 +0.315 +0.042 +0.427 +0.328 +0.043 +0.448 +0.341 +0.044 +0.470 +0.356 +0.045 +0.492 +0.371 +0.046 +0.514 +0.386 +0.047 +0.536 +0.401 +0.048 +0.558 +0.415 +0.049 +0.431 +0.050 +0.449 +0.051 +0.468 +0.052 +0.487 +0.053 +0.505 +0.054 +0.524 +0.055 +0.543 +0.056 +0.562 +0.057 +0.581 +MNRAS 000, 1–12 (2022) + +Tilted discs in six poorly studied CVs +13 +Table A3. List of discovered eclipsing variables with no superhump signatures. Equatorial coordinates come from Gaia DR3 and are in the J2000 epoch. +Name +RA +Dec +TESS Sector +Porb +TYC 8920-22-1 +06h56m39.s36 +−67◦02′16.′′8 +39 +0.d09081(18) +Gaia DR3 5755874037751559424 +09h02m57.s71 +−07◦59′19.′′9 +35 +0.d15585(53) +ATO J213.4592-29.8897 +14h13m50.s22 +−29◦53′23.′′1 +38 +0.d13879(43) +3XLSS J231521.7-541842 +23h15m21.s72 +−54◦18′43.′′1 +28 +0.d14966(54) +Gaia DR3 5499736649371768192 +06h27m51.s08 +−53◦45′17.′′7 +39 +0.d15838(53) +MNRAS 000, 1–12 (2022) + diff --git a/q9FAT4oBgHgl3EQffR3S/content/tmp_files/load_file.txt b/q9FAT4oBgHgl3EQffR3S/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0198c37e64f7a5bc6c58d9cb15446f7790432424 --- /dev/null +++ b/q9FAT4oBgHgl3EQffR3S/content/tmp_files/load_file.txt @@ -0,0 +1,1647 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf,len=1646 +page_content='MNRAS 000, 1–12 (2022) Preprint 23 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 Tilted discs in six poorly studied cataclysmic variables Stefan Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Stefanov,1,2★ Atanas K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Stefanov3 1Institute of Astronomy and National Astronomical Observatory, Bulgarian Academy of Sciences, 72 Tsarigradsko Shose Boulevard, 1784 Sofia, Bulgaria 2Department of Astronomy, Sofia University "St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Kliment Ohridski", 5 James Bourchier Boulevard, 1164 Sofia, Bulgaria 3Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' in original form ZZZ ABSTRACT In this work, we search for negative superhumps (nSHs) in poorly studied cataclysmic variables using TESS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' We find three eclipsing binaries with nSH signatures: HBHA 4204-09, Gaia DR3 5931071148325476992, and SDSS J090113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='51+144704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' The last one exhibits IW And-like behaviour in archival ZTF data, and appears to have shallow, grazing eclipses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' In addition, we detect nSH signatures in two non-eclipsing systems: KQ Mon and Gaia DR3 4684361817175293440, by identifying the orbital period from the superorbital-dependent irradiation of the secondary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' We discover nSH signatures in one more system, [PK2008] HalphaJ103959, by using an orbital period from another work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' An improved mass ratio – nSH deficit relation 𝑞(𝜀−) is suggested by us, which agrees with independent measurements on nova-like variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' With this relation, we estimate the mass ratios of all systems in our sample, and determine the orbital inclinations for the three that are eclipsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' All systems with discovered nSHs in this work are excellent targets for follow-up spectroscopic studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Key words: stars: activity – binaries: close – novae, cataclysmic variables – stars: individual: HBHA 4204-09, Gaia DR3 4684361817175293440, KQ Mon, SDSS J090113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='51+144704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='6, Gaia DR3 5931071148325476992, [PK2008] HalphaJ103959 1 INTRODUCTION Cataclysmic variables (CVs) are binary systems that consist of a white-dwarf (WD) primary and a Roche-lobe filling secondary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Mat- ter from the secondary flows through the first Lagrangian point and accretes on to the primary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' In the case of a non-magnetic or a very weakly magnetic primary, this mass transfer happens through an accretion disc (Hellier 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' In systems with mass-transfer rates of �M ≃ 1 – 5 ×10−9 M⊙yr−1, thermal instabilities arise in the accretion disc and cause repeating quasi-periodic outbursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' These outbursts usually occur once about every few months, last several days, and can increase the system brightness with up to ∼ 5 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' CVs with recorded outbursts are termed dwarf novae (DNe), whereas CVs with no recorded outbursts are termed nova-likes (NLs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' In NLs, most of the flux originates from the accretion disc, which is in a hot steady state and is much brighter than the two system components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' The orbital periods of this type of variables can range from ∼ 1 h to more than 10 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Not many CVs, however, are observed in the period range of 2–3 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' This phenomenon is called the ”period gap” and is explained by transitions in evolutionary stages of this type of variables (see Warner 1995 for an encyclopedic description of CVs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' NLs can change their brightness on time-scales from seconds to millennia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Some systems have drops in brightness of several magni- tudes, which can last from months to years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' This behaviour is most commonly observed in systems with orbital periods (𝑃orb) near the upper edge of the period gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Such drops in brightness are cate- ★ E-mail: sstefanov@nao-rozhen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='org gorised as a low state of type VY Scl (King & Cannizzo 1998) and can also be displayed by magnetic CVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' VY Scl low states are likely caused by the reduction or the complete cessation of mass transfer in the system, which significantly decreases the flux coming from the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' They are believed to be associated with the magnetic activity of the secondary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Star spots emerging on the first Lagrangian point may suppress mass transfer in the system (Livio & Pringle 1994), and the radius of the secondary itself can be affected by magnetic activity (Howell 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Yet, the exact mechanism of mass-transfer cessation during VY Scl episodes remains unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Apart from low states and other long-term trends in brightness, CVs display an abundance of photometric variability on shorter time- scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Roche-lobe geometry requires that the secondary takes a char- acteristic teardrop-like shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' As it orbits the barycentre, it presents different projections of itself to the observer, which introduces a photometric variability of period 𝑃orb/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' A similar effect can oc- cur when the secondary is strongly irradiated by the accretion disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' In that case, the visibility of the irradiated side of the secondary is dependent on the orbital phase of the system, and a light-curve modulation of period 𝑃orb takes place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Some CVs exhibit variations in brightness that have periods slightly shorter or slightly longer than 𝑃orb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' These variations are called ”superhumps” and are believed to be caused by a precessing accretion disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Superhumps can be of either positive (pSH) or neg- ative (nSH) type, depending on the sign of 𝑃SH − 𝑃orb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' They are well-studied and commonly seen in SU UMa stars, a DN subclass (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Kato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' 2009, 2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' as well as in NLs (Bruch 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' For NLs in particular, Bruch gave a sample of 46 systems, 13 have had © 2022 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='08581v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='SR] 20 Jan 2023 2 Stefanov & Stefanov pSHs, 16 have had nSHs and 17 have had superhumps of both types at some point in the past (but not necessarily at the same time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Each superhump type is associated with processes of different nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' pSHs are believed to be caused by an apsidally precessing accretion disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' In this case, the 3:1 resonance induces tidal deforma- tions, the heat from which causing periodic changes in disc brightness (Whitehurst 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Hirose & Osaki 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Lubow 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' On the other hand, nSHs can be explained with a retrograde nodal precession of a tilted accretion disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' The tilt allows for the mass-transfer stream to go deeper in the gravitational well of the primary, and thus to release more energy upon impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' The point of impact on the disc is commonly referred to as the ”bright spot”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' The sweeping of the bright spot across the disc faces introduces an additional photometric variability that has a period equal to the beating of 𝑃orb and the disc precession period 𝑃prec (Wood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Montgomery 2009a), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' 1 𝑃nSH = 1 𝑃orb + 1 𝑃prec .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (1) Superhumps of both types can be used to estimate some physical properties of these systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' The nSH deficit 𝜀− is defined as 𝜀− = 𝑃nSH − 𝑃orb 𝑃orb (2) and has been shown to correlate with the mass ratio of the system 𝑞 = 𝑀1/𝑀2 in several works (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Wood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Montgomery 2009a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' A detailed study of nSHs can be found in Kimura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (2020b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Kimura & Osaki (2021), where Kepler photometry of the NL system KIC 9406652 was analysed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' The light curve of this partic- ular object has identifiable 𝑃orb, 𝑃nSH signals as well as superorbital ones (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' 𝑃prec signatures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' In this work, we present our results from a search for nSHs in poorly studied CVs that are similar to KIC 9406652.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Section 2 presents our methods for searching and data reduction, and gives a list of objects with discovered nSH signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Section 3 contains a literature review and discussion of each system we found to have nSH behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' In Section 4, we attempt to estimate some physical parameters in said systems, and in Section 5, we summarise the findings of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' 2 ANALYSIS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='1 Data from TESS The Transiting Exoplanet Survey Satellite (TESS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Ricker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' 2015) mission is an all-sky survey in the red-infrared that continues to provide with long-term measurements of remarkable photometric precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' The TESS Science Processing Operations Center pipeline (SPOC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Jenkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' 2016) offers light curves from two different reduction techniques: Simple Aperture Photometry (SAP) and Pre- Search Data Conditioning Simple Aperture Photometry (PDCSAP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' A comprehensive comparison between the two is given in Kine- muchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' PDCSAP tries to reduce effects of instrumental nature, but can sometimes introduce systematics in periodograms, and analysis should proceed with care.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Bruch (2022) found in par- ticular that the additional conditioning in PDCSAP may distort DNe light curves, and chose to use the simpler SAP technique in order to search for periodic variations in CVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' We use SAP light curves too in all analysis to follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='2 Photometric features of tilted accretion discs Negative superhumps are direct evidence for a titled accretion disc, but finding their signatures is only possible in systems of known Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' A model CV system at an orbital phase 𝜑orb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5, as it would be seen by an observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Four precession phases 𝜑prec of a disc with tilt 𝜃 = 6◦ are illustrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' The orbital plane of the system is defined by dotted lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' It divides space into two half-spaces: the near (above the plane) and the far one (below the plane), with respect to the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' The precession phase is defined such that the system is the brightest at 𝜑prec = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' In this orientation, the disc has the largest projected area at 𝜑prec = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Conversely, at 𝜑prec = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5, it has the smallest projection, but faces towards the secondary and irradiates it the most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Kimura & Osaki (2021), Figure 9 gives a full description of CV configurations in titled-disc regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' 𝑃orb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' This is a strong restriction, since not many CVs have had their orbital periods measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' To expand the population of stars with known 𝑃orb, we searched for systems with several significant peaks in the power spectrum, in a frequency region above the period gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' In the case of two neighbouring prominent peaks, it could be that those are signatures of 𝑃pSH, 𝑃orb and not 𝑃nSH, 𝑃orb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Nevertheless, this degeneracy can be lifted with the following rationale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' In systems with a precessing tilted accretion disc, the disc orienta- tion changes with respect to the secondary for different orbital phases 𝜑orb and different disc precession phases 𝜑prec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' The former is defined such that the secondary is at inferior conjunction at 𝜑orb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' the latter is defined1 such that the light maximum in the disc precession cycle is at 𝜑prec = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' The observed irradiation of the secondary by the bright disc varies with both 𝜑prec and 𝜑orb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Consider a non- eclipsing system at 𝜑orb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' The orbital plane of the system divides space into two half-spaces, one of which the observer finds themselves in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' One part of the system resides in the same half- space as the observer, and the other part is in the opposite half-space (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' on the other side of the orbital plane with respect to the ob- server).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' We shall refer to those as the ”near half-space” and the ”far half-space”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' As an example, in Figure 1, the part of the accretion disc that lies in the near half-space is: its rear side at 𝜑prec = 0, its right side at 𝜑prec = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='25 and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' For an observer, the near half-space of a system is more photomet- rically accessible than the far half-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='2 At 𝜑prec = 0, the luminous disc reveals the most of itself to the observer, and the average sys- tem brightness across 𝜑orb is the greatest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' However, the irradiated region of the secondary is in the far half-space, and thus the 𝜑orb variation in brightness is minimal in amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' In the opposite case of 𝜑prec = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5, the observer sees the smallest possible projection of the disc, and the average system brightness across 𝜑orb is the smallest – but the irradiated region of the secondary is now in the near half-space, and the 𝜑orb variation in brightness is maximal in amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' 1 These definitions are consistent with Kimura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' 2 This is only untrue in the special case of 𝑖 = 90◦, when the observer lies in the orbital plane, and both half-spaces are thus equally accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' MNRAS 000, 1–12 (2022) Ppree Ppree Ppree PpreeTilted discs in six poorly studied CVs 3 At the same time, nSHs introduce additional complexities in vari- ability that need to be accounted for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Kimura & Osaki (2021) discuss this issue and carry out the following procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' A given light curve is initially split into subsets of different time intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Then, for each subset, they: (1) fold by 𝑃nSH and construct an average light-curve profile of the nSH, (2) subtract said profile from each subset, (3) split the subset into different 𝜑prec windows, (4) fold each window by 𝑃orb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' This technique results in multiple orbital phase curves, each corresponding to a different 𝜑prec window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' If these phase curves show a 𝜑prec-dependent irradiation of the secondary, the system has a pre- cessing tilted accretion disc and the observed superhump is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' It is this consideration that could lift the pSH-nSH degeneracy in the power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' In order to address the nSH contamination, we use a variant of the nSH-subtraction technique by Kimura & Osaki (2021) with the fol- lowing adjustments: all data is smoothed by a fourth-order Savitzky- Golay filter (Savitzky & Golay 1964) of window size 10 d, and no sep- arate subsets are considered;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' in (1), nSH light-curve profiles are con- structed with median filters of window size 1101;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='3 in (3), four 𝜑prec intervals are considered with centres at 𝜑prec = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='00, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='50, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='75 and of width 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='3 Target selection The International Variable Star Index (VSX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Watson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' 2006, ac- cessed 2022 June) is perhaps the most extensive catalogue of known variable stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' We took all objects from the VSX labelled as CV or as NL (𝑛 = 1249), and then sought all for which there were avail- able TESS SPOC light curves of 120-second cadence (𝑛 = 180).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Lomb-Scargle periodograms (LS periodogram;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Lomb 1976;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Scar- gle 1982) of range between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='125 – 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='000 d−1 were constructed for those systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Periodograms were then manually searched for the si- multaneous presence of at least two neighbouring periodicities in the region above the period gap, as well as for one periodicity near their expected beat period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' This was done to select NLs with signatures of all 𝑃prec, 𝑃orb, 𝑃nSH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' For most stars, long-term photometry from the All-Sky Automated Survey for Supernovae (ASAS-SN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Shappee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Kochanek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' 2017) and from the Catalina Sky Survey (CSS, Drake et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' 2009) was available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' We attempted to construct LS periodograms using photometry from said surveys, but data was found to be sparse and of too long cadence to be usable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' We report on the discovery of nSH behaviour in six poorly studied CVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Three of them are eclipsing systems, which enabled us to directly determine 𝑃orb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' For two other systems, 𝑃orb was identified with the use of 𝜑prec-dependent irradiation of the secondary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' The last CV was found to have 𝑃prec and 𝑃nSH signatures, but not a 𝑃orb one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Our derived value of 𝑃orb by Equation (1), however, agrees well with the spectroscopic measurement of Pretorius & Knigge (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' All six objects are discussed individually in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' During inspection, we also found five new eclipsing CVs with no superhump behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Their measured orbital periods are provided in Table A3, and their orbital phase curves are shown in Figure A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' 3 REVIEW AND RESULTS The following sections provide literature review, discussion and in- terpretation of data for all CVs with discovered nSH behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Each 3 We found that this window size worked generally well for all systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' 4 That is, the intervals 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='95 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='20 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='30, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='45 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='55, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='70 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' CV system has an associated figure containing: (1) available sky- survey data, (2) TESS photometry from sectors with prominent nSH behaviour together with corresponding LS periodograms, (3) orbital phase plots of data in the four 𝜑prec regions discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Measured periodicities of each system are given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' All measurements agree well with Equation (1) within uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='1 HBHA 4204-09 HBHA 4204-095 (Figure 2) is discovered by ASAS-SN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' It was clas- sified as a CV by Jayasinghe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (2018) and by ALeRCE (Förster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' 2021) in data from the Zwicky Transient Facility (ZTF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Bellm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' This object is part of the ”Catalogue of Stars in the Northern Milky Way Having H-alpha in Emission” (Kohoutek & Wehmeyer 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' The Gaia DR3 distance estimate is 478 ± 3 pc and ASAS-SN photometry gives a mean brightness of 𝑚𝑉 = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='19 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' We report the presence of previously unknown V-shaped eclipses in HBHA 4204-09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Using them, we identify the periodogram peaks corresponding to 𝑃orb, 𝑃nSH, 𝑃prec (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Aside from these periodicities, the power spectrum contains a strong signal at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='d070655(58), which matches 𝑃orb/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' A collection of peaks at around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='d155 is observed, which may be indicative of a pSH sig- nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Additional photometry of HBHA 4204-09 can be found in TESS Sectors 55 and 56, but no superhumps are present in those data sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Due to its high orbital inclination, the near and the far half-spaces defined by the orbital plane are comparably accessible to the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' The portion of the secondary in the far half-space is most irradiated at 𝜑prec = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='00, while the portion in the near half-space is most irradiated at 𝜑prec = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' The orbital profiles in panels (d) and (f) of Figure 2 show stronger secondary irradiation at aforementioned 𝜑prec, which is expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='2 Gaia DR3 4684361817175293440 Gaia DR3 4684361817175293440, hereinafter Gaia-4684366 (Fig- ure 3) was discovered and classified as a NL type CV by Bajer (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' The Gaia DR3 distance estimate is 1062+29 −30 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' On the long- term ASAS-SN curve, a 1-mag fall in brightness can be observed around BTJD 800 – BTJD 1700.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' A panel with ASAS-SN photom- etry in this time period is shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' The observed drop in brightness has a smaller amplitude from what is expected in classic VY Scl low states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Quasi-cyclic variations of 𝑃 ∼ 20 d resembling Z Cam outbursts appear after the start of the low state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' The system later returns to normal brightness and outbursts are replaced with a standstill lasting for ∼ 300 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' This standstill is followed by another Z Cam outburst episode, after which no more outbursts of this type are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Our LS periodogram of Gaia-468436 shows three peaks with peri- ods matching Equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' We interpret them as 𝑃orb, 𝑃nSH,𝑃prec in a system with a tilted precessing disc (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' We find two ad- ditional peaks at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='d07372(14) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='d07702(15) that match 𝑃nSH/2 and 𝑃orb/2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' In Figure 3(d)–(g), it can be seen that the light maximum of orbital-phase curves gradually shifts to earlier 𝜑orb as the disc precession cycle advances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' This is direct evidence for a retrogradely precessing tilted disc (Kimura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' 5 The VSX identifier of this source is ASASSN-V J210752.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='24+440542.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' 6 The VSX identifier of this source is BMAM-V424.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' MNRAS 000, 1–12 (2022) 4 Stefanov & Stefanov Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' List of CVs with discovered nSHs using the methods described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' All periodicities in this table were measured on a Lomb-Scargle periodogram of range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='125–16 d−1 and of ten-fold oversampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' All measured 𝑃prec in this table agree within uncertainty with the expected values by Equation (1) using measured 𝑃orb and 𝑃nSH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Equatorial coordinates come from Gaia DR3 and are in the J2000 epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Name RA Dec TESS Sector 𝑃orb 𝑃nSH 𝑃prec |𝜀− | HBHA 4204-09 21h07m52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='s24 +44◦05′42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='′′0 15, 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='d14128(22) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='d13657(22) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='d11(18) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0333(22) Gaia DR3 4684361817175293440 00h49m59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='s93 −76◦08′27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='′′5 28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='d15401(53) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='d14750(52) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='d40(27) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0423(48) KQ Mon 07h31m21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='s13 −10◦21′49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='′′4 34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='d13456(40) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='d12894(38) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='d12(24) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0418(41) SDSS J090113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='51+144704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='6 09h01m13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='s51 +14◦47′04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='′′7 44 – 46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='d14631(17) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='d13991(17) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='d198(70) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0437(16) Gaia DR3 5931071148325476992 16h36m03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='s63 −52◦33′32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='′′6 39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='d14827(46) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='d14248(43) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='d57(30) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0391(42) [PK2008] HalphaJ103959 10h39m59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='s98 −47◦01′26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='′′3 36, 37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='d1577(2)† 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='d15285(29) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='d94(26) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0308(22) †Orbital period measured spectroscopically by Pretorius & Knigge (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' 0 500 1000 1500 2000 2500 BTJD (d) 14 15 Mag itude (mag) (a) 1720 1740 1760 BTJD (d) −200 0 200 Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' flu) (e − s −1 ) (b) 0 5 10 15 Freque cy (d −1 ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' po(er (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 Orbital phase −50 0 50 Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' flu) (e − s −1 ) (d) φ prec = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 Orbital phase −50 0 50 Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' flu) (e − s −1 ) (e) φ prec = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 Orbital phase −50 0 50 Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' flu) (e − s −1 ) (f) φ prec = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 Orbital phase −50 0 50 Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' flu) (e − s −1 ) (g) φ prec = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='75 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Photometry and analysis of HBHA 4204-09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (a) Long-term photometry from: ASAS-SN 𝑔 (purple downward triangles), ASAS-SN 𝑉 (green upward triangles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Temporal coverage of TESS: Sector 15 (light blue), Sector 16 (yellow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (b) Residual (mean-subtracted) SAP flux from TESS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (c) Associated LS periodogram of (b), with indications of 𝑃prec, 𝑃orb, 𝑃nSH signals from Table 1 (blue dashes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (d)–(g) Binned orbital profiles around disc precession phases 𝜑prec = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='00, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='50, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='75 with nSH subtraction (blue squares) and without one (black circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' The typical standard deviation of data in bins is 25 e− s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' This system has a high inclination, and the parts of the secondary in both half-spaces are accessible to the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' The secondary is expected to be the most irradiated in panels (d) and (f), where the observed out-of-eclipse profile is non-flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' In panels (e) and (g), where no irradiation of the secondary is expected, the out-of-eclipse profile is mostly flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='3 KQ Mon KQ Mon (Figure 5) was classified as a NL-type CV by Bond (1979) using low-resolution spectra in the optical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Its orbital period was measured in Schmidtobreick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (2005) to be 𝑃orb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='d1283(17) by analysing two nights of time-resolved spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Later, Wolfe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (2013) examined far-ultraviolet spectra of KQ Mon from the International Ultraviolet Explorer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' The mass of the primary was esti- mated to be 𝑀1 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='6 𝑀⊙ with the use of synthetic spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' The same work argued that the primary contributes little to the total system flux, and is overwhelmed by the flux of a steady-state accretion disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' It was concluded that the system is located at a distance of 144–159 pc, with an inclination of 𝑖 ≤ 60◦ and an accretion rate in the order of �𝑀 ∼ 10−9 𝑀⊙ yr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' The Gaia DR3 distance is 628±8 pc, which disagrees with their estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Our measured value for 𝑃nSH matches the 𝑃orb given in Schmidto- breick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' What we measure as 𝑃orb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='d13456(40) would have corresponded to a pSH signal in their interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' But then, no other signals in the periodogram would have been expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' We, however, measure a strong third signal at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='d12(24), which is self- consistent with the other two by Equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' In addition, we observe a 𝜑prec-dependent amplitude of the orbital phase curve, which could be explained by a varying irradiation of the secondary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' In Figure 3 of Schmidtobreick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (2005), a strong aliasing pattern can be seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' The authors chose an orbital period 𝑃orb among four possible signals, two of which agree with our measurements of 𝑃orb, 𝑃nSH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' With all this in mind, we think that the correct value of 𝑃orb is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='d13456(40), and that there is presence of a tilted accretion disc in this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' MNRAS 000, 1–12 (2022) Tilted discs in six poorly studied CVs 5 0 500 1000 1500 2000 2500 BTJD (d) 14 15 16 Magnit(de ( ag) (a) 2065 2070 2075 2080 2085 BTJD (d) −25 0 25 Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' fl(x (e − s −1 ) (b) 0 5 10 15 Freq(ency (d −1 ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 Nor .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' po)er (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 Orbital phase −10 0 10 Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' fl(x (e − s −1 ) (d) φ prec = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 Orbital phase −10 0 10 Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' fl(x (e − s −1 ) (e) φ prec = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 Orbital phase −10 0 10 Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' fl(x (e − s −1 ) (f) φ prec = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 Orbital phase −10 0 10 Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' fl(x (e − s −1 ) (g) φ prec = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='75 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Photometry and analysis of Gaia-468436.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (a) Long-term photometry from: ASAS-SN 𝑔 (purple downward triangles), ASAS-SN 𝑉 (green upward tri- angles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Temporal coverage of TESS: Sector 28 (light blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (b) Residual (mean-subtracted) SAP flux from TESS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (c) Associated LS periodogram of (b), with indications of 𝑃prec, 𝑃orb, 𝑃nSH signals from Table 1 (blue dashes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (d)–(g) Binned orbital profiles around disc precession phases 𝜑prec = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='00, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='50, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='75 with nSH subtraction (blue squares) and without one (black circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' The typical standard deviation of data in bins is 4 e− s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' The effect of variable irradiation in panels (d)–(g) is similar to the one observed in KIC 9406652 (Kimura & Osaki 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' 1000 1200 1400 1600 1800 BTJD (d) 14 15 16 Magnitude (mag) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Z Cam-like episodes of Gaia-468436 in ASAS-SN 𝑔 (blue downward triangles) and ASAS-SN 𝑉 (green upward triangles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' The Z Cam behaviour begins after a ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 mag fall in brightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Emerging oscillations have a variable amplitude of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='8 mag and are quasi-periodic with 𝑃 ∼ 20 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' At about BTJD 1500, a brightening takes place, which is followed by a standstill at a level of 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' At about BTJD 1760, the standstill is replaced with another oscillatory episode that has outbursts of similar period and amplitude as the former ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' No more Z Cam episodes were observed in Gaia-468436 in this data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='4 SDSS J090113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='51+144704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='6 SDSS J090113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='51+144704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='6, hereinafter SDSS-090113 (Figure 6) first appeared in Szkody et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (2009) where it was classified as a CV due to accretion disc features in its spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' This system was included in the catalogue of bright WDs of Raddi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Gaia DR3 estimated the distance to SDSS-090113 to be 1482+100 −116 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Later, Mösenlechner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (2022) included this system in their time-series analysis study of subdwarf A-type stars using Kepler K2 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' They discovered a periodicity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='d146, which was suggested to be the orbital period 𝑃orb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' SDSS-090113 has no recorded low states and its brightness varies around 𝑚𝑉 = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='2 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Between BTJD 1600 and BTJD 2300, we recognise an episode of anomalous Z Cam-type outbursts repeating once about every 25 days (Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' These outbursts begin after a brightening, which is one of the defining features of the IW And- phenomenon systems (Kato 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' This can be explained by a tilted disc that causes the accretion stream to enter inner disc regions, and thus to disrupt the accretion cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' In this new type of accretion, the inner disc is in a hot state, while the outer disc repeats outbursts (Kimura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' 2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Figure 6(d)–(g) shows what seems to be the presence of graz- ing eclipses in the orbital curve of SDSS-090113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' They vary in depth and width, and for some phases of 𝑃prec they disappear com- pletely, similar to ES Dra (Kato 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Through them, we iden- tify 𝑃orb, 𝑃nSH, 𝑃prec (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Two additional peaks are found at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='d071531(52) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='d073161(40) that match 𝑃nSH/2 and 𝑃orb/2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' MNRAS 000, 1–12 (2022) 6 Stefanov & Stefanov 0 500 1000 1500 2000 2500 BTJD (d) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 Mag itude (mag) (a) 2230 2240 2250 BTJD (d) −100 0 100 Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' flu) (e − s −1 ) (b) 0 5 10 15 Freque cy (d −1 ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' po(er (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 Orbital phase 0 50 Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' flu) (e − s −1 ) (d) φ prec = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 Orbital phase 0 50 Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' flu) (e − s −1 ) (e) φ prec = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 Orbital phase 0 50 Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' flu) (e − s −1 ) (f) φ prec = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 Orbital phase 0 50 Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' flu) (e − s −1 ) (g) φ prec = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='75 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Photometry and analysis of KQ Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (a) Long-term photometry from: ASAS-SN 𝑔 (purple downward triangles), ASAS-SN 𝑉 (green upward triangles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Temporal coverage of TESS: Sector 34 (light blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (b) Residual (mean-subtracted) SAP flux from TESS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (c) Associated LS periodogram of (b), with indications of 𝑃prec, 𝑃orb, 𝑃nSH signals from Table 1 (blue dashes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (d)–(g) Binned orbital profiles around disc precession phases 𝜑prec = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='00, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='50, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='75 with nSH subtraction (blue squares) and without one (black circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' The typical standard deviation of data in bins is 16 e− s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' The blue and the black curves differ due to the significant nSH contribution to the observed system flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' In blue curves of different 𝜑prec, there seems to be a change of shape and amplitude near 𝜑orb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5, which is expected, but could be also due to noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Kimura & Osaki (2021) provide models for such orbital curves that could explain these observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 Gaia DR3 5931071148325476992 Gaia DR3 5931071148325476992, hereinafter Gaia-5931077 (Fig- ure 8) is a poorly studied CV that was discovered in plates by Prestgard (2020) from the Digitized Sky Survey8 and the Super- COSMOS H𝛼 survey (Parker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' The NOMAD catalogue (Zacharias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' 2004) gives an apparent magnitude of 𝑚𝑉 = 16 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' No ASAS-SN photometry is available for this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Its TESS brightness reads 𝑚TESS = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='02 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' There is an X-ray source (1RXS J163605.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='9-523335) at a distance of 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='8 arcsec, which is likely associated with Gaia-593107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' In addition, we find two bright sources of brightness 𝑚TESS = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='44 and 𝑚TESS = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='16 mag in the aperture mask, that are expected to severely contaminate the light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' This issue, however, is resolved by the apparent variability of Gaia-593107 in the discovery images9 of Prestgard, on the basis of which we attribute the tilted-disc behaviour to this specific system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Our analysis of TESS light curves shows Gaia-593107 to be an eclipsing variable with an orbital period of 𝑃orb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='d14248(43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' A peak at 𝑃orb/2 is present as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' This allows us to locate 𝑃orb, 𝑃nSH, 𝑃prec (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' A change in eclipse depth is observed in different phases of the determined 𝜑prec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' 7 The VSX identifier of this source is USNO-A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 0300-28957281.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' 8 ESO Online Digitized Sky Survey: http://archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='eso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='org/dss/dss (accessed 2022 October).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' 9 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='aavso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='org/vsx_docs/1544030/3344/USNO-A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0% 200300-28957281.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='png 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='6 [PK2008] HalphaJ103959 [PK2008] HalphaJ103959, hereinafter PK-103959 (Figure 9) was classified as a CV in Pretorius & Knigge (2008), where spectro- scopic and photometric analyses of the system were carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' The orbital period of PK-103959 was measured to be 𝑃orb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='d1577(2) in the same work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Catalina and ASAS-SN photometry has a mean brightness of 𝑚𝑉 =15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='7 mag, with no low states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' A gradual in- crease in brightness can be seen in the period between -1000 and 2000 BTJD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' We find signatures of 𝑃nSH and 𝑃prec (Table 1), but no peaks at the 𝑃orb by Pretorius & Knigge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' However, there are two other visible peaks at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='d07885(14) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='d07639(13), which correspond to 𝑃orb/2 by Pretorius & Knigge and 𝑃nSH/2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' 4 DISCUSSION 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='1 Mass-ratio estimates By using smoothed particle hydrodynamic (SPH) simulations of tilted accretion discs, Wood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (2009) found that the relation between the mass ratio and the nSH deficit is well-represented by 𝑞(𝜀−) = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='192|𝜀−|0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 +10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='37|𝜀−| −99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='83|𝜀−|1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 +451.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='1|𝜀−|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (3) This result has been supported by other works using related SPH simulations (Montgomery 2009a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Thomas & Wood 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' To com- pare Equation (3) with observations, we searched for NL objects in MNRAS 000, 1–12 (2022) Tilted discs in six poorly studied CVs 7 −3000 −2000 −1000 0 1000 2000 3000 BTJD (d) 15 16 17 Magnit(de ( ag) (a) 2520 2540 2560 BTJD (d) −25 0 25 Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' fl(x (e − s −1 ) (b) 0 5 10 15 Freq(ency (d −1 ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 Nor .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' po)er (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 Orbital phase −5 0 5 Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' fl(x (e − s −1 ) (d) φ prec = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 Orbital phase −5 0 5 Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' fl(x (e − s −1 ) (e) φ prec = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 Orbital phase −5 0 5 Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' fl(x (e − s −1 ) (f) φ prec = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 Orbital phase −5 0 5 Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' fl(x (e − s −1 ) (g) φ prec = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='75 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Photometry and analysis of SDSS-090113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (a) Long-term photometry from: ASAS-SN 𝑔 (purple downward triangles), ASAS-SN 𝑉 (green upward triangles), Catalina 𝑉 (pink squares).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Temporal coverage of TESS: Sector 44 (light blue), Sector 45 (yellow), Sector 46 (light blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (b) Residual (mean-subtracted) SAP flux from TESS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (c) Associated LS periodogram of (b), with indications of 𝑃prec, 𝑃orb, 𝑃nSH signals from Table 1 (blue dashes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Data from (b) was smoothed by a fourth-order Savitzky-Golay filter of window size 10 d before constructing the periodogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' This was done solely for the sake of clear identification of 𝑃prec by the reader, and not for periodicity measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (d)–(g) Binned orbital profiles around disc precession phases 𝜑prec = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='00, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='50, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='75 with nSH subtraction (blue squares) and without one (black circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' The typical standard deviation of data in bins is 4 e− s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' SDSS-090113 appears to have shallow grazing eclipses that are barely detectable for some 𝜑prec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Observed eclipses vary in depth and width for different 𝜑prec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' This could be explained by a secondary that partially covers the tilted disc only when the projected area of the disc is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' 1400 1600 1800 2000 2200 BTJD (d) 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 Magnitude (mag) Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' IW And episodes of SDSS-090113 in ZTF 𝑔 (teal circles) and ZTF 𝑟 (magenta diamonds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Three seasons of photometry are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' The first season starts with oscillations that are terminated by brightening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' This is one of the defining features of the IW And phenomenon (Kato 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Kato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' The second season shows the beginning of a new oscillatory episode that is variable in amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' The episode continues in the third season and abruptly ends at BTJD 2315.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' literature for which superhump deficits and mass ratios were mea- sured independently from one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Our reasoning is that NLs share three main similarities with our discovered CVs: (1) their 𝑃orb are of the same order, (2) they have steady, hot and luminous discs, (3) samples of both populations exhibit VY-Scl behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Using the sample of NLs with nSH signatures, given in Bruch (2023), we were able to find twelve such objects, which we list in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Figure 10(a) compares their measurements with the 𝑞(𝜀−) relations provided by Montgomery (2009a) and Wood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' We share the concern that both works underestimate 𝑞(𝜀−) with respect to past measure- ments in literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' There exists a different approach that could estimate mass ratios using nSHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' By making some assumptions, a 𝑞(𝜀−) relation can be derived in the following manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Through linear perturbation theory, Papaloizou & Terquem (1995) derived the precession rate 𝜔prec of a differentially rotating fluid disc with a mass profile Σ(𝑟): 𝜔prec = −3 4 𝐺𝑀2 𝑎3 ∫ Σ𝑟3d𝑟 ∫ ΣΩ𝑟3d𝑟 cos 𝜃, (4) where 𝑎 is the orbital separation, Ω(𝑟) = √︁ 𝐺𝑀1/𝑟3 is the Keplerian angular velocity profile of the disc, and 𝜃 is the disc tilt with respect to the orbital plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' For a power-law mass profile Σ(𝑟) ∝ 𝑟𝑛, Osaki MNRAS 000, 1–12 (2022) 8 Stefanov & Stefanov 2370 2380 BTJD (d) −50 0 50 Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' flux (e − s −1 ) (a) 0 5 10 15 Frequenc( (d −1 ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' ower (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 Orbital hase −20 0 20 Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' flux (e − s −1 ) (c) φ prec = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 Orbital hase −20 0 20 Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' flux (e − s −1 ) (d) φ prec = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 Orbital hase −20 0 20 Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' flux (e − s −1 ) (e) φ prec = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 Orbital hase −20 0 20 Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' flux (e − s −1 ) (f) φ prec = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='75 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Photometry and analysis of Gaia-593107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (a) Residual (mean-subtracted) SAP flux from TESS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (b) Associated LS periodogram of (a), with indications of 𝑃prec, 𝑃orb, 𝑃nSH signals from Table 1 (blue dashes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (c)–(f) Binned orbital profiles around disc precession phases 𝜑prec = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='00, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='50, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='75 with nSH subtraction (blue squares) and without one (black circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' The typical standard deviation of data in bins is 8 e− s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Similar to the other eclipsing binaries in our sample, the secondary is expected to be the most irradiated in panels (c) and (e), where the observed out-of-eclipse profile is non-flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' In panels (d) and (f), these profiles are mostly flat, and the system brightness does not seem to increase near 𝜑orb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' −3000 −2000 −1000 0 1000 2000 3000 BTJD (d) 15 16 Magnitude (mag) (a) 2290 2300 2310 2320 2330 BTJD (d) 0 50 Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' flu( (e − s −1 ) (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 Frequenc) (d −1 ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0 N rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' p wer (c) Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Photometry and analysis of PK-103959.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (a) Long-term photometry from: ASAS-SN 𝑔 (purple downward triangles), ASAS-SN 𝑉 (green upward triangles), Catalina 𝑉 (pink squares).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Temporal coverage of TESS: Sector 36 (light blue), Sector 37 (yellow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (b) Residual (mean-subtracted) SAP flux from TESS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (c) Associated LS periodogram of (b), with indications of 𝑃prec, 𝑃orb, 𝑃nSH signals from Table 1 (blue dashes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' & Kato (2013) derived that 𝜈prec 𝜈orb = −3 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 + 𝑛 4 + 𝑛 𝑞 √︁ 1 + 𝑞 � 𝑅𝑑 𝑎 �1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 cos 𝜃, (5) where 𝜈 = 𝜔/2𝜋 and 𝑅𝑑 is the disc radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='10 Using Equations (1), (2) and a mass profile of a steady-state disc given by 𝑛 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='75 (Shakura & Sunyaev 1973), Equation (5) can be reduced to 𝜀− 1 + 𝜀− = −21 52 𝑞 √︁ 1 + 𝑞 � 𝑅𝑑 𝑎 �1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 cos 𝜃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (6) A similar derivation can be found in Montgomery (2009b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' This shows that 𝜀− depends on three parameters: the mass ratio 𝑞, the disc tilt 𝜃 and the fractional disc radius 𝑅𝑑/𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' The third can be reasoned to be a function of 𝑞 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Suppose that in our systems with discovered nSHs, accretion discs are in steady state most of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Then, 𝑅𝑑 approaches the tidal truncation radius 𝑟tidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Paczynski (1977, Table 1) provided a functional dependence 𝑟tidal(𝑞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Later, Warner (1995) proposed the approximation 𝑟tidal = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='60𝑎 1 + 𝑞 (7) for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='03 < 𝑞 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' This is a good approximation in all regions but near 𝑞 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='7, where 𝑟tidal(𝑞) is underestimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Using it would reduce Equation (6) to 𝜀− 1 + 𝜀− = − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='188𝑞 (1 + 𝑞)2 cos 𝜃, (8) which does not describe well observational data for 𝑞 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='4 (see dot- ted line in Figure 10(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Because of this, we do not use Equation (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Instead, we linearly interpolate between data given in Paczynski (1977, Table 1) in order to evaluate 𝑅𝑑/𝑎 in Equation (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' The other independent variable in Equation (6) is the disc tilt 𝜃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Smak (2009) predicts that disc tilts should not exceed 𝜃max = 7◦ for CVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' In their photometric analysis of KIC 9406652, Kimura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (2020b) concluded that 𝜃 varies between 0–3◦ over the course of 1500 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Such range of 𝜃 allows the assumption cos 𝜃 ≃ 1, which is accurate to within one per cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' This motivates us to compute a 𝑞(𝜀−) curve for cos 𝜃 = 1 and compare it against measurements of Table 2 objects (Figure 10(b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Table A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' The 𝑞(𝜀−) relation becomes two-fold degenerate in 𝑞 from about |𝜀−| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='048.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' 10 We note that precession is retrograde, which implies 𝜔prec, 𝜈prec < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' MNRAS 000, 1–12 (2022) Tilted discs in six poorly studied CVs 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='06 |ε − | 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='9 q (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='06 |ε − | 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='9 q 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='9r tidal 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='1r tidal (b) Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Different 𝑞(𝜀−) relations against NL variables with 𝜀−, 𝑞 measurements from Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (a) Solid line: Montgomery (2009a), Dashed line: Wood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Both relations underestimate 𝑞(𝜀−) for given measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (b) Dotted line: Equation (8), derived from the approximation by Warner (1995) used in Equation (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' This fails to accurately describe Paczynski (1977) in 𝑞 regimes near 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Blue curve: a computed 𝑞(𝜀−) relation from our treatment in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='1 that makes use of Paczynski (1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Shaded region: solutions between 𝑅𝑑 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='9–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='1𝑟tidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' All measurements belong to this region within uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' A list of NLs with independently measured superhump deficit 𝜀− and mass ratio 𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Equatorial coordinates come from Gaia DR3 and are in the J2000 epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Other references: (𝑎) Gies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (𝑏) Africano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (1978);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (𝑐) Smak (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (𝑑) Subebekova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (𝑒) Skillman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (1995);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' ( 𝑓 ) Bruch (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (𝑔) Rodríguez-Gil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (ℎ) Kozhevnikov (2007);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (𝑖) Araujo-Betancor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (2003);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' ( 𝑗) Boyd et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (𝑘) Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (2002);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (𝑙) Huber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (1998);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (𝑚) Taylor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (1998);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (𝑛) Hoard & Szkody (1997);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (𝑜) Patterson (1999);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (𝑝) Arenas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (2000);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (𝑞) Peters & Thorstensen (2006);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (𝑟) Patterson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (1997);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (𝑠) Neustroev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (2011);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (𝑡) de Miguel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (𝑢) Szkody & Howell (1993);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (𝑣) Bruch (2023);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (𝑤) Gülsecen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (2009);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (𝑥) Hellier (1993);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' (𝑦) Bruch (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Name RA Dec 𝑞 𝑃orb 𝑃nSH |𝜀− | KIC 9406652 19h31m29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='s15 +45◦59′06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='′′1 𝑎0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='83 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='07 𝑎0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='25450(2) 𝑎0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='23971(2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0581(1) RW Tri 02h25m36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='s16 +28◦05′50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='′′9 𝑑0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='60 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='03 𝑏0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='23188324(4) 𝑐0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='2203(14) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='050(6) MV Lyr 19h07m16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='s29 +44◦01′07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='′′9 𝑒0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='43+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='19 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='13 𝑒0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='1329(4) 𝑓 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='12816(1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='036(3) KR Aur 06h15m43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='s92 +28◦35′08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='′′6 𝑔0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='39+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='04 𝑔0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='16277164(5) ℎ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='1571(2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='035(1) DW UMa 10h33m52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='s88 +58◦46′54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='′′7 𝑖0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='12 𝑖0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='136606499(3) 𝑗0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='132626(9) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='02914(7) TT Ari 02h06m53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='s08 +15◦17′41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='′′9 𝑘0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='04 𝑘0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='1375504(17) 𝑓 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='132921(2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='03366(2) V592 Cas 00h20m52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='s22 +55◦42′16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='′′2 𝑙0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='19+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='09 𝑚0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='115063(1) 𝑚0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='11193(5) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0272(4) BH Lyn 08h22m36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='s05 +51◦05′24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='′′6 𝑛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='45+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='10 𝑛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='15587520(5) 𝑜0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='1490(11) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='044(7) V603 Aql 18h48m54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='s64 +00◦35′02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='′′9 𝑝0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='24 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='05 𝑞0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='13820103(8) 𝑟0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='1341(3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='030(2) UX UMa 13h36m40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='s95 +51◦54′49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='′′4 𝑠0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='43 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='07 𝑡0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='19667118(19) 𝑡0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='186700(11) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='05070(6) AY Psc 01h36m55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='s46 +07◦16′29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='′′3 𝑢0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='45 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='2 𝑣0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='217320654(4) 𝑤0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='20645(75) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='050(3) TV Col 05h29m25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='s53 −32◦49′03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='′′9 𝑥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='15 𝑦0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='22860010(2) 𝑦0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='215995(1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='055140(4) There are several systems that lie far from our computed curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' However, they would all agree with a 𝑞(𝜀−) relation where 𝑅𝑑 is between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='9–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='1𝑟tidal (also in Figure 10(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' It becomes apparent that the curve is much more sensitive to changes in 𝑅𝑑 than to changes in 𝜃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' On this basis, we compute three curves for different 𝑅𝑑 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='9𝑟tidal, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0𝑟tidal, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='1𝑟tidal and then we graphically solve 𝑞 for our measured 𝜀−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Our mass-ratio estimates are listed in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Systems with smaller |𝜀−| tend to be better constrained in 𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' For example, PK-103959 has the lowest |𝜀−| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0308(22) in our sam- ple, and its mass ratio shows little variation for different values of 𝑅d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Conversely, SDSS-090113 has the largest measured |𝜀−| in our sam- ple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' For different 𝑅d, its 𝑞 estimates differ significantly with respect to statistical uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Estimates of the mass ratio 𝑞 of Table 1 systems for disc radii 𝑅𝑑 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='9𝑟tidal, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0𝑟tidal, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='1𝑟tidal from Paczynski (1977), using the methods described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Upper bounds of 𝑞 entering the region of two-fold degeneracy are labeled with an asterisk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Name 𝑞(𝜀−) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='9𝑟tidal 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0𝑟tidal 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='1𝑟tidal HBHA 4204-09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='38+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='28+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='22+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='02 Gaia-468436 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='46+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='35+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='02 Gaia-593107 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='52+0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='05 PK-103959 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='33+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='25+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='20+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='02 MNRAS 000, 1–12 (2022) 10 Stefanov & Stefanov 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='2 Orbital inclination estimates The eclipse width at half depth Δ𝜑orb is a reasonable measure of the primary-eclipse duration in the assumption of an axisymmetric disc (Warner 1995, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' This duration can be used to determine the orbital inclination 𝑖 by the relationship sin2 𝑖 ≈ 1 − 𝑅𝐿(2)2 cos2(2𝜋𝜑𝑝) , (9) where 𝑅𝐿(2) is the radius of the secondary star, and ±𝜑𝑝 are the phases of mid-immersion and mid-emergence of the primary star (Horne 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Warner 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='11 The Roche-lobe radius approximation by Eggleton (1983) 𝑅𝐿(2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='49𝑞2/3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='6𝑞2/3 + ln�1 + 𝑞1/3� (10) can be used to substitute 𝑅𝐿(2) in Equation (9), such that the relation can retrieve 𝑖 using only Δ𝜑orb and the superhump-derived 𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' We use this relation to constrain 𝑖 for HBHA 4204-09, SDSS-090113 and Gaia-593107, which are eclipsing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' We measured Δ𝜑orb on nSH-subtracted data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' HBHA 4204-09 has a deep eclipse, with Δ𝜑orb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='049(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Us- ing our value of 𝑞 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='29(3), we compute 𝑖 = 77(1)◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' On the other hand, SDSS-090113 shows grazing eclipses of variable depth and width that disappear completely for some 𝜑prec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' For this reason, we can assume Δ𝜑orb ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' With our value of 𝑞 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='47(4), we compute 𝑖 = 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='◦6(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' For the last eclipsing system in our sample, Gaia-593107, we measure Δ𝜑orb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='06(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' This eclipse width, combined with 𝑞 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='38(8), gives an orbital inclination 𝑖 = 76(3)◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' 5 CONCLUSIONS In this work we present results from a search for nSHs in poorly studied CVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' We initially cross-matched TESS light-curve data with the VSX catalogue for objects labelled as CV or NL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' We manually inspected LS periodograms of objects from this query (𝑛 = 180), and then selected targets with at least two neighbouring periodicities above the period gap, including one large-period signal that matches the beating of the former two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' This resulted in six systems with nSHs, which we list in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Spectroscopic measurements of 𝑃orb were available for only one of the six aforementioned systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' For the rest, we used a couple of methods to recognise 𝑃orb signatures in their LS periodograms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Three systems had their orbital period determined by the pres- ence of eclipses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' For the last two, 𝑃orb was identified using the 𝜑prec-dependent irradiation of the secondary caused by the precess- ing tilted disc (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' For all systems, the light maximum in the orbital phase profile was found to shift to earlier 𝜑orb, as 𝜑prec ad- vances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' This is strong evidence of nSHs (Kimura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' 2020b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Kimura & Osaki 2021) and supports our findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' For SDSS-090113, a pe- culiar behaviour was observed in ZTF photometry (Figure 7), which is similar to what is observed in IW And stars (Kato 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' For Gaia-46843, two Z Cam-like episodes in ASAS-SN data were found, which take place after drops in brightness of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5 mag (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Determined periodicities from TESS photometry can constrain some physical parameters of CV systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' The dependence between the nSH deficit 𝜀− and the system mass ratio 𝑞 has been explored in several works already (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Wood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Montgomery 2009a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' 11 From here, 𝜑𝑝 ≡ Δ𝜑orb/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Referenced 𝑞(𝜀−) relations, however, underestimate independent measurements of 𝑞 and 𝜀−, which gives some ground for concern (Figure 10(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' We tried to come to a better 𝑞(𝜀−) relation by us- ing the precession rate of a differentially rotating steady-state disc that extends to the maximum tidal truncation radius 𝑟tidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' This, in combination with the 𝑟tidal(𝑞) relation given in Paczynski (1977, Table 1), resulted in better agreement with independent observations (Figure 10(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Mass-ratio estimates using this method are given in Table 3 for different disk radii 𝑅𝑑 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='9𝑟tidal, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0𝑟tidal, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='1𝑟tidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' For the three systems that are eclipsing, we used the eclipse-mapping techniques described by Horne (1985);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Warner (1995) to compute the orbital inclination 𝑖 with our values of 𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' There are several subtle points that bear discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' SAP light curves from TESS-SPOC may contain instrumental effects that could affect LS periodogram measurements;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' and photometry itself may be contaminated by nearby sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' To address the former, we re- peated our methods on PDCSAP data, and found small differences in comparison to Table 1 measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Regarding the latter, we used mean-subtracted fluxes in all analysis, which mitigates effects by non-variable contaminating objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' None of our systems have bright sources in their vicinity, except the case of Gaia-593107, which is discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Our variant of the nSH subtraction method in Kimura & Osaki (2021) shares the same shortcomings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' If the nSH profile is time- dependent, it cannot be fully subtracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' In addition, the mass-transfer stream could happen to obstruct some parts of the disc, causing the light maximum to occur at earlier orbital phases 𝜑orb (Kimura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Inhomogeneities in the stellar surface brightness of the secondary can produce similar effects, shifting the light maximum to earlier or to later 𝜑orb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' The technique of using irradiation of the secondary in order to determine 𝑃orb is entirely based on photometry, and spectroscopic measurements of 𝑃orb could support its feasibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' In addition, radial- velocity analysis would put constraints on mass ratios, and would test the 𝑞(𝜀−) relation we consider in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' The newly discovered systems in this work are therefore strongly encouraged for follow-up spectroscopic observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' ACKNOWLEDGEMENTS This work includes data collected by the TESS mission and made use of lightkurve, a Python package for Kepler and TESS data analysis (Lightkurve Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Funding for the TESS mission is provided by the NASA’s Science Mission Directorate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' This work has made use of the NASA/IPAC Infrared Science Archive, which is funded by the National Aeronautics and Space Administration and operated by the California Institute of Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' The CSS survey is funded by the National Aeronautics and Space Administration under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' NNG05GF22G issued through the Science Mission Directorate Near-Earth Objects Observations Pro- gram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' The CRTS survey is supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' National Science Foundation under grants AST-0909182 and AST-1313422.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' We used the following Python packages for data analysis and visu- alisation: NumPy (Harris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' 2020), SciPy (Virtanen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' 2020), pandas (McKinney 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' The Pandas Development Team 2022), Matplotlib (Hunter 2007) and uncertainties (Lebigot 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' We are grateful to R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Zamanov and to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Kurtenkov for their advice during the preparation of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' We thank the anonymous referee for their time and their effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' We acknowledge the grants KΠ-06-H28/2 and KΠ-06-M58/2 from the Bulgarian National Sci- ence Fund.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Both authors contributed equally to this work.' metadata={'source': 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+page_content=', 2002, ApJ, 569, 418 Zacharias N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=', Monet D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=', Levine S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=', Urban S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=', Gaume R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=', Wycoff G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=', 2004, in American Astronomical Society Meeting Abstracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='15 de Miguel E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=', 2016, MNRAS, 457, 1447 APPENDIX A: EXTRA MATERIAL This appendix contains orbital phase curves (Figure A1) and a list of 𝑃orb measurements (Table A3) of discovered eclipsing binaries with no nSH behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Our calculated 𝑞(𝜀−) curves using the tidal truncation disc radii relation 𝑟tidal(𝑞) in Paczynski (1977) are shown in Table A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Computed values can be approximated by |𝜀−| = 𝑎𝑞 (1 + 𝑞)𝑏 , (A1) MNRAS 000, 1–12 (2022) 12 Stefanov & Stefanov Table A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Equation (A1) fit coefficients 𝑎 and 𝑏 that approximate Table A2 data for different values of 𝑅𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' 𝑅𝑑 𝑎 𝑏 |𝜀− | interval 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='9𝑟tidal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='155 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='706 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='006, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='041] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0𝑟tidal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='182 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='694 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='006, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='048] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='1𝑟tidal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='210 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='680 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='006, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='057] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='75 Orbital phas −5 0 R s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' flux (e − s −1 ) (a) TYC 8920-22-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' fl)x ( − s −1 ) (c) A TO J213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='4592-29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='8897 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='75 Orbital phase −5 0 5 Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' flux (e − s −1 ) (d) 3XLSS J231521.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='7-541842 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='75 Orbital phase −10 0 Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' flux (e − s −1 ) (e) Gaia DR3 5499736649371768192 Figure A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Orbital phase curves of Table A3 systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Light curves were smoothed by a fourth-order Savitzky-Golay filter of window size 10 d before folding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Orbital phases are offset with +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='50 for the sake of clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' where 𝑎 and 𝑏 are fit coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Values of best fits are given in Table A1 for 𝑅𝑑 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='9𝑟tidal, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0𝑟tidal, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='1𝑟tidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' This paper has been typeset from a TEX/LATEX file prepared by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Table A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' Tabular form of our derived 𝑞(𝜀−) relation for 𝑅𝑑 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='9−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='1𝑟tidal using 𝑟tidal from Paczynski (1977, Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content=' |𝜀− | 𝑞(𝜀−) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='9𝑟tidal 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FAT4oBgHgl3EQffR3S/content/2301.08581v1.pdf'} +page_content='0𝑟tidal 1.' metadata={'source': 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a/qNAzT4oBgHgl3EQfcPwr/content/tmp_files/2301.01398v1.pdf.txt b/qNAzT4oBgHgl3EQfcPwr/content/tmp_files/2301.01398v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6e229efd6b26060c5c27cb0fee791a1689ae557f --- /dev/null +++ b/qNAzT4oBgHgl3EQfcPwr/content/tmp_files/2301.01398v1.pdf.txt @@ -0,0 +1,1511 @@ +Cost Inference for Feedback Dynamic Games from +Noisy Partial State Observations and Incomplete Trajectories +Jingqi Li +University of California, Berkeley +Berkeley, United States +jingqili@berkeley.edu +Chih-Yuan Chiu +University of California, Berkeley +Berkeley, United States +chihyuan_chiu@berkeley.edu +Lasse Peters +Delft University of Technology +Delft, Netherlands +l.peters@tudelft.nl +Somayeh Sojoudi +University of California, Berkeley +Berkeley, United States +sojoudi@berkeley.edu +Claire Tomlin +University of California, Berkeley +Berkeley, United States +tomlin@eecs.berkeley.edu +David Fridovich-Keil +University of Texas, Austin +Austin, United States +dfk@utexas.edu +ABSTRACT +In multi-agent dynamic games, the Nash equilibrium state trajec- +tory of each agent is determined by its cost function and the infor- +mation pattern of the game. However, the cost and trajectory of each +agent may be unavailable to the other agents. Prior work on using +partial observations to infer the costs in dynamic games assumes +an open-loop information pattern. In this work, we demonstrate +that the feedback Nash equilibrium concept is more expressive and +encodes more complex behavior. It is desirable to develop specific +tools for inferring players’ objectives in feedback games. Therefore, +we consider the dynamic game cost inference problem under the +feedback information pattern, using only partial state observations +and incomplete trajectory data. To this end, we first propose an +inverse feedback game loss function, whose minimizer yields a +feedback Nash equilibrium state trajectory closest to the observa- +tion data. We characterize the landscape and differentiability of the +loss function. Given the difficulty of obtaining the exact gradient, +our main contribution is an efficient gradient approximator, which +enables a novel inverse feedback game solver that minimizes the +loss using first-order optimization. In thorough empirical evalua- +tions, we demonstrate that our algorithm converges reliably and +has better robustness and generalization performance than the +open-loop baseline method when the observation data reflects a +group of players acting in a feedback Nash game. +KEYWORDS +Inverse Games, Dynamic Game Theory, Nash Equilibrium +ACM Reference Format: +Jingqi Li, Chih-Yuan Chiu, Lasse Peters, Somayeh Sojoudi, Claire Tomlin, +and David Fridovich-Keil. 2023. Cost Inference for Feedback Dynamic Games +from Noisy Partial State Observations and Incomplete Trajectories. In Proc. +of the 22nd International Conference on Autonomous Agents and Multiagent +Systems (AAMAS 2023), London, United Kingdom, May 29 – June 2, 2023, +IFAAMAS, 10 pages. +1 +INTRODUCTION +The safety and efficiency of urban traffic relies heavily on the ability +of each participant to predict the effects of their actions on others’ +Proc. of the 22nd International Conference on Autonomous Agents and Multiagent Sys- +tems (AAMAS 2023), A. Ricci, W. Yeoh, N. Agmon, B. An (eds.), May 29 – June 2, 2023, +London, United Kingdom. © 2023 International Foundation for Autonomous Agents +and Multiagent Systems (www.ifaamas.org). All rights reserved. +decisions [23, 33]. For example, drivers on a highway may wish +to halt an overtaking maneuver if they believe the other drivers +are aggressive, and some drivers may decelerate their cars to avoid +collision if they believe that another driver wishes to merge. +A powerful paradigm for modeling the interdependence of deci- +sions in multi-agent settings is provided general-sum dynamic games +[2, 13]. A Nash equilibrium solution of a game-theoretic model can +be used to simultaneously predict the actions of all agents in the +scene. This equilibrium solution is particularly expressive when +the game possesses a feedback information structure. In this case, +each equilibrium strategy explicitly accounts for the dynamically +evolving information available to each player over time. +Despite the theoretical attractiveness of this modeling paradigm, +in reality, autonomous agents often have only limited information +available about the world around them. For example, in urban traffic +an autonomous agent typically has incomplete knowledge of the +objectives of other players. To address this challenge, recent works +on inverse dynamic game theory [21, 28, 31] recover these objectives +from past trajectory data. Moreover, in realistic applications, only +noisy sensor measurements of agents’ states are available. This +partial observability further complicates the inverse game problem, +and existing work [28] treats this case in the open-loop information +structure. +In this work, we present a gradient-based solver for inverse +dynamic games, under the state feedback information structure. +Our solver can recover objectives from partial state observations +of incomplete trajectories. Both of these effects are common in +robotics due to noisy perception and occlusions. We show that +our algorithm converges reliably in practice, and demonstrate the +superior robustness and generalization performance as compared +with a baseline method which learns cost functions under the open- +loop assumption [28], when the observation data is from a group +of players pursuing a feedback Nash equilibrium strategy. +Our contributions are threefold. Firstly, we characterize the solu- +tion set of the inverse feedback dynamic game problem. In particu- +lar, we show that the set of the global minima could be nonconvex +and disconnected, and discuss regularization schemes to mitigate +this problem. Secondly, we show the differentiability of the loss +function in linear quadratic games and propose a computationally +efficient procedure to approximate the gradient for nonlinear games. +Finally, we propose an efficient first-order coordinate-descent solver +arXiv:2301.01398v1 [cs.MA] 4 Jan 2023 + +for the inverse feedback game problem, using noisy partial obser- +vations of an incomplete expert state trajectory. Experimental re- +sults show that our method reliably converges for inverse feedback +games with nonlinear dynamics and is able to learn nonconvex +costs. Moreover, the converged cost function can accurately predict +the feedback Nash equilibrium state trajectories even for unseen +initial states. +2 +RELATED WORK +2.1 +Non-cooperative Dynamic Games +Non-cooperative dynamic game theory [2, 13] provides a formal +framework for analyzing strategic interaction in a multi-agent set- +ting [2, 6, 17]. In non-cooperative games, each player minimizes +its own individual cost function; since players’ costs may not be +mutually aligned, the resulting equilibrium behavior is generally +competitive. Among different equilibrium concepts, the Nash equi- +librium has been extensively studied because of its representative +power of capturing many non-cooperative behaviors arising in +real-world multi-agent systems [9, 32]. +Recent advances in the literature aim to develop efficient solu- +tions to Nash equilibrium problems in dynamic games. Though +the solutions for the open-loop and feedback Nash equilibrium in +linear quadratic (LQ) games are well understood [2], for nonlinear +games there is no closed-form solution in general. The work [30] +characterizes the local Nash solution concept for open-loop Nash +equilibrium. In the feedback setting, numerous approaches have +been proposed under various special cases [14, 35]. A value iter- +ation based approach for computing feedback Nash equilibria of +nonlinear games without constraints is introduced in [10]. Recently, +a set of KKT conditions for feedback Nash equilibria in constrained +nonlinear games is derived in [16]. Computing a feedback Nash +equilibrium is challenging due to the nested KKT conditions in +different time steps. +Our work draws upon the ILQGames [8] framework, which at +each iteration solves a linear-quadratic game that approximates the +original game. The construction of the approximate game parallels +the iterative linearization and quadraticization methods of iterative +LQR [18], and the dynamic programming equations that charac- +terize equilibrium strategies in linear quadratic dynamic games +[2]. This approach differs from the ALGames [5] method, which +computes an open-loop Nash equilibrium strategy. +2.2 +Inverse Non-cooperative Dynamic Games +In contrast to the forward game problem of computing a strategy +in dynamic games, an inverse game problem amounts to finding +objectives for all agents such that the corresponding strategic (e.g., +Nash equilibrium) interactions reproduce expert demonstrations. +The inverse game problem is important because it paves the way +for an agent to understand the preferences which explain other +agents’ behavior, which may facilitate more efficient multi-agent +interaction and coordination. +The problem of inverse infinite-horizon LQ games is considered +in [12], where the set of cost functions whose feedback Nash equi- +librium strategies coincide with an expert strategy is derived. In +[31, 36], the two-player inverse LQ game is solved by transforming +the problem to an inverse optimal control under the assumption +that the control input data of one player is known. Two methods +based on the KKT conditions of an open-loop Nash equilibrium +are proposed for open-loop general-sum differential games in [24]. +Several necessary conditions for open-loop Nash equilibria are pro- +posed in [22] and used for developing an inverse game solution for +some classes of open-loop games. +Recently, an efficient bilevel optimization framework [28] based +on the open-loop Nash equilibrium KKT conditions was proposed +for solving inverse games with an open-loop Nash assumption. +Another line of work on inferring costs in open-loop games [1, 7, 11] +proposes to minimize the residual violation of the KKT conditions. +This KKT residual framework assumes the knowledge of complete +trajectory data and is a convex problem. Given the difficulty of +evaluating KKT conditions for feedback Nash equilibria in nonlinear +games [16], the extension of the KKT residual method to feedback +nonlinear games may be subject to numerical difficulty. +A bilevel optimization approach for inverse feedback game prob- +lem is proposed in [25], with the assumption that both the expert +state and control trajectories are observed without noise. In addi- +tion, an inverse game solver is proposed in [20] where they infer the +players’ cost functions with the assumption that the expert strategy +follows a new concept called Maximum Entropy Nash Equilibrium. +To the best of the authors’ knowledge, there is no work on inferring +cost functions of nonlinear dynamic games under feedback Nash +equilibrium condition, from noisy partial state observation and +incomplete trajectory data. +3 +PRELIMINARIES +Consider an 𝑁-player,𝑇-stage, deterministic, discrete-time dynamic +game, with a state 𝑥𝑖 +𝑡 ∈ R𝑛𝑖 and control input 𝑢𝑖 +𝑡 ∈ R𝑚𝑖 for each +player 𝑖 ∈ [𝑁] := {1, · · · , 𝑁 }, 𝑡 ∈ [𝑇]. Let the dimension of the +joint state and control input be 𝑛 := �𝑁 +𝑖=1 𝑛𝑖 and 𝑚 := �𝑁 +𝑖=1 𝑚𝑖, +respectively. We denote by 𝑥𝑡 := [𝑥1 +𝑡 , . . . ,𝑥𝑁 +𝑡 ] ∈ R𝑛 and 𝑢𝑡 := +[𝑢1 +𝑡 , . . . ,𝑢𝑁 +𝑡 ] ∈ R𝑚 the joint state and joint control at time 𝑡 ∈ [𝑇], +respectively. The joint dynamics for the system is given by the +differentiable dynamics map 𝑓𝑡 (·, ·) : R𝑛 × R𝑚 → R𝑛: +𝑥𝑡+1 = 𝑓𝑡 (𝑥𝑡,𝑢𝑡), +∀𝑡 = 1, · · · ,𝑇 . +(1) +We denote by f := {𝑓𝑡 }𝑇 +𝑡=1 the set of dynamics across all the time +instances within horizon𝑇. We define x := {𝑥𝑡 }𝑇 +𝑡=1 and u := {𝑢𝑡 }𝑇 +𝑡=1 +to be a state trajectory and control trajectory, respectively, if 𝑥𝑡+1 = +𝑓 (𝑥𝑡,𝑢𝑡), for each 𝑡 ∈ [𝑇]. The objective of each agent 𝑖 is to +minimize its overall cost, given by the sum of its running costs +𝑔𝑖 +𝑡 : R𝑛 × R𝑚 → R over the time horizon: +𝐽𝑖 (x, u) := +𝑇 +∑︁ +𝑡=1 +𝑔𝑖 +𝑡 (𝑥𝑡,𝑢𝑡) +(2) +Define 𝑔𝑡 := {𝑔1 +𝑡 ,𝑔2 +𝑡 , · · · ,𝑔𝑁 +𝑡 }, 𝑡 ∈ [𝑇]. We denote by g := {𝑔𝑡 }𝑇 +𝑡=1 +the set of cost functions for all the agents within horizon 𝑇. +To minimize (2), each player uses their observations of the envi- +ronment to design a sequence of control inputs to deploy during +the discrete time interval [𝑇]. The information available to each +player at each time characterizes the information pattern of the +dynamic game, and plays a major role in shaping the optimal re- +sponses of each player [2]. Below, we explore two such information +patterns—feedback and open-loop. + +3.1 +Nash Solutions in Feedback Strategies +Under the state feedback information pattern, each player observes +the state 𝑥𝑡 at each time 𝑡, and uses this information to design a +feedback strategy 𝛾𝑖 +𝑡 : R𝑛 → R𝑚𝑖 , given by: 𝑢𝑖 +𝑡 := 𝛾𝑖 +𝑡 (𝑥𝑡), for each +𝑖 ∈ [𝑁] and 𝑡 ∈ [𝑇]. Let 𝛾𝑡 (𝑥𝑡) := [𝛾1 +𝑡 (𝑥𝑡),𝛾2 +𝑡 (𝑥𝑡), . . . ,𝛾𝑁 +𝑡 (𝑥𝑡)] ∈ +R𝑚. +Following the notation of [2], we denote by Γ𝑖 +𝑡 the set of all state +feedback strategies of player 𝑖, for each 𝑖 ∈ [𝑁]. Under this feedback +information pattern, the Nash equilibrium of the dynamic game is +as defined below. +Definition 1 (Feedback Nash Eqilibrium (FBNE) [2, Ch. 6]). +The set of control strategies {𝛾1∗ +𝑡 , · · · ,𝛾𝑁∗ +𝑡 +}𝑇 +𝑡=1 is called a feedback +Nash equilibrium if no player is incentivized to unilaterally alter its +strategy. Formally: +𝑊 𝑖∗ +𝑡 +� +𝑥𝑡, [𝛾1∗ +𝑡 (𝑥𝑡), . . . ,𝛾𝑖∗ +𝑡 (𝑥𝑡), . . . ,𝛾𝑁∗ +𝑡 +(𝑥𝑡)] +� +(3) +≤ 𝑊 𝑖∗ +𝑡 +� +𝑥𝑡, [𝛾1∗ +𝑡 (𝑥𝑡), . . . ,𝛾𝑖 +𝑡 (𝑥𝑡), . . . ,𝛾𝑁 ∗ +𝑡 +(𝑥𝑡)] +� +, ∀𝛾𝑖 +𝑡 ∈ Γ𝑖 +𝑡 , ∀𝑡 ∈ [𝑇]. +where 𝑊 𝑖∗ +𝑡 (·, ·) : R𝑛 × R𝑚 → R, 𝑡 ∈ [𝑇] is the optimal state-action +function defined as follows, +𝑊 𝑖∗ +𝑇 (𝑥𝑇,𝑢𝑇 ) := 𝑔𝑖 +𝑇 (𝑥𝑇,𝑢𝑇 ) +𝑊 𝑖∗ +𝑡 (𝑥𝑡,𝑢𝑡) := 𝑔𝑖 +𝑡 (𝑥𝑡,𝑢𝑡) + 𝑉 𝑖∗ +𝑡+1(𝑥𝑡+1), ∀𝑡 ∈ [𝑇 − 1], +𝑉 𝑖∗ +𝑡 (𝑥𝑡) := 𝑊 𝑖∗ +𝑡 (𝑥𝑡, [𝛾1 +𝑡 +∗(𝑥𝑡), . . . ,𝛾𝑁 +𝑡 +∗(𝑥𝑡)]), ∀𝑡 ∈ [𝑇]. +(4) +We define x and u to be a FBNE state trajectory and a FBNE +control trajectory, respectively, if𝑢𝑖 +𝑡 = 𝛾𝑖∗ +𝑡 (𝑥𝑡), for each 𝑖 ∈ [𝑁] and +𝑡 ∈ [𝑇]. We denote by 𝜉(f, g) the set of all FBNE state trajectories +in the game defined by the dynamics f and cost functions g. +Remark 1 (Strong Time Consistency). The FBNE conditions of +(3) implicitly enforce strong time-consistency [2, Def. 5.14] of the equi- +librium strategies. That is, FBNE does not admit arbitrary feedback +strategies, but imposes the additional condition that those strategies +must also be in equilibrium for any subgame starting at a later stage +from an arbitrary state. +3.2 +Nash Solutions in Open-loop Strategies +In contrast, under the open-loop information pattern, each player +only observes the initial state 𝑥1. In this case, the strategy for each +player 𝑖 ∈ [𝑁] is a map from 𝑥1 to {𝑢𝑖 +1,𝑢𝑖 +2, · · · ,𝑢𝑖 +𝑇 }, which we +denote by 𝜙𝑖 (·) : R𝑛 → R𝑚𝑖 × · · · × R𝑚𝑖 +���������������������������������� +𝑇 +. Let Φ𝑖 be the set of all +open-loop strategies of the player 𝑖, 𝑖 ∈ [𝑁]. The corresponding +open-loop Nash equilibrium is defined as follows. +Definition 2 (Open-Loop Nash Eqilibrium (OLNE) [2, +Ch. 6]). The tuple of control strategies {𝜙∗ +1, · · · ,𝜙∗ +𝑁 } is called an +open-loop Nash equilibrium if no player is incentivized to unilaterally +alter its sequence of control inputs. Formally: +𝐽𝑖 � +x, [𝜙1∗(𝑥1), · · · ,𝜙𝑖∗(𝑥1), · · · ,𝜙𝑁 ∗(𝑥1)] +� +(5) +≤ 𝐽𝑖 � +x, [𝜙1∗(𝑥1), · · · ,𝜙𝑖 (𝑥1), · · · ,𝜙𝑁 ∗(𝑥1)] +� +, ∀𝜙𝑖 ∈ Φ𝑖, ∀𝑥1 ∈ R𝑛. +Remark 2. The OLNE definition does not imply the strong time- +consistence as in the feedback counterpart [2]. +0 +1 +2 +3 +px +0 +1 +2 +3 +4 +py +Example 1, trajectories +player 1, FBNE +player 2, FBNE +player 1, OLNE +player 2, OLNE +0 +5 +10 +time +0 +1 +2 +3 +4 +5 +position +Example 1, FBNE +player 1, px +player 1, py +player 2, px +player 2, py +0 +5 +10 +time +0 +1 +2 +3 +4 +5 +position +Example 1, OLNE +player 1, px +player 1, py +player 2, px +player 2, py +1 +1.5 +2 +2.5 +3 +px +0 +1 +2 +3 +4 +py +Example 2, trajectories +0 +5 +10 +time +0 +1 +2 +3 +4 +5 +position +Example 2, FBNE +0 +5 +10 +time +0 +1 +2 +3 +4 +5 +position +Example 2, OLNE +-10 +-5 +0 +5 +10 +px +-10 +-5 +0 +5 +10 +py +Example 3, trajectories +0 +20 +40 +60 +time +0 +1 +2 +3 +4 +5 +position +Example 3, FBNE +0 +20 +40 +60 +time +-10 +-5 +0 +5 +10 +position +Example 3, OLNE +Fig. 1: Examples of cost functions that yield trajectories that are dif- +ferent under the OLNE and FBNE assumptions. +3.3 +Feedback vs. Open-loop Nash Equilibria +In this subsection, we demonstrate the difference between open- +loop and feedback Nash equilibria and show the necessity of de- +veloping specific solutions for cost inference problems with the +feedback information pattern, instead of applying existing work +with the open-loop assumption [29]. To this end, we introduce be- +low several linear-quadratic (LQ) games where the open-loop Nash +equilibrium (OLNE) and feedback Nash equilibrium (FBNE) state +trajectories differ substantially. LQ games are a class of dynamic +games with dynamics and player objectives of the form in (6) and +(7), respectively, +𝑥𝑡+1 = 𝐴𝑡𝑥𝑡 + +∑︁ +𝑖 ∈[𝑁 ] +𝐵𝑖 +𝑡𝑢𝑖 +𝑡, ∀𝑡 ∈ [𝑇], +(6) +𝑔𝑖 +𝑡 (𝑥𝑡,𝑢𝑡) = 1 +2 (𝑥⊤ +𝑡 𝑄𝑖 +𝑡𝑥𝑡 + +∑︁ +𝑗 ∈[𝑁 ] +𝑢 𝑗 +𝑡 +⊤𝑅𝑖𝑗 +𝑡 𝑢 𝑗 +𝑡 ), ∀𝑡 ∈ [𝑇], ∀𝑖 ∈ [𝑁], (7) +where matrices {𝐴𝑡, 𝐵𝑖 +𝑡 }, positive semidefinite matrix 𝑄𝑖 +𝑡 and posi- +tive definite matrix 𝑅𝑖𝑗 +𝑡 are defined with appropriate dimensions, +for each 𝑖, 𝑗 ∈ [𝑁] and 𝑡 ∈ [𝑇]. +Case Study: We consider a two-player LQ game with a state +vector 𝑥𝑡 = [𝑝1 +𝑥,𝑡, 𝑝1 +𝑦,𝑡, 𝑝2 +𝑥,𝑡, 𝑝2 +𝑦,𝑡], where 𝑝𝑖 +𝑥,𝑡 and 𝑝𝑖 +𝑦,𝑡 are the x- +and y-coordinates of agent 𝑖 ∈ {1, 2}, respectively. Let 𝑢𝑖 +𝑡 ∈ R2 be +the control input for the 𝑖-th agent, 𝑖 ∈ {1, 2}. In this setting, we +consider a class of games in which the first agent wants to drive +the second agent to the origin, while the second agent wants to +catch the first agent. The agents’ joint dynamics and costs at time +𝑡 ∈ [𝑇] are specified as follows: +𝑥𝑡+1 = +�𝐼2 +0 +0 +𝐼2 +� +𝑥𝑡 + +�𝐼2 +0 +� +𝑢1 +𝑡 + +� 0 +𝐼2 +� +𝑢2 +𝑡 , +𝑔1 +𝑡 (𝑥𝑡,𝑢𝑡) = ∥𝑝2 +𝑥,𝑡 ∥2 +2 + ∥𝑝2 +𝑦,𝑡 ∥2 +2 + ∥𝑢1 +𝑡 ∥2 +2, +𝑔2 +𝑡 (𝑥𝑡,𝑢𝑡) = ∥𝑝2 +𝑥,𝑡 − 𝑝1 +𝑥,𝑡 ∥2 +2 + ∥𝑝2 +𝑦,𝑡 − 𝑝1 +𝑦,𝑡 ∥2 +2 + ∥𝑢2 +𝑡 ∥2 +2, +(8) + +where 𝐼2 is the 2 × 2 identity matrix. We visualize the unique FBNE +and OLNE state trajectories of this example in the first row in Fig. 1. +If we modify the cost function of the first player such that it wants +to lead the 𝑥- and 𝑦-position of the second player to be aligned with +each other, i.e., +ˆ𝑔1 +𝑡 (𝑥𝑡,𝑢𝑡) := ∥𝑝2 +𝑥,𝑡 − 𝑝2 +𝑦,𝑡 ∥2 +2 + ∥𝑢1 +𝑡 ∥2 +2, +(9) +then, the unique FBNE and OLNE state trajectories are still different, +as shown in the second row of Fig. 1. Moreover, observations of +players may be noisy in practice. To illustrate this, we consider a +task where the two agents want to catch each other, but the first +player’s observation of the second player’s position is inaccurate. +We modify the first player’s cost in (8) as follows: +ˆˆ𝑔1 +𝑡 (𝑥𝑡,𝑢𝑡) := ∥𝑝1 +𝑥,𝑡 − 2𝑝2 +𝑥,𝑡 ∥2 +2 + ∥𝑝1 +𝑦,𝑡 − 2𝑝2 +𝑦,𝑡 ∥2 +2 + ∥𝑢1 +𝑡 ∥2 +2. +(10) +The third row of Fig. 1 reveals that the FBNE state trajectory is +robust to inaccurate observations, but the unique OLNE state tra- +jectory is not. +Thus, it is readily apparent that the OLNE and FBNE state strate- +gies can be substantially different even for fixed cost functions. This +difference in expressive power may be understood as a consequence +of the strong time consistency property, which is enforced in the +feedback information structure but not in the open-loop setting, +per Remarks 1 and 2. A similar problem arises in the cost inference +problem, where the existing OLNE cost inference algorithms may +fail to infer the correct cost function in feedback games. +4 +PROBLEM STATEMENT +Let x be an expert FBNE state trajectory under the nonlinear dy- +namics f but unknown cost functions {𝑔𝑖 +𝑡 }𝑇,𝑁 +𝑡=1,𝑖=1. Let T ⊆ [𝑇] be +the set of observed time indices of the trajectory x. We denote by +yT := {𝑦𝑡 }𝑡 ∈T the observation data of x, where 𝑦𝑡 ∈ Rℓ is a partial +observation of the state, composed of certain coordinates of 𝑥𝑡 cor- +rupted by noise. The task is to infer the cost function of each player +such that those inferred costs jointly yield a FBNE state trajectory +that is as close as possible to the observed trajectory. We param- +eterize the cost of the player 𝑖 by a vector 𝜃𝑖 ∈ R𝑑𝑖 , and let 𝜃 := +[𝜃1,𝜃2, . . . ,𝜃𝑁 ] ∈ R𝑑. Denote by 𝑔𝑖 +𝑡,𝜃 (𝑥𝑡,𝑢𝑡) = �𝑑𝑖 +𝑗=1 𝜃𝑖 +𝑗𝑏𝑖 +𝑡,𝑗 (𝑥𝑡,𝑢𝑡) +player 𝑖’s parameterized cost at time 𝑡 ∈ [𝑇], for some basis func- +tions {{𝑏𝑖 +𝑡,𝑗 }𝑑𝑖 +𝑗=1}𝑇,𝑁 +𝑡=1,𝑖=1. Define g𝜃 := {𝑔𝑖 +𝑡,𝜃 }𝑇,𝑁 +𝑡=1,𝑖=1. Formally, this +problem is of the form: +min +𝜃,𝑥1,ˆx +− 𝑝(yT|ˆx) +s.t. +ˆx ∈ 𝜉(f, g𝜃,𝑥1), +(11) +where 𝑝(·|·) is the likelihood function corresponding to a known +sensor model and 𝜉(f, g𝜃,𝑥1) represents the set of state trajectories +from the initial condition 𝑥1 ∈ R𝑛 following a FBNE strategy, under +the cost set g𝜃. Due to the noisy partial observation, 𝑥1 is not +assumed to be known and instead needs to be inferred as well in +(11). Note that the above formulation can also be extended to the +cases where multiple partially observed incomplete trajectories +from different initial conditions are available. +Running example: We consider a highway platooning scenario +where player 1 wants to guide player 2 to a particular lane of the +road. The joint state vector is𝑥𝑡 = [𝑝1 +𝑥,𝑡, 𝑝1 +𝑦,𝑡, 𝛽1 +𝑡 , 𝑣1 +𝑡 , 𝑝2 +𝑥,𝑡, 𝑝2 +𝑦,𝑡, 𝛽2 +𝑡 , 𝑣2 +𝑡 ]. +Fig. 2: Visualization of the running example. +The time horizon 𝑇 = 40. The dynamics model for the player 𝑖 is: + +𝑝𝑖 +𝑥,𝑡+1 +𝑝𝑖 +𝑦,𝑡+1 +𝛽𝑖 +𝑡+1 +𝑣𝑖 +𝑡+1 + += + +𝑝𝑖 +𝑥,𝑡 +𝑝𝑖 +𝑦,𝑡 +𝛽𝑖 +𝑡 +𝑣𝑖 +𝑡 + ++ Δ𝑇 + +𝑣𝑖 +𝑡 cos(𝛽𝑖 +𝑡) +𝑣𝑖 +𝑡 sin(𝛽𝑖 +𝑡) +𝜔𝑖 +𝑡 +𝑎𝑖 +𝑡 + +(12) +where Δ𝑇 is a time discretization constant and 𝑢𝑖 +𝑡 = [𝜔𝑖 +𝑡,𝑎𝑖 +𝑡] ∈ R2 +is the control input for player 𝑖 ∈ [𝑁]. Let 𝑝∗𝑥 be the target lane +that player 1 wants to guide player 2 to. We parameterize the cost +function of the player 𝑖 by 𝜃𝑖 ∈ R2, +𝑔1 +𝑡,𝜃 (𝑥𝑡,𝑢𝑡) = 𝜃1 +1 ∥𝑝1 +𝑥,𝑡 ∥2 +2 + 𝜃1 +2 ∥𝑝2 +𝑥,𝑡 − 𝑝∗ +𝑥 ∥2 +2 + ∥𝑢1 +𝑡 ∥2 +2 +(13) +𝑔2 +𝑡,𝜃 (𝑥𝑡,𝑢𝑡) = 𝜃2 +1 ∥𝑝2 +𝑥,𝑡 − 𝑝1 +𝑥,𝑡 ∥2 +2 + 𝜃2 +2 ∥𝑣2 +𝑡 − 1∥2 +2 + ∥𝑢2 +𝑡 ∥2 +2, ∀𝑡 ∈ [𝑇]. +The ground truth solution is𝜃∗ = [0, 8, 4, 4]. We assume that there is +a period of occlusion happening from the time index𝑡 = 11 to𝑡 = 19, +and the observed time index set is T = {1, 2, . . . , 10, 20, 21, . . . , 40}. +Also, it may be difficult for a human driver to measure other ve- +hicles’ velocity accurately, and therefore we assume that partial +observation data yT excludes the velocity of both cars in the data +set, and is further subject to Gaussian noise of standard deviation 𝜎. +The initial condition 𝑥1 is not known and needs to be inferred. We +visualize the ground truth solution in the first subplot of Fig. 2 and +the noisy incomplete trajectory data in the second subplot of Fig. 2. +The many challenges of the above problem include: (a) partial +observation; (b) noisy and incomplete expert trajectory data; and (c) +the difficulty of evaluating and differentiating the objective in (11), +due to the challenge of computing a FBNE strategy in nonlinear +games [16]. In the following sections, we will characterize the +complexity of this inverse feedback game problem and propose an +efficient solution. +5 +RESULTS: FROM CHARACTERIZATION TO +COMPUTATION +In this section, we first characterize the complexity of the inverse +feedback game problem (11). In particular, we will show the non- +convexity of the loss function and the existence of multiple isolated +global minima. Based on this observation, we discuss regulariza- +tion schemes that can mitigate this issue. Our main contribution is +to characterize the differentiability of the inverse feedback game +loss function in (11). Finally, we present a gradient approximation +scheme that can be used in a first-order optimization formulation. + +Fig. 3: Visualization of the loss function 𝐿(𝜃,𝑥1) of the LQ game spec- +ified in (16) and (17), and its 𝐿2 regularization, with an initial condi- +tion 𝑥1 = 1. We adopt Gaussian likelihood function. The yellow hy- +perplane is drawn according to 2𝑄1 +𝑄2 = 3. With 𝐿2 regularization, +the number of global minima is reduced. +5.1 +Characterization of the Inverse Feedback +Dynamic Game Problem +The inverse feedback dynamic game problem (11) is a constrained +optimization problem, which is hard to solve due to the nonconvex- +ity of the set 𝜉(f, g𝜃,𝑥1). With a slight abuse of notation, we denote +by ˆx(f, g𝜃,𝑥1) ∈ 𝜉(f, g𝜃,𝑥1) a FBNE state trajectory. To simplify +the problem, we transform (11) to an unconstrained problem by +substituting a forward game solution ˆx(f, g𝜃,𝑥1) into the likelihood +function 𝑝(yT|ˆx), as follows: +ˆ𝐿(𝜃,𝑥1) := −𝑝(yT|ˆx(f, g𝜃,𝑥1)). +(14) +The minimizer of (14) is a local optimum to the original problem +(11) and becomes global when 𝜉(f, g𝜃,𝑥1) contains only a single +element. +Before we dive into the nonlinear setting, let us first consider a +simplified LQ case to highlight the main challenges associated with +the optimization of this loss. In the LQ case, the evaluation of the +loss (14) is straightforward if there exists a closed-form expression +for 𝑝(yT|ˆx), e.g., under a Gaussian observation model. Even in that +setting, however, it is important to realize that the problem remains +nonconvex, as shown in Fig. 3. The following proposition makes +this challenge explicit, and the proof can be found in the Appendix. +Proposition 1. There exists an inverse LQ game problem (11): +(a) whose global minima are isolated, and (b) for which there exist +multiple cost functions that exactly match expert data from any initial +condition, when there is no observation noise. +Remark 3. Proposition 1 does not imply that any inverse LQ game +problem will suffer from the multiple global minima issue. Instead, +Proposition 1 suggests that simply normalizing the cost vector does +not rule out the possibility of having multiple global solutions. That is, +there exist two cost parameter vectors which are linearly independent, +but generate the same FBNE state trajectories for any given initial +state. This non-injective mapping from the cost parameter space to +the FBNE state trajectory space is a fundamental problem in inverse +feedback games, and is not particular to the formulation (11). In +practice, this multiple global minima issue could be mitigated by +adding 𝐿2 regularization, as visualized in Fig. 3. +Though being nonconvex, the loss function ˆ𝐿(𝜃,𝑥1) is differen- +tiable with respect to both 𝜃 and 𝑥1 under the condition of Theorem +3.2 in [16], which follows from the implicit function theorem [15]. +Inspired by the success of gradient-based methods in non-convex +optimization with differentiable objective functions [3, 26, 34], one +natural idea is to apply gradient descent to minimize ˆ𝐿(𝜃,𝑥1). In +what follows, we discuss efficient ways to evaluate and differentiate +ˆ𝐿(𝜃,𝑥1) in nonlinear games. +5.2 +Efficient Computation for a FBNE State +Trajectory in Nonlinear Games +It is easy to evaluate ˆ𝐿(𝜃,𝑥1) for LQ games, but when dynamics are +nonlinear or objectives are non-quadratic, the problem becomes +more challenging [16]. In forward games, this challenge can be +addressed by using the ILQGames algorithm [8], which finds ap- +proximate local FBNE solutions in smooth non-LQ dynamic games. +Given the effectiveness of this approximation scheme in those do- +mains, we also adopt it as a submodule for evaluating the loss +ˆ𝐿(𝜃,𝑥1). Akin to the ILQR method [18, 19], in each step of the +ILQGames algorithm, the system dynamics 𝑥𝑡+1 = 𝑓 (𝑥𝑡,𝑢𝑡) and +the costs {𝑔𝑖 +𝑡 (𝑥,𝑢)}𝑇,𝑁 +𝑡=1,𝑖=1 are linearized and quadraticized, respec- +tively, around a state trajectory x and a control trajectory u. A +FBNE strategy for each player of the derived LQ game is then used +to update the state and control trajectories. This iteration continues +until a convergence criterion is satisfied. +To be more specific, we approximate ˆ𝐿(𝜃,𝑥1) by a new loss +function ˜𝐿(𝜃,𝑥1) defined as, +ˆ𝐿(𝜃,𝑥1) ≃ ˜𝐿(𝜃,𝑥1) := −𝑝 �yT|x(˜f𝜃, ˜g𝜃,𝑥1)� +(15) +where x(˜f𝜃, ˜g𝜃,𝑥1) represents a FBNE state trajectory from initial +condition 𝑥1, for the LQ game defined by the linearized dynamics ˜f𝜃, +quadraticized cost set ˜g𝜃 := {˜g𝑖 +𝑡,𝜃 }𝑇,𝑁 +𝑡=1,𝑖=1 at the converged solution +returned by ILQGames solver. Note that the linearized dynamics ˜f𝜃 +depend upon 𝜃 via the state trajectory about which f is linearized; +this trajectory is simulated under the feedback policy returned by +ILQGames, where the policy depends upon costs g𝜃. +5.3 +Differentiating the Loss in the Inverse +Feedback Game Problem +The challenge of computing a feedback Nash equilibrium strategy +not only makes the evaluation of the loss function ˆ𝐿(𝜃,𝑥1) hard, but +also renders differentiation difficult. In this work, we approximate +the gradient of ˆ𝐿(𝜃,𝑥1) using a similar idea as the ILQGames algo- +rithm in the previous section. In other words, we propose to use +the LQ approximation of the nonlinear game specified by ˜f𝜃 and +˜g𝜃 to derive an approximation to the gradient of ˆ𝐿(𝜃,𝑥1). Note that +˜𝑔𝑖 +𝑡,𝜃 (𝑥,𝑢) = �𝑑𝑖 +𝑗=1 𝜃𝑖 +𝑗 ˜𝑏𝑖 +𝑡,𝑗,𝜃 (𝑥,𝑢), where ˜𝑏𝑖 +𝑡,𝑗,𝜃 (𝑥,𝑢) : R𝑛 × R𝑚 → R +is the 𝑗-th quadraticized cost basis function. By the chain rule, we +have +𝜕 ˜𝐿(𝜃,𝑥1) +𝜕𝜃𝑖 +𝑗 += −∇x𝑝(yT|x) +���x(˜f𝜃,˜g𝜃,𝑥1) · 𝜕x(˜f𝜃, ˜g𝜃,𝑥1) +𝜕𝜃𝑖 +𝑗 +, +𝜕x(˜f𝜃, ˜g𝜃,𝑥1) +𝜕𝜃𝑖 +𝑗 += +� +∇˜f𝜃 x(˜f𝜃, ˜g𝜃,𝑥1) 𝜕˜f𝜃 +𝜕𝜃𝑖 +𝑗 ++ ∇˜g𝜃 x(˜f𝜃, ˜g𝜃,𝑥1) 𝜕˜g𝜃 +𝜕𝜃𝑖 +𝑗 +� +. + +No regularization +×10-4 +No regularization +0.8 +(0, α1) +1 +0.6 +(Iα‘)T +0.4 +0.5 +0.2 +0 +0.5 +1 +1.5 +0.4 +0.6 +0.8 +1 +1.2 +Q1 +Q1 +L2 regularization +×10~4 +L2 regularization +[z‘]l-01+(I") +0.8 +0.6 +0.4 +0.5 +0.2 +0.5 +0 +1 +1.5 +0 +0.4 +0.6 +0.8 +1 +1.2 +Q1 +0 +Q1The complexity of differentiating ˜𝐿(𝜃,𝑥1) comes from the fact that +the linearized dynamics and the quadraticized costs are functions +of 𝜃 implicitly, which makes the total derivative hard to compute. +We propose to approximate the above gradient by treating the +linearized ˜f𝜃 and each quadraticized cost basis function ˜𝑏𝑖 +𝑡,𝑗,𝜃 as +constants with respect to 𝜃, denoted by ˜f and ˜𝑏𝑖 +𝑡,𝑗, and only com- +pute the partial derivative with respect to 𝜃, rather than the total +derivative: +𝜕 ˜𝐿(𝜃,𝑥1) +𝜕𝜃𝑖 +𝑗 +≃ −∇x𝑝(yT|x) +���x(˜f,˜g𝜃,𝑥1)· +𝜕x(˜f, {�𝑑𝑖 +𝑗=1 𝜃𝑖 +𝑗 ˜𝑏𝑖 +𝑡,𝑗 }𝑇,𝑁 +𝑡=1,𝑖=1,𝑥1) +𝜕𝜃𝑖 +𝑗 +. +This is based on the observation that at the convergence of the +forward ILQGames solver, the linearized dynamics are a good ap- +proximation of the full nonlinear dynamics f, so long as the cost +parameter being perturbed remains sufficiently small. We adopt a +similar approximation for the gradient ∇𝑥1 ˜𝐿(𝜃,𝑥1) by fixing the lin- +earized dynamics and quadraticized costs and obtaining the partial +derivative with respect to 𝑥1. +In summary, we approximate ∇ ˆ𝐿(𝜃,𝑥1) by ∇ ˜𝐿(𝜃,𝑥1). In practice, +∇ ˜𝐿(𝜃,𝑥1) can be efficiently computed by automatic differentiation +[27, Ch. 8]. As exemplified in Fig. 4, the proposed gradient approxi- +mation is virtually always a descent direction and therefore aligns +well with the true gradient of ˆ𝐿(𝜃,𝑥1). +5.4 +An Inverse Feedback Game Solver +In this subsection, we present a solver for the inverse feedback +game problem (11). In what follows, we first discuss how the three +challenges mentioned in Section 4 are handled in our solver. We +then introduce the proposed solver in Algorithm 1. +The first two challenges on noisy partial observation and in- +complete trajectory data are handled by maintaining an estimate +of the full initial condition and a noise-free state-input trajectory. +As shown in Section 6, this procedure of joint reconstruction and +filtering enables our solver to reliably recover player costs even in +scenarios of substantial partial observability. The third difficulty of +evaluating and differentiating the objective function in the inverse +feedback game problem is mitigated by the efficient approximation +outlined in Section 5.3. To jointly infer the initial condition, the +cost and the state-input trajectory, we first adopt the coordinate +gradient descent method, where gradient descent steps are first +taken over the initial condition ˆ𝑥1, and then taken over the cost +parameter. We update the estimate of the noise-free full state-input +trajectory by computing a FBNE state trajectory from the inferred +initial condition and the cost. +We summarize our proposed solver in Algorithm 1. At the 𝑘-th it- +eration, we first compute an approximate FBNE state trajectory ˜𝑥 (𝑘) +and the associated LQ approximation via the ILQGames algorithm +of [8]. Using this LQ approximation, we estimate ∇𝑥1 ˆ𝐿(𝜃,𝑥 (𝑘) +1 +) us- +ing the procedure outlined in Section 5.3. We then update the initial +condition 𝑥 (𝑘) +1 +by a step of gradient descent, where the stepsize +is chosen by a suitable linesearch technique [27, Ch. 3] such that +the loss ˆ𝐿(𝜃,𝑥1) is sufficiently decreased. Given the updated initial +condition 𝑥 (𝑘+1) +1 +, we find a new approximate FBNE state trajectory +Algorithm 1: Inverse Iterative LQ (i2LQ) Games +Data: Horizon 𝑇 > 0, initial solution 𝜃 (0) ∈ R𝑑, observed time +index set T ⊆ [𝑇 ], observation data yT, max iteration +number 𝐾 , tolerance 𝜖. +Result: Inferred cost parameter ˆ𝜃 and ˆ𝑥1 +1 for 𝑘 = 0, 1, . . . , 𝐾 do +2 +( ˜x(𝑘), { ˜𝛾𝑖 +𝑡 }𝑇,𝑁 +𝑡=1,𝑖=1, ˜f𝜃 (𝑘) , ˜g𝜃 (𝑘) ) ← ILQGames(f, g𝜃 (𝑘) ,𝑥 (𝑘) +1 +) +3 +∇𝑥1 ˆ𝐿(𝜃 (𝑘),𝑥 (𝑘) +1 +) ← evaluated using ˜f𝜃 (𝑘) and ˜g𝜃 (𝑘) via +Gradient Approximation in Section 5.3 +4 +𝑥 (𝑘+1) +1 +← 𝑥 (𝑘) +1 +− 𝜂∇𝑥1 ˆ𝐿(𝜃 (𝑘),𝑥 (𝑘) +1 +) with line search over 𝜂 +5 +( ˇ𝑥 (𝑘), {ˇ𝛾𝑖 +𝑡 }𝑇,𝑁 +𝑡=1,𝑖=1, ˇf𝜃 (𝑘) , ˇg𝜃 (𝑘) ) ← ILQGames�f, g𝜃 (𝑘) ,𝑥 (𝑘+1) +1 +� +6 +∇𝜃 ˆ𝐿(𝜃 (𝑘),𝑥 (𝑘+1) +1 +) ← evaluated using ˇf𝜃 (𝑘) and ˇg𝜃 (𝑘) via +Gradient Approximation in Section 5.3 +7 +𝜃 (𝑘+1) ← 𝜃 (𝑘) − 𝜂′∇𝜃 ˆ𝐿(𝜃 (𝑘),𝑥 (𝑘+1) +1 +) with line search over 𝜂′ +8 +Return (𝜃 (𝑘+1),𝑥 (𝑘+1) +1 +) if ∥𝜃 (𝑘) − 𝜃 (𝑘−1) ∥2 ≤ 𝜖 or Return +(𝜃 (𝑘′),𝑥 (𝑘′) +1 +), where 𝑘′ ← arg min𝑘 ˜𝐿(𝜃 (𝑘),𝑥 (𝑘) +1 +), if +iteration number 𝑘 reaches 𝐾. +9 end +via the ILQGames algorithm again, which is then used to estimate +∇𝜃 ˆ𝐿(𝜃 (𝑘),𝑥 (𝑘+1) +1 +) via the procedure in Section 5.3. With this gradi- +ent, we update 𝜃 (𝑘) by one step of gradient descent with linesearch. +We repeat this procedure until, at convergence, we find a locally +optimal solution ( ˆ𝜃, ˆ𝑥1). +6 +EXPERIMENTS +In this section, we adopt the open-loop solution method of [28] as +the baseline method and compare it to Algorithm 1. In particular, +we evaluate Algorithm 1 in several Monte Carlo studies which aim +to justify the following claims. +• The proposed gradient approximation often aligns with a +descent direction in the loss function. +• Algorithm 1 is more robust than the open-loop baseline +method [28] with respect to noise in, and incomplete obser- +vations of, the expert demonstration trajectory. +• The cost functions inferred by Algorithm 1 can be general- +ized to predict trajectories from unseen initial conditions. +• Algorithm 1 can infer nonconvex costs in nonlinear games. +6.1 +Gradient Approximation Quality +We continue the 2-vehicle platooning example defined in (12) and +(13). We measure the performance of Algorithm 1 in two settings, in- +complete expert trajectory data with noisy partial state observation, +and complete expert trajectory data with noisy full observation. +In the first case, each player’s partial observation only contains +its x-position, y-position and heading angle. The time index set +of the incomplete trajectory is T = [𝑇] \ {11, 12, . . . , 19}. In the +second case, the expert data includes the noisy observation of all +the states of both players at all 𝑡 ∈ [𝑇]. The ground truth expert +state trajectory follows a FBNE strategy from the initial condi- +tion 𝑥1 = [0, 0.5, 𝜋 +2 , 1, 1, 0, 𝜋 +2 , 1] and the target lane is 𝑝∗𝑥 = 0.0. +At each variance level 𝜎 ∈ {0.004, 0.008, . . . , 0.04}, we generate +10 noisy observations of the ground truth expert trajectory, with +isotropic zero-mean Gaussian noise. For each noisy expert data set + +yT, we minimize the negative log-likelihood objective in (11), i.e., +� +𝑡 ∈T ∥𝑦𝑡 − ℎ(𝑥𝑡)∥2 +2, where ℎ(·) : R𝑛 → Rℓ maps a state 𝑥𝑡 to its +partial observation. +As shown in Fig. 4, the loss decreases monotonically on the +average. This indicates that the gradient approximation proposed +in Section 5.3 provides a reliable descent direction. The inverse +feedback game problem becomes challenging when there is only +partial state observation and incomplete trajectory data, and the +quality of inferred costs may degrade when the observation noise +is high. +6.2 +Robustness, Generalization and the Ability +to Infer Nonconvex Costs +We continue the previous 2-vehicle example and compare Algo- +rithm 1 and the baseline in a Monte Carlo study, where we infer the +costs under 10 different levels of Gaussian noise with increasing +variance. In particular, we evaluate three metrics in Fig. 5: (a) the +distance between the noisy expert data and the FBNE state trajec- +tory which results from players’ inferred costs; (b) the distance +between the computed FBNE state trajectory (under the players’ +inferred costs) and the ground truth expert data. An example of +such a comparison is shown in Fig. 6. Finally, we evaluate (c) the +distance between the inferred FBNE state trajectories and the FBNE +state trajectory under the ground truth costs for some randomly +sampled initial conditions, which is also visualized in Fig. 7. Collec- +tively, the results demonstrate that Algorithm 1 has better robustness +and generalization performance than the open-loop baseline when +the expert data follows the FBNE assumption. +To show that Algorithm 1 can infer nonconvex cost functions, we +extend the previous 2-vehicle platooning example and assume that +the 2-vehicle team encounters a third vehicle and the follower wants +to stay close to the leader without colliding with the third vehicle. +We model this scenario as a 3-vehicle game with a 12 dimensional +state space and a horizon 𝑇 = 30. The dynamics for each vehicle is +the same as (12) and the costs are as follows, +𝑔1 +𝑡,𝜃 (𝑥𝑡,𝑢𝑡) =𝜃1 +1 ∥𝑝1 +𝑥,𝑡 ∥2 +2 + 𝜃1 +2 ∥𝑝2 +𝑥,𝑡 − 𝑝∗ +𝑥 ∥2 +2 + ∥𝑣1 +𝑡 − 2∥2 +2 ++ ∥𝛽1 +𝑡 − 𝜋 +2 ∥2 +2 + ∥𝑢1 +𝑡 ∥2 +2 +𝑔2 +𝑡,𝜃 (𝑥𝑡,𝑢𝑡) =𝜃2 +1 ∥𝑝2 +𝑥,𝑡 ∥2 +2 + ∥𝛽2 +𝑡 − 𝜋 +2 ∥2 +2 + 𝜃2 +2 ∥𝑝2 +𝑥,𝑡 − 𝑝1 +𝑥,𝑡 ∥2 +2 + ∥𝑣2 +𝑡 − 2∥2 +2 +− 1 +2 log(∥𝑝2 +𝑥,𝑡 − 𝑝3 +𝑥,𝑡 ∥2 +2 + ∥𝑝2 +𝑦,𝑡 − 𝑝3 +𝑦,𝑡 ∥2 +2) + ∥𝑢2 +𝑡 ∥2 +2 +𝑔3 +𝑡,𝜃 (𝑥𝑡,𝑢𝑡) =𝜃3 +1 ∥𝑝3 +𝑥,𝑡 − 1 +2 ∥2 +2 + ∥𝑢3 +𝑡 ∥2 +2 +where the ground truth 𝜃∗ ∈ R5 is [0, 4, 0, 4, 2]. The ground truth +expert state trajectory follows a FBNE strategy from the initial +condition 𝑥1 = [0, 1, 𝜋 +2 , 2, 0.3, 0, 𝜋 +2 , 2, 0.5, 0.5, 𝜋 +2 , 2], where the last +four elements encode the state of the third vehicle. The target lane +in the expert data is 𝑝∗𝑥 = 0.2. +Similar to the 2-vehicle experiment, we consider two settings, +incomplete trajectory data with partial state observation and com- +plete trajectory data with full state observation. The partial state +observation includes all the states of each vehicle except for the +velocity of all the vehicles, and the time indices set of the incom- +plete trajectory is T = [𝑇] \ {11, 12, . . . , 19}. The nonconvex cost +of player 2 causes numerical problems in the baseline KKT OLNE +Fig. 4: Convergence of Algorithm 1 with the Gradient Approxima- +tion proposed in Section 5.3. The loss decreases monotonically on +the average. The bold lines and shaded areas represent the mean val- +ues and their standard error, i.e., the variance divided by the square +root of the sample size, respectively. +Fig. 5: 2-vehicle platooning scenario. The bold lines and shaded ar- +eas represent the mean values and their standard error, i.e., the vari- +ance divided by the square root of the sample size, respectively. As +the noise variance growing, the converged loss value increases, as +shown in the red curves. However, Algorithm 1 is still able to learn +a more accurate cost and has less generalization error than the base- +line, as shown in the blue and yellow curves, respectively. +Fig. 6: Full and partial, noisy observation of the expert trajectories. +Dashed lines represent predicted trajectories which result from in- +ferred costs, and solid lines are ground truth. The trajectories pre- +dicted by Algorithm 1 are closer to the ground truth than the base- +line. +solver [28]. Thus, we add an 𝐿2 regularization 10−4∥𝜃 ∥2 +2 to the loss +ˆ𝐿(𝜃,𝑥1) and summarize the Monte Carlo study in Fig. 8, where we +see Algorithm 1 is also able to learn better cost functions reflecting +the true intentions of each vehicle in feedback games, even with +only partial state observations and incomplete trajectory data. + +Fig. 7: Generalization performance comparison. 𝑝∗𝑥 is the target lane +position that player 1 wants to guide player 2 toward. All the costs +are inferred from partial observations and incomplete trajectory +data, with different noise variance specified in each of the subplot. +The trajectories predicted by Algorithm 1 are closer to the ground +truth than the baseline. +Fig. 8: 3-vehicle platooning scenario. The bold lines and shaded ar- +eas represent the mean values and their standard error, i.e., the vari- +ance divided by the square root of the sample size, respectively. As +the noise variance growing, the converged loss value increases on +the average, as shown in the red curves. However, Algorithm 1 is +still able to learn a more accurate cost and has less generalization +error than the baseline, as shown in the blue and yellow curves, re- +spectively. +7 +CONCLUSION +In this work, we propose an efficient cost inference algorithm for +inverse feedback nonlinear games, with only partial state observa- +tion and incomplete trajectory data. Empirical results show that +the proposed solver converges reliably for inverse games with non- +convex costs and has superior generalization performance than a +state-of-the-art open-loop baseline method when the expert demon- +stration reflects a group of agents acting in a dynamic feedback +game. There are many future directions. We can investigate under +what conditions the cost can be inferred exactly in feedback games. +The active and online inference are also promising directions. In +addition, we are eager to extend this work to settings of closed-loop +interaction. In such an extension, rather than merely inferring the +objectives of observed players, this information would be used to +guide the decision-making of an autonomous agent in that scene. +APPENDIX +Proof of Proposition 1. Proposition 1 claims that there exists +an inverse LQ game, which has isolated global minima and the +induced FBNE state trajectories of those solutions match the expert +demonstration. Here, we show such a counterexample, which sup- +ports the claim. Consider a 2-player horizon-3 LQ game with the +linear dynamics +𝑥𝑡+1 = 𝑥𝑡 + 𝑢1 +𝑡 + 𝑢2 +𝑡 , 𝑡 ∈ {1, 2, 3}, +(16) +and the cost +𝑔1 +𝑡 (𝑥𝑡,𝑢𝑡) = 1 +2 (𝑄1∥𝑥𝑡 ∥2 +2 + ∥𝑢1 +𝑡 ∥2 +2), 𝑡 ∈ {1, 2}, +𝑔2 +𝑡 (𝑥𝑡,𝑢𝑡) = 1 +2 (𝑄2∥𝑥𝑡 ∥2 +2 + 2∥𝑢2 +𝑡 ∥2 +2), 𝑡 ∈ {1, 2}, +𝑔1 +3(𝑥3,𝑢3) = 1 +2𝑄1∥𝑥3∥2 +2, 𝑔2 +3(𝑥3,𝑢3) = 1 +2𝑄2∥𝑥3∥2 +2. +(17) +We assume that the ground truth solutions are 𝑄1 = 1, 𝑄2 = 1. +We will show there is also one extra solution ˆ𝑄1 = 1 +2 and ˆ𝑄2 = 2, +which yields the same FBNE state trajectory as the ground truth for +any initial condition. We follow the same definition of the variable +{𝑍𝑖 +𝑡 }3,2 +𝑡=1,𝑖=1 as in [2]. By definition, we have𝑍𝑖 +𝑡 ≥ 𝑄𝑖 > 0, when𝑄1 ∈ +R+ and 𝑄2 ∈ R+. Following the notations in FBNE condition in +Corollary 6.1 of [2], we consider the feedback matrices {𝑃𝑖 +𝑡 }2,2 +𝑡=1,𝑖=1, +�𝑃1 +𝑡 +𝑃2 +𝑡 +� += +�1 + 𝑍1 +𝑡+1 +𝑍1 +𝑡+1 +𝑍2 +𝑡+1 +2 + 𝑍2 +𝑡+1 +� +������������������������������������������������ +𝐺𝑖 +𝑡 +�𝑍1 +𝑡+1 +𝑍2 +𝑡+1 +� +, ∀𝑡 ∈ {1, 2}, +(18) +where the matrix 𝐺𝑖 +𝑡 is invertible because det(𝐺𝑖 +𝑡) = 2 + 𝑍2 +𝑡+1 + +2𝑍1 +𝑡+1 > 0. The above analysis suggests that the FBNE state tra- +jectory for all 𝑄1 > 0 and 𝑄2 > 0 are uniquely determined. We +consider the time instant 𝑡 = 2, and observe +�𝑃1 +2 +𝑃2 +2 +� += +�1 + 𝑄1 +𝑄1 +𝑄2 +2 + 𝑄2 +�−1�𝑄1 +𝑄2 +� += +1 +2 + 2𝑄1 + 𝑄2 +�2𝑄1 +𝑄2 +� +. +(19) +We then have the closed-loop dynamics 𝑥3 = (1 − 𝑃1 +2 − 𝑃2 +2)𝑥2 = +2 +2+2𝑄1+𝑄2 𝑥2, which yields that for two pairs of positive variables +(𝑄1,𝑄2) and ( ˆ𝑄1, ˆ𝑄2), a necessary condition for them to have the +same FBNE trajectory is that 2𝑄1 + 𝑄2 = 2 ˆ𝑄1 + ˆ𝑄2. We have +𝑍1 +2 = 𝑄1 + +𝑄1+(2𝑄1)2 +(2+2𝑄1+𝑄2)2 , 𝑍2 +2 = 𝑄2 + +𝑄2+2(𝑄2)2 +(2+2𝑄1+𝑄2)2 . Similarly, for +the time instant 𝑡 = 1, we have 𝑥2 = (1 − 𝑃1 +1 − 𝑃2 +1)𝑥1 = +2 +2+2𝑍 1 +2+𝑍 2 +2 𝑥1. +A necessary condition for ( ˆ𝑄1, ˆ𝑄2) to have the same FBNE state +trajectory as (𝑄1,𝑄2) is that the following 2 equations are satisfied, +2𝑄1 + 𝑄2 = 2 ˆ𝑄1 + ˆ𝑄2 +2�𝑄1 + +𝑄1 + (2𝑄1)2 +(2 + 2𝑄1 + 𝑄2)2 +� + 𝑄2 + +𝑄2 + 2(𝑄2)2 +(2 + 2𝑄1 + 𝑄2)2 += 2� ˆ𝑄1 + +ˆ𝑄1 + (2 ˆ𝑄1)2 +(2 + 2 ˆ𝑄1 + ˆ𝑄2)2 +� + ˆ𝑄2 + +ˆ𝑄2 + 2( ˆ𝑄2)2 +(2 + 2 ˆ𝑄1 + ˆ𝑄2)2 . +(20) +We substitute 𝑄1 = 1, 𝑄2 = 1 and ˆ𝑄2 = 3−2 ˆ𝑄1 into the second row +of (20), which is reduced to a 2-degree polynomial of ˆ𝑄2. By the +fundamental theorem of algebra [4], there exist at most 2 solutions +for ˆ𝑄2. The two pairs of ( ˆ𝑄1, ˆ𝑄2) satisfying (20) are (1, 1) and ( 1 +2, 2). +The two global minima are isolated. Since the dimension of the + +state 𝑥𝑡 is 1, for all initial states 𝑥1 ∈ R, the FBNE state trajectories +under the costs specified by the two pairs cost parameters (1, 1) +and ( 1 +2, 2) coincide with each other. +□ +REFERENCES +[1] Chaitanya Awasthi and Andrew Lamperski. Inverse differential games +with mixed inequality constraints. In 2020 American control conference +(ACC), pages 2182–2187. IEEE, 2020. +[2] Tamer Başar and Geert Jan Olsder. Dynamic Noncooperative Game +Theory. SIAM, 1998. +[3] Stephen Boyd, Stephen P Boyd, and Lieven Vandenberghe. Convex +optimization. Cambridge university press, 2004. +[4] Augustin-Louis Cauchy. Cours d’analyse de l’ecole royale polytech- +nique, 1re partie. Analyse algébrique. 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Automatica, + +145:110534, 2022. + diff --git a/qNAzT4oBgHgl3EQfcPwr/content/tmp_files/load_file.txt b/qNAzT4oBgHgl3EQfcPwr/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a63ca8dc96adb54df30c8d928831605e1f2ae0e6 --- /dev/null +++ b/qNAzT4oBgHgl3EQfcPwr/content/tmp_files/load_file.txt @@ -0,0 +1,592 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf,len=591 +page_content='Cost Inference for Feedback Dynamic Games from Noisy Partial State Observations and Incomplete Trajectories Jingqi Li University of California, Berkeley Berkeley, United States jingqili@berkeley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='edu Chih-Yuan Chiu University of California, Berkeley Berkeley, United States chihyuan_chiu@berkeley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='edu Lasse Peters Delft University of Technology Delft, Netherlands l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='peters@tudelft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='nl Somayeh Sojoudi University of California, Berkeley Berkeley, United States sojoudi@berkeley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='edu Claire Tomlin University of California, Berkeley Berkeley, United States tomlin@eecs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='berkeley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='edu David Fridovich-Keil University of Texas, Austin Austin, United States dfk@utexas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='edu ABSTRACT In multi-agent dynamic games, the Nash equilibrium state trajec- tory of each agent is determined by its cost function and the infor- mation pattern of the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' However, the cost and trajectory of each agent may be unavailable to the other agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Prior work on using partial observations to infer the costs in dynamic games assumes an open-loop information pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' In this work, we demonstrate that the feedback Nash equilibrium concept is more expressive and encodes more complex behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' It is desirable to develop specific tools for inferring players’ objectives in feedback games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Therefore, we consider the dynamic game cost inference problem under the feedback information pattern, using only partial state observations and incomplete trajectory data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' To this end, we first propose an inverse feedback game loss function, whose minimizer yields a feedback Nash equilibrium state trajectory closest to the observa- tion data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' We characterize the landscape and differentiability of the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Given the difficulty of obtaining the exact gradient, our main contribution is an efficient gradient approximator, which enables a novel inverse feedback game solver that minimizes the loss using first-order optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' In thorough empirical evalua- tions, we demonstrate that our algorithm converges reliably and has better robustness and generalization performance than the open-loop baseline method when the observation data reflects a group of players acting in a feedback Nash game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' KEYWORDS Inverse Games, Dynamic Game Theory, Nash Equilibrium ACM Reference Format: Jingqi Li, Chih-Yuan Chiu, Lasse Peters, Somayeh Sojoudi, Claire Tomlin, and David Fridovich-Keil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Cost Inference for Feedback Dynamic Games from Noisy Partial State Observations and Incomplete Trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' of the 22nd International Conference on Autonomous Agents and Multiagent Systems (AAMAS 2023), London, United Kingdom, May 29 – June 2, 2023, IFAAMAS, 10 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' 1 INTRODUCTION The safety and efficiency of urban traffic relies heavily on the ability of each participant to predict the effects of their actions on others’ Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' of the 22nd International Conference on Autonomous Agents and Multiagent Sys- tems (AAMAS 2023), A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Ricci, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Yeoh, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Agmon, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' An (eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' ), May 29 – June 2, 2023, London, United Kingdom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' © 2023 International Foundation for Autonomous Agents and Multiagent Systems (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='ifaamas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' decisions [23, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' For example, drivers on a highway may wish to halt an overtaking maneuver if they believe the other drivers are aggressive, and some drivers may decelerate their cars to avoid collision if they believe that another driver wishes to merge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' A powerful paradigm for modeling the interdependence of deci- sions in multi-agent settings is provided general-sum dynamic games [2, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' A Nash equilibrium solution of a game-theoretic model can be used to simultaneously predict the actions of all agents in the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' This equilibrium solution is particularly expressive when the game possesses a feedback information structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' In this case, each equilibrium strategy explicitly accounts for the dynamically evolving information available to each player over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Despite the theoretical attractiveness of this modeling paradigm, in reality, autonomous agents often have only limited information available about the world around them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' For example, in urban traffic an autonomous agent typically has incomplete knowledge of the objectives of other players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' To address this challenge, recent works on inverse dynamic game theory [21, 28, 31] recover these objectives from past trajectory data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Moreover, in realistic applications, only noisy sensor measurements of agents’ states are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' This partial observability further complicates the inverse game problem, and existing work [28] treats this case in the open-loop information structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' In this work, we present a gradient-based solver for inverse dynamic games, under the state feedback information structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Our solver can recover objectives from partial state observations of incomplete trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Both of these effects are common in robotics due to noisy perception and occlusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' We show that our algorithm converges reliably in practice, and demonstrate the superior robustness and generalization performance as compared with a baseline method which learns cost functions under the open- loop assumption [28], when the observation data is from a group of players pursuing a feedback Nash equilibrium strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Our contributions are threefold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Firstly, we characterize the solu- tion set of the inverse feedback dynamic game problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' In particu- lar, we show that the set of the global minima could be nonconvex and disconnected, and discuss regularization schemes to mitigate this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Secondly, we show the differentiability of the loss function in linear quadratic games and propose a computationally efficient procedure to approximate the gradient for nonlinear games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Finally, we propose an efficient first-order coordinate-descent solver arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='01398v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='MA] 4 Jan 2023 for the inverse feedback game problem, using noisy partial obser- vations of an incomplete expert state trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Experimental re- sults show that our method reliably converges for inverse feedback games with nonlinear dynamics and is able to learn nonconvex costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Moreover, the converged cost function can accurately predict the feedback Nash equilibrium state trajectories even for unseen initial states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' 2 RELATED WORK 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='1 Non-cooperative Dynamic Games Non-cooperative dynamic game theory [2, 13] provides a formal framework for analyzing strategic interaction in a multi-agent set- ting [2, 6, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' In non-cooperative games, each player minimizes its own individual cost function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' since players’ costs may not be mutually aligned, the resulting equilibrium behavior is generally competitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Among different equilibrium concepts, the Nash equi- librium has been extensively studied because of its representative power of capturing many non-cooperative behaviors arising in real-world multi-agent systems [9, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Recent advances in the literature aim to develop efficient solu- tions to Nash equilibrium problems in dynamic games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Though the solutions for the open-loop and feedback Nash equilibrium in linear quadratic (LQ) games are well understood [2], for nonlinear games there is no closed-form solution in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' The work [30] characterizes the local Nash solution concept for open-loop Nash equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' In the feedback setting, numerous approaches have been proposed under various special cases [14, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' A value iter- ation based approach for computing feedback Nash equilibria of nonlinear games without constraints is introduced in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Recently, a set of KKT conditions for feedback Nash equilibria in constrained nonlinear games is derived in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Computing a feedback Nash equilibrium is challenging due to the nested KKT conditions in different time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Our work draws upon the ILQGames [8] framework, which at each iteration solves a linear-quadratic game that approximates the original game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' The construction of the approximate game parallels the iterative linearization and quadraticization methods of iterative LQR [18], and the dynamic programming equations that charac- terize equilibrium strategies in linear quadratic dynamic games [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' This approach differs from the ALGames [5] method, which computes an open-loop Nash equilibrium strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='2 Inverse Non-cooperative Dynamic Games In contrast to the forward game problem of computing a strategy in dynamic games, an inverse game problem amounts to finding objectives for all agents such that the corresponding strategic (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=', Nash equilibrium) interactions reproduce expert demonstrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' The inverse game problem is important because it paves the way for an agent to understand the preferences which explain other agents’ behavior, which may facilitate more efficient multi-agent interaction and coordination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' The problem of inverse infinite-horizon LQ games is considered in [12], where the set of cost functions whose feedback Nash equi- librium strategies coincide with an expert strategy is derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' In [31, 36], the two-player inverse LQ game is solved by transforming the problem to an inverse optimal control under the assumption that the control input data of one player is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Two methods based on the KKT conditions of an open-loop Nash equilibrium are proposed for open-loop general-sum differential games in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Several necessary conditions for open-loop Nash equilibria are pro- posed in [22] and used for developing an inverse game solution for some classes of open-loop games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Recently, an efficient bilevel optimization framework [28] based on the open-loop Nash equilibrium KKT conditions was proposed for solving inverse games with an open-loop Nash assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Another line of work on inferring costs in open-loop games [1, 7, 11] proposes to minimize the residual violation of the KKT conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' This KKT residual framework assumes the knowledge of complete trajectory data and is a convex problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Given the difficulty of evaluating KKT conditions for feedback Nash equilibria in nonlinear games [16], the extension of the KKT residual method to feedback nonlinear games may be subject to numerical difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' A bilevel optimization approach for inverse feedback game prob- lem is proposed in [25], with the assumption that both the expert state and control trajectories are observed without noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' In addi- tion, an inverse game solver is proposed in [20] where they infer the players’ cost functions with the assumption that the expert strategy follows a new concept called Maximum Entropy Nash Equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' To the best of the authors’ knowledge, there is no work on inferring cost functions of nonlinear dynamic games under feedback Nash equilibrium condition, from noisy partial state observation and incomplete trajectory data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' 3 PRELIMINARIES Consider an 𝑁-player,𝑇-stage, deterministic, discrete-time dynamic game, with a state 𝑥𝑖 𝑡 ∈ R𝑛𝑖 and control input 𝑢𝑖 𝑡 ∈ R𝑚𝑖 for each player 𝑖 ∈ [𝑁] := {1, · · · , 𝑁 }, 𝑡 ∈ [𝑇].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Let the dimension of the joint state and control input be 𝑛 := �𝑁 𝑖=1 𝑛𝑖 and 𝑚 := �𝑁 𝑖=1 𝑚𝑖, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' We denote by 𝑥𝑡 := [𝑥1 𝑡 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' ,𝑥𝑁 𝑡 ] ∈ R𝑛 and 𝑢𝑡 := [𝑢1 𝑡 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' ,𝑢𝑁 𝑡 ] ∈ R𝑚 the joint state and joint control at time 𝑡 ∈ [𝑇], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' The joint dynamics for the system is given by the differentiable dynamics map 𝑓𝑡 (·, ·) : R𝑛 × R𝑚 → R𝑛: 𝑥𝑡+1 = 𝑓𝑡 (𝑥𝑡,𝑢𝑡), ∀𝑡 = 1, · · · ,𝑇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' (1) We denote by f := {𝑓𝑡 }𝑇 𝑡=1 the set of dynamics across all the time instances within horizon𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' We define x := {𝑥𝑡 }𝑇 𝑡=1 and u := {𝑢𝑡 }𝑇 𝑡=1 to be a state trajectory and control trajectory, respectively, if 𝑥𝑡+1 = 𝑓 (𝑥𝑡,𝑢𝑡), for each 𝑡 ∈ [𝑇].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' The objective of each agent 𝑖 is to minimize its overall cost, given by the sum of its running costs 𝑔𝑖 𝑡 : R𝑛 × R𝑚 → R over the time horizon: 𝐽𝑖 (x, u) := 𝑇 ∑︁ 𝑡=1 𝑔𝑖 𝑡 (𝑥𝑡,𝑢𝑡) (2) Define 𝑔𝑡 := {𝑔1 𝑡 ,𝑔2 𝑡 , · · · ,𝑔𝑁 𝑡 }, 𝑡 ∈ [𝑇].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' We denote by g := {𝑔𝑡 }𝑇 𝑡=1 the set of cost functions for all the agents within horizon 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' To minimize (2), each player uses their observations of the envi- ronment to design a sequence of control inputs to deploy during the discrete time interval [𝑇].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' The information available to each player at each time characterizes the information pattern of the dynamic game, and plays a major role in shaping the optimal re- sponses of each player [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Below, we explore two such information patterns—feedback and open-loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='1 Nash Solutions in Feedback Strategies Under the state feedback information pattern, each player observes the state 𝑥𝑡 at each time 𝑡, and uses this information to design a feedback strategy 𝛾𝑖 𝑡 : R𝑛 → R𝑚𝑖 , given by: 𝑢𝑖 𝑡 := 𝛾𝑖 𝑡 (𝑥𝑡), for each 𝑖 ∈ [𝑁] and 𝑡 ∈ [𝑇].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Let 𝛾𝑡 (𝑥𝑡) := [𝛾1 𝑡 (𝑥𝑡),𝛾2 𝑡 (𝑥𝑡), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' ,𝛾𝑁 𝑡 (𝑥𝑡)] ∈ R𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Following the notation of [2], we denote by Γ𝑖 𝑡 the set of all state feedback strategies of player 𝑖, for each 𝑖 ∈ [𝑁].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Under this feedback information pattern, the Nash equilibrium of the dynamic game is as defined below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Definition 1 (Feedback Nash Eqilibrium (FBNE) [2, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' 6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' The set of control strategies {𝛾1∗ 𝑡 , · · · ,𝛾𝑁∗ 𝑡 }𝑇 𝑡=1 is called a feedback Nash equilibrium if no player is incentivized to unilaterally alter its strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Formally: 𝑊 𝑖∗ 𝑡 � 𝑥𝑡, [𝛾1∗ 𝑡 (𝑥𝑡), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' ,𝛾𝑖∗ 𝑡 (𝑥𝑡), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' ,𝛾𝑁∗ 𝑡 (𝑥𝑡)] � (3) ≤ 𝑊 𝑖∗ 𝑡 � 𝑥𝑡, [𝛾1∗ 𝑡 (𝑥𝑡), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' ,𝛾𝑖 𝑡 (𝑥𝑡), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' ,𝛾𝑁 ∗ 𝑡 (𝑥𝑡)] � , ∀𝛾𝑖 𝑡 ∈ Γ𝑖 𝑡 , ∀𝑡 ∈ [𝑇].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' where 𝑊 𝑖∗ 𝑡 (·, ·) : R𝑛 × R𝑚 → R, 𝑡 ∈ [𝑇] is the optimal state-action function defined as follows, 𝑊 𝑖∗ 𝑇 (𝑥𝑇,𝑢𝑇 ) := 𝑔𝑖 𝑇 (𝑥𝑇,𝑢𝑇 ) 𝑊 𝑖∗ 𝑡 (𝑥𝑡,𝑢𝑡) := 𝑔𝑖 𝑡 (𝑥𝑡,𝑢𝑡) + 𝑉 𝑖∗ 𝑡+1(𝑥𝑡+1), ∀𝑡 ∈ [𝑇 − 1], 𝑉 𝑖∗ 𝑡 (𝑥𝑡) := 𝑊 𝑖∗ 𝑡 (𝑥𝑡, [𝛾1 𝑡 ∗(𝑥𝑡), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' ,𝛾𝑁 𝑡 ∗(𝑥𝑡)]), ∀𝑡 ∈ [𝑇].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' (4) We define x and u to be a FBNE state trajectory and a FBNE control trajectory, respectively, if𝑢𝑖 𝑡 = 𝛾𝑖∗ 𝑡 (𝑥𝑡), for each 𝑖 ∈ [𝑁] and 𝑡 ∈ [𝑇].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' We denote by 𝜉(f, g) the set of all FBNE state trajectories in the game defined by the dynamics f and cost functions g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Remark 1 (Strong Time Consistency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' The FBNE conditions of (3) implicitly enforce strong time-consistency [2, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='14] of the equi- librium strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' That is, FBNE does not admit arbitrary feedback strategies, but imposes the additional condition that those strategies must also be in equilibrium for any subgame starting at a later stage from an arbitrary state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='2 Nash Solutions in Open-loop Strategies In contrast, under the open-loop information pattern, each player only observes the initial state 𝑥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' In this case, the strategy for each player 𝑖 ∈ [𝑁] is a map from 𝑥1 to {𝑢𝑖 1,𝑢𝑖 2, · · · ,𝑢𝑖 𝑇 }, which we denote by 𝜙𝑖 (·) : R𝑛 → R𝑚𝑖 × · · · × R𝑚𝑖 ���������������������������������� 𝑇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Let Φ𝑖 be the set of all open-loop strategies of the player 𝑖, 𝑖 ∈ [𝑁].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' The corresponding open-loop Nash equilibrium is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Definition 2 (Open-Loop Nash Eqilibrium (OLNE) [2, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' 6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' The tuple of control strategies {𝜙∗ 1, · · · ,𝜙∗ 𝑁 } is called an open-loop Nash equilibrium if no player is incentivized to unilaterally alter its sequence of control inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Formally: 𝐽𝑖 � x, [𝜙1∗(𝑥1), · · · ,𝜙𝑖∗(𝑥1), · · · ,𝜙𝑁 ∗(𝑥1)] � (5) ≤ 𝐽𝑖 � x, [𝜙1∗(𝑥1), · · · ,𝜙𝑖 (𝑥1), · · · ,𝜙𝑁 ∗(𝑥1)] � , ∀𝜙𝑖 ∈ Φ𝑖, ∀𝑥1 ∈ R𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' The OLNE definition does not imply the strong time- consistence as in the feedback counterpart [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' 0 1 2 3 px 0 1 2 3 4 py Example 1, trajectories player 1, FBNE player 2, FBNE player 1, OLNE player 2, OLNE 0 5 10 time 0 1 2 3 4 5 position Example 1, FBNE player 1, px player 1, py player 2, px player 2, py 0 5 10 time 0 1 2 3 4 5 position Example 1, OLNE player 1, px player 1, py player 2, px player 2, py 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='5 3 px 0 1 2 3 4 py Example 2, trajectories 0 5 10 time 0 1 2 3 4 5 position Example 2, FBNE 0 5 10 time 0 1 2 3 4 5 position Example 2, OLNE 10 5 0 5 10 px 10 5 0 5 10 py Example 3, trajectories 0 20 40 60 time 0 1 2 3 4 5 position Example 3, FBNE 0 20 40 60 time 10 5 0 5 10 position Example 3, OLNE Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' 1: Examples of cost functions that yield trajectories that are dif- ferent under the OLNE and FBNE assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='3 Feedback vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Open-loop Nash Equilibria In this subsection, we demonstrate the difference between open- loop and feedback Nash equilibria and show the necessity of de- veloping specific solutions for cost inference problems with the feedback information pattern, instead of applying existing work with the open-loop assumption [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' To this end, we introduce be- low several linear-quadratic (LQ) games where the open-loop Nash equilibrium (OLNE) and feedback Nash equilibrium (FBNE) state trajectories differ substantially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' LQ games are a class of dynamic games with dynamics and player objectives of the form in (6) and (7), respectively, 𝑥𝑡+1 = 𝐴𝑡𝑥𝑡 + ∑︁ 𝑖 ∈[𝑁 ] 𝐵𝑖 𝑡𝑢𝑖 𝑡, ∀𝑡 ∈ [𝑇], (6) 𝑔𝑖 𝑡 (𝑥𝑡,𝑢𝑡) = 1 2 (𝑥⊤ 𝑡 𝑄𝑖 𝑡𝑥𝑡 + ∑︁ 𝑗 ∈[𝑁 ] 𝑢 𝑗 𝑡 ⊤𝑅𝑖𝑗 𝑡 𝑢 𝑗 𝑡 ), ∀𝑡 ∈ [𝑇], ∀𝑖 ∈ [𝑁], (7) where matrices {𝐴𝑡, 𝐵𝑖 𝑡 }, positive semidefinite matrix 𝑄𝑖 𝑡 and posi- tive definite matrix 𝑅𝑖𝑗 𝑡 are defined with appropriate dimensions, for each 𝑖, 𝑗 ∈ [𝑁] and 𝑡 ∈ [𝑇].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Case Study: We consider a two-player LQ game with a state vector 𝑥𝑡 = [𝑝1 𝑥,𝑡, 𝑝1 𝑦,𝑡, 𝑝2 𝑥,𝑡, 𝑝2 𝑦,𝑡], where 𝑝𝑖 𝑥,𝑡 and 𝑝𝑖 𝑦,𝑡 are the x- and y-coordinates of agent 𝑖 ∈ {1, 2}, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Let 𝑢𝑖 𝑡 ∈ R2 be the control input for the 𝑖-th agent, 𝑖 ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' In this setting, we consider a class of games in which the first agent wants to drive the second agent to the origin, while the second agent wants to catch the first agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' The agents’ joint dynamics and costs at time 𝑡 ∈ [𝑇] are specified as follows: 𝑥𝑡+1 = �𝐼2 0 0 𝐼2 � 𝑥𝑡 + �𝐼2 0 � 𝑢1 𝑡 + � 0 𝐼2 � 𝑢2 𝑡 , 𝑔1 𝑡 (𝑥𝑡,𝑢𝑡) = ∥𝑝2 𝑥,𝑡 ∥2 2 + ∥𝑝2 𝑦,𝑡 ∥2 2 + ∥𝑢1 𝑡 ∥2 2, 𝑔2 𝑡 (𝑥𝑡,𝑢𝑡) = ∥𝑝2 𝑥,𝑡 − 𝑝1 𝑥,𝑡 ∥2 2 + ∥𝑝2 𝑦,𝑡 − 𝑝1 𝑦,𝑡 ∥2 2 + ∥𝑢2 𝑡 ∥2 2, (8) where 𝐼2 is the 2 × 2 identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' We visualize the unique FBNE and OLNE state trajectories of this example in the first row in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' If we modify the cost function of the first player such that it wants to lead the 𝑥- and 𝑦-position of the second player to be aligned with each other, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=', ˆ𝑔1 𝑡 (𝑥𝑡,𝑢𝑡) := ∥𝑝2 𝑥,𝑡 − 𝑝2 𝑦,𝑡 ∥2 2 + ∥𝑢1 𝑡 ∥2 2, (9) then, the unique FBNE and OLNE state trajectories are still different, as shown in the second row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Moreover, observations of players may be noisy in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' To illustrate this, we consider a task where the two agents want to catch each other, but the first player’s observation of the second player’s position is inaccurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' We modify the first player’s cost in (8) as follows: ˆˆ𝑔1 𝑡 (𝑥𝑡,𝑢𝑡) := ∥𝑝1 𝑥,𝑡 − 2𝑝2 𝑥,𝑡 ∥2 2 + ∥𝑝1 𝑦,𝑡 − 2𝑝2 𝑦,𝑡 ∥2 2 + ∥𝑢1 𝑡 ∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' (10) The third row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' 1 reveals that the FBNE state trajectory is robust to inaccurate observations, but the unique OLNE state tra- jectory is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Thus, it is readily apparent that the OLNE and FBNE state strate- gies can be substantially different even for fixed cost functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' This difference in expressive power may be understood as a consequence of the strong time consistency property, which is enforced in the feedback information structure but not in the open-loop setting, per Remarks 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' A similar problem arises in the cost inference problem, where the existing OLNE cost inference algorithms may fail to infer the correct cost function in feedback games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' 4 PROBLEM STATEMENT Let x be an expert FBNE state trajectory under the nonlinear dy- namics f but unknown cost functions {𝑔𝑖 𝑡 }𝑇,𝑁 𝑡=1,𝑖=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Let T ⊆ [𝑇] be the set of observed time indices of the trajectory x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' We denote by yT := {𝑦𝑡 }𝑡 ∈T the observation data of x, where 𝑦𝑡 ∈ Rℓ is a partial observation of the state, composed of certain coordinates of 𝑥𝑡 cor- rupted by noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' The task is to infer the cost function of each player such that those inferred costs jointly yield a FBNE state trajectory that is as close as possible to the observed trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' We param- eterize the cost of the player 𝑖 by a vector 𝜃𝑖 ∈ R𝑑𝑖 , and let 𝜃 := [𝜃1,𝜃2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' ,𝜃𝑁 ] ∈ R𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Denote by 𝑔𝑖 𝑡,𝜃 (𝑥𝑡,𝑢𝑡) = �𝑑𝑖 𝑗=1 𝜃𝑖 𝑗𝑏𝑖 𝑡,𝑗 (𝑥𝑡,𝑢𝑡) player 𝑖’s parameterized cost at time 𝑡 ∈ [𝑇], for some basis func- tions {{𝑏𝑖 𝑡,𝑗 }𝑑𝑖 𝑗=1}𝑇,𝑁 𝑡=1,𝑖=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Define g𝜃 := {𝑔𝑖 𝑡,𝜃 }𝑇,𝑁 𝑡=1,𝑖=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Formally, this problem is of the form: min 𝜃,𝑥1,ˆx − 𝑝(yT|ˆx) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' ˆx ∈ 𝜉(f, g𝜃,𝑥1), (11) where 𝑝(·|·) is the likelihood function corresponding to a known sensor model and 𝜉(f, g𝜃,𝑥1) represents the set of state trajectories from the initial condition 𝑥1 ∈ R𝑛 following a FBNE strategy, under the cost set g𝜃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Due to the noisy partial observation, 𝑥1 is not assumed to be known and instead needs to be inferred as well in (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Note that the above formulation can also be extended to the cases where multiple partially observed incomplete trajectories from different initial conditions are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Running example: We consider a highway platooning scenario where player 1 wants to guide player 2 to a particular lane of the road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' The joint state vector is𝑥𝑡 = [𝑝1 𝑥,𝑡, 𝑝1 𝑦,𝑡, 𝛽1 𝑡 , 𝑣1 𝑡 , 𝑝2 𝑥,𝑡, 𝑝2 𝑦,𝑡, 𝛽2 𝑡 , 𝑣2 𝑡 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' 2: Visualization of the running example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' The time horizon 𝑇 = 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' The dynamics model for the player 𝑖 is: \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 𝑝𝑖 𝑥,𝑡+1 𝑝𝑖 𝑦,𝑡+1 𝛽𝑖 𝑡+1 𝑣𝑖 𝑡+1 \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb = \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 𝑝𝑖 𝑥,𝑡 𝑝𝑖 𝑦,𝑡 𝛽𝑖 𝑡 𝑣𝑖 𝑡 \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb + Δ𝑇 \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 𝑣𝑖 𝑡 cos(𝛽𝑖 𝑡) 𝑣𝑖 𝑡 sin(𝛽𝑖 𝑡) 𝜔𝑖 𝑡 𝑎𝑖 𝑡 \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb (12) where Δ𝑇 is a time discretization constant and 𝑢𝑖 𝑡 = [𝜔𝑖 𝑡,𝑎𝑖 𝑡] ∈ R2 is the control input for player 𝑖 ∈ [𝑁].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Let 𝑝∗𝑥 be the target lane that player 1 wants to guide player 2 to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' We parameterize the cost function of the player 𝑖 by 𝜃𝑖 ∈ R2, 𝑔1 𝑡,𝜃 (𝑥𝑡,𝑢𝑡) = 𝜃1 1 ∥𝑝1 𝑥,𝑡 ∥2 2 + 𝜃1 2 ∥𝑝2 𝑥,𝑡 − 𝑝∗ 𝑥 ∥2 2 + ∥𝑢1 𝑡 ∥2 2 (13) 𝑔2 𝑡,𝜃 (𝑥𝑡,𝑢𝑡) = 𝜃2 1 ∥𝑝2 𝑥,𝑡 − 𝑝1 𝑥,𝑡 ∥2 2 + 𝜃2 2 ∥𝑣2 𝑡 − 1∥2 2 + ∥𝑢2 𝑡 ∥2 2, ∀𝑡 ∈ [𝑇].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' The ground truth solution is𝜃∗ = [0, 8, 4, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' We assume that there is a period of occlusion happening from the time index𝑡 = 11 to𝑡 = 19, and the observed time index set is T = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' , 10, 20, 21, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' , 40}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Also, it may be difficult for a human driver to measure other ve- hicles’ velocity accurately, and therefore we assume that partial observation data yT excludes the velocity of both cars in the data set, and is further subject to Gaussian noise of standard deviation 𝜎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' The initial condition 𝑥1 is not known and needs to be inferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' We visualize the ground truth solution in the first subplot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' 2 and the noisy incomplete trajectory data in the second subplot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' The many challenges of the above problem include: (a) partial observation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' (b) noisy and incomplete expert trajectory data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' and (c) the difficulty of evaluating and differentiating the objective in (11), due to the challenge of computing a FBNE strategy in nonlinear games [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' In the following sections, we will characterize the complexity of this inverse feedback game problem and propose an efficient solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' 5 RESULTS: FROM CHARACTERIZATION TO COMPUTATION In this section, we first characterize the complexity of the inverse feedback game problem (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' In particular, we will show the non- convexity of the loss function and the existence of multiple isolated global minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Based on this observation, we discuss regulariza- tion schemes that can mitigate this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Our main contribution is to characterize the differentiability of the inverse feedback game loss function in (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Finally, we present a gradient approximation scheme that can be used in a first-order optimization formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' 3: Visualization of the loss function 𝐿(𝜃,𝑥1) of the LQ game spec- ified in (16) and (17), and its 𝐿2 regularization, with an initial condi- tion 𝑥1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' We adopt Gaussian likelihood function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' The yellow hy- perplane is drawn according to 2𝑄1 +𝑄2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' With 𝐿2 regularization, the number of global minima is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='1 Characterization of the Inverse Feedback Dynamic Game Problem The inverse feedback dynamic game problem (11) is a constrained optimization problem, which is hard to solve due to the nonconvex- ity of the set 𝜉(f, g𝜃,𝑥1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' With a slight abuse of notation, we denote by ˆx(f, g𝜃,𝑥1) ∈ 𝜉(f, g𝜃,𝑥1) a FBNE state trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' To simplify the problem, we transform (11) to an unconstrained problem by substituting a forward game solution ˆx(f, g𝜃,𝑥1) into the likelihood function 𝑝(yT|ˆx), as follows: ˆ𝐿(𝜃,𝑥1) := −𝑝(yT|ˆx(f, g𝜃,𝑥1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' (14) The minimizer of (14) is a local optimum to the original problem (11) and becomes global when 𝜉(f, g𝜃,𝑥1) contains only a single element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Before we dive into the nonlinear setting, let us first consider a simplified LQ case to highlight the main challenges associated with the optimization of this loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' In the LQ case, the evaluation of the loss (14) is straightforward if there exists a closed-form expression for 𝑝(yT|ˆx), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=', under a Gaussian observation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Even in that setting, however, it is important to realize that the problem remains nonconvex, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' The following proposition makes this challenge explicit, and the proof can be found in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' There exists an inverse LQ game problem (11): (a) whose global minima are isolated, and (b) for which there exist multiple cost functions that exactly match expert data from any initial condition, when there is no observation noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Proposition 1 does not imply that any inverse LQ game problem will suffer from the multiple global minima issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Instead, Proposition 1 suggests that simply normalizing the cost vector does not rule out the possibility of having multiple global solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' That is, there exist two cost parameter vectors which are linearly independent, but generate the same FBNE state trajectories for any given initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' This non-injective mapping from the cost parameter space to the FBNE state trajectory space is a fundamental problem in inverse feedback games, and is not particular to the formulation (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' In practice, this multiple global minima issue could be mitigated by adding 𝐿2 regularization, as visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Though being nonconvex, the loss function ˆ𝐿(𝜃,𝑥1) is differen- tiable with respect to both 𝜃 and 𝑥1 under the condition of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='2 in [16], which follows from the implicit function theorem [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Inspired by the success of gradient-based methods in non-convex optimization with differentiable objective functions [3, 26, 34], one natural idea is to apply gradient descent to minimize ˆ𝐿(𝜃,𝑥1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' In what follows, we discuss efficient ways to evaluate and differentiate ˆ𝐿(𝜃,𝑥1) in nonlinear games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='2 Efficient Computation for a FBNE State Trajectory in Nonlinear Games It is easy to evaluate ˆ𝐿(𝜃,𝑥1) for LQ games, but when dynamics are nonlinear or objectives are non-quadratic, the problem becomes more challenging [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' In forward games, this challenge can be addressed by using the ILQGames algorithm [8], which finds ap- proximate local FBNE solutions in smooth non-LQ dynamic games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Given the effectiveness of this approximation scheme in those do- mains, we also adopt it as a submodule for evaluating the loss ˆ𝐿(𝜃,𝑥1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Akin to the ILQR method [18, 19], in each step of the ILQGames algorithm, the system dynamics 𝑥𝑡+1 = 𝑓 (𝑥𝑡,𝑢𝑡) and the costs {𝑔𝑖 𝑡 (𝑥,𝑢)}𝑇,𝑁 𝑡=1,𝑖=1 are linearized and quadraticized, respec- tively, around a state trajectory x and a control trajectory u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' A FBNE strategy for each player of the derived LQ game is then used to update the state and control trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' This iteration continues until a convergence criterion is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' To be more specific, we approximate ˆ𝐿(𝜃,𝑥1) by a new loss function ˜𝐿(𝜃,𝑥1) defined as, ˆ𝐿(𝜃,𝑥1) ≃ ˜𝐿(𝜃,𝑥1) := −𝑝 �yT|x(˜f𝜃, ˜g𝜃,𝑥1)� (15) where x(˜f𝜃, ˜g𝜃,𝑥1) represents a FBNE state trajectory from initial condition 𝑥1, for the LQ game defined by the linearized dynamics ˜f𝜃, quadraticized cost set ˜g𝜃 := {˜g𝑖 𝑡,𝜃 }𝑇,𝑁 𝑡=1,𝑖=1 at the converged solution returned by ILQGames solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Note that the linearized dynamics ˜f𝜃 depend upon 𝜃 via the state trajectory about which f is linearized;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' this trajectory is simulated under the feedback policy returned by ILQGames, where the policy depends upon costs g𝜃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='3 Differentiating the Loss in the Inverse Feedback Game Problem The challenge of computing a feedback Nash equilibrium strategy not only makes the evaluation of the loss function ˆ𝐿(𝜃,𝑥1) hard, but also renders differentiation difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' In this work, we approximate the gradient of ˆ𝐿(𝜃,𝑥1) using a similar idea as the ILQGames algo- rithm in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' In other words, we propose to use the LQ approximation of the nonlinear game specified by ˜f𝜃 and ˜g𝜃 to derive an approximation to the gradient of ˆ𝐿(𝜃,𝑥1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Note that ˜𝑔𝑖 𝑡,𝜃 (𝑥,𝑢) = �𝑑𝑖 𝑗=1 𝜃𝑖 𝑗 ˜𝑏𝑖 𝑡,𝑗,𝜃 (𝑥,𝑢), where ˜𝑏𝑖 𝑡,𝑗,𝜃 (𝑥,𝑢) : R𝑛 × R𝑚 → R is the 𝑗-th quadraticized cost basis function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' By the chain rule, we have 𝜕 ˜𝐿(𝜃,𝑥1) 𝜕𝜃𝑖 𝑗 = −∇x𝑝(yT|x) ���x(˜f𝜃,˜g𝜃,𝑥1) · 𝜕x(˜f𝜃, ˜g𝜃,𝑥1) 𝜕𝜃𝑖 𝑗 , 𝜕x(˜f𝜃, ˜g𝜃,𝑥1) 𝜕𝜃𝑖 𝑗 = � ∇˜f𝜃 x(˜f𝜃, ˜g𝜃,𝑥1) 𝜕˜f𝜃 𝜕𝜃𝑖 𝑗 + ∇˜g𝜃 x(˜f𝜃, ˜g𝜃,𝑥1) 𝜕˜g𝜃 𝜕𝜃𝑖 𝑗 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' No regularization ×10-4 No regularization 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='8 (0, α1) 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='6 (Iα‘)T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='2 Q1 Q1 L2 regularization ×10~4 L2 regularization [z‘]l-01+(I") 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='5 0 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='2 Q1 0 Q1The complexity of differentiating ˜𝐿(𝜃,𝑥1) comes from the fact that the linearized dynamics and the quadraticized costs are functions of 𝜃 implicitly, which makes the total derivative hard to compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' We propose to approximate the above gradient by treating the linearized ˜f𝜃 and each quadraticized cost basis function ˜𝑏𝑖 𝑡,𝑗,𝜃 as constants with respect to 𝜃, denoted by ˜f and ˜𝑏𝑖 𝑡,𝑗, and only com- pute the partial derivative with respect to 𝜃, rather than the total derivative: 𝜕 ˜𝐿(𝜃,𝑥1) 𝜕𝜃𝑖 𝑗 ≃ −∇x𝑝(yT|x) ���x(˜f,˜g𝜃,𝑥1)· 𝜕x(˜f, {�𝑑𝑖 𝑗=1 𝜃𝑖 𝑗 ˜𝑏𝑖 𝑡,𝑗 }𝑇,𝑁 𝑡=1,𝑖=1,𝑥1) 𝜕𝜃𝑖 𝑗 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' This is based on the observation that at the convergence of the forward ILQGames solver, the linearized dynamics are a good ap- proximation of the full nonlinear dynamics f, so long as the cost parameter being perturbed remains sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' We adopt a similar approximation for the gradient ∇𝑥1 ˜𝐿(𝜃,𝑥1) by fixing the lin- earized dynamics and quadraticized costs and obtaining the partial derivative with respect to 𝑥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' In summary, we approximate ∇ ˆ𝐿(𝜃,𝑥1) by ∇ ˜𝐿(𝜃,𝑥1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' In practice, ∇ ˜𝐿(𝜃,𝑥1) can be efficiently computed by automatic differentiation [27, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' As exemplified in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' 4, the proposed gradient approxi- mation is virtually always a descent direction and therefore aligns well with the true gradient of ˆ𝐿(𝜃,𝑥1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='4 An Inverse Feedback Game Solver In this subsection, we present a solver for the inverse feedback game problem (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' In what follows, we first discuss how the three challenges mentioned in Section 4 are handled in our solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' We then introduce the proposed solver in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' The first two challenges on noisy partial observation and in- complete trajectory data are handled by maintaining an estimate of the full initial condition and a noise-free state-input trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' As shown in Section 6, this procedure of joint reconstruction and filtering enables our solver to reliably recover player costs even in scenarios of substantial partial observability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' The third difficulty of evaluating and differentiating the objective function in the inverse feedback game problem is mitigated by the efficient approximation outlined in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' To jointly infer the initial condition, the cost and the state-input trajectory, we first adopt the coordinate gradient descent method, where gradient descent steps are first taken over the initial condition ˆ𝑥1, and then taken over the cost parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' We update the estimate of the noise-free full state-input trajectory by computing a FBNE state trajectory from the inferred initial condition and the cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' We summarize our proposed solver in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' At the 𝑘-th it- eration, we first compute an approximate FBNE state trajectory ˜𝑥 (𝑘) and the associated LQ approximation via the ILQGames algorithm of [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Using this LQ approximation, we estimate ∇𝑥1 ˆ𝐿(𝜃,𝑥 (𝑘) 1 ) us- ing the procedure outlined in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' We then update the initial condition 𝑥 (𝑘) 1 by a step of gradient descent, where the stepsize is chosen by a suitable linesearch technique [27, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' 3] such that the loss ˆ𝐿(𝜃,𝑥1) is sufficiently decreased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Given the updated initial condition 𝑥 (𝑘+1) 1 , we find a new approximate FBNE state trajectory Algorithm 1: Inverse Iterative LQ (i2LQ) Games Data: Horizon 𝑇 > 0, initial solution 𝜃 (0) ∈ R𝑑, observed time index set T ⊆ [𝑇 ], observation data yT, max iteration number 𝐾 , tolerance 𝜖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Result: Inferred cost parameter ˆ𝜃 and ˆ𝑥1 1 for 𝑘 = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' , 𝐾 do 2 ( ˜x(𝑘), { ˜𝛾𝑖 𝑡 }𝑇,𝑁 𝑡=1,𝑖=1, ˜f𝜃 (𝑘) , ˜g𝜃 (𝑘) ) ← ILQGames(f, g𝜃 (𝑘) ,𝑥 (𝑘) 1 ) 3 ∇𝑥1 ˆ𝐿(𝜃 (𝑘),𝑥 (𝑘) 1 ) ← evaluated using ˜f𝜃 (𝑘) and ˜g𝜃 (𝑘) via Gradient Approximation in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='3 4 𝑥 (𝑘+1) 1 ← 𝑥 (𝑘) 1 − 𝜂∇𝑥1 ˆ𝐿(𝜃 (𝑘),𝑥 (𝑘) 1 ) with line search over 𝜂 5 ( ˇ𝑥 (𝑘), {ˇ𝛾𝑖 𝑡 }𝑇,𝑁 𝑡=1,𝑖=1, ˇf𝜃 (𝑘) , ˇg𝜃 (𝑘) ) ← ILQGames�f, g𝜃 (𝑘) ,𝑥 (𝑘+1) 1 � 6 ∇𝜃 ˆ𝐿(𝜃 (𝑘),𝑥 (𝑘+1) 1 ) ← evaluated using ˇf𝜃 (𝑘) and ˇg𝜃 (𝑘) via Gradient Approximation in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='3 7 𝜃 (𝑘+1) ← 𝜃 (𝑘) − 𝜂′∇𝜃 ˆ𝐿(𝜃 (𝑘),𝑥 (𝑘+1) 1 ) with line search over 𝜂′ 8 Return (𝜃 (𝑘+1),𝑥 (𝑘+1) 1 ) if ∥𝜃 (𝑘) − 𝜃 (𝑘−1) ∥2 ≤ 𝜖 or Return (𝜃 (𝑘′),𝑥 (𝑘′) 1 ), where 𝑘′ ← arg min𝑘 ˜𝐿(𝜃 (𝑘),𝑥 (𝑘) 1 ), if iteration number 𝑘 reaches 𝐾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' 9 end via the ILQGames algorithm again, which is then used to estimate ∇𝜃 ˆ𝐿(𝜃 (𝑘),𝑥 (𝑘+1) 1 ) via the procedure in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' With this gradi- ent, we update 𝜃 (𝑘) by one step of gradient descent with linesearch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' We repeat this procedure until, at convergence, we find a locally optimal solution ( ˆ𝜃, ˆ𝑥1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' 6 EXPERIMENTS In this section, we adopt the open-loop solution method of [28] as the baseline method and compare it to Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' In particular, we evaluate Algorithm 1 in several Monte Carlo studies which aim to justify the following claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' The proposed gradient approximation often aligns with a descent direction in the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Algorithm 1 is more robust than the open-loop baseline method [28] with respect to noise in, and incomplete obser- vations of, the expert demonstration trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' The cost functions inferred by Algorithm 1 can be general- ized to predict trajectories from unseen initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Algorithm 1 can infer nonconvex costs in nonlinear games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='1 Gradient Approximation Quality We continue the 2-vehicle platooning example defined in (12) and (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' We measure the performance of Algorithm 1 in two settings, in- complete expert trajectory data with noisy partial state observation, and complete expert trajectory data with noisy full observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' In the first case, each player’s partial observation only contains its x-position, y-position and heading angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' The time index set of the incomplete trajectory is T = [𝑇] \\ {11, 12, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' , 19}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' In the second case, the expert data includes the noisy observation of all the states of both players at all 𝑡 ∈ [𝑇].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' The ground truth expert state trajectory follows a FBNE strategy from the initial condi- tion 𝑥1 = [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='5, 𝜋 2 , 1, 1, 0, 𝜋 2 , 1] and the target lane is 𝑝∗𝑥 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' At each variance level 𝜎 ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='004, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='008, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='04}, we generate 10 noisy observations of the ground truth expert trajectory, with isotropic zero-mean Gaussian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' For each noisy expert data set yT, we minimize the negative log-likelihood objective in (11), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=', � 𝑡 ∈T ∥𝑦𝑡 − ℎ(𝑥𝑡)∥2 2, where ℎ(·) : R𝑛 → Rℓ maps a state 𝑥𝑡 to its partial observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' 4, the loss decreases monotonically on the average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' This indicates that the gradient approximation proposed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='3 provides a reliable descent direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' The inverse feedback game problem becomes challenging when there is only partial state observation and incomplete trajectory data, and the quality of inferred costs may degrade when the observation noise is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='2 Robustness, Generalization and the Ability to Infer Nonconvex Costs We continue the previous 2-vehicle example and compare Algo- rithm 1 and the baseline in a Monte Carlo study, where we infer the costs under 10 different levels of Gaussian noise with increasing variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' In particular, we evaluate three metrics in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' 5: (a) the distance between the noisy expert data and the FBNE state trajec- tory which results from players’ inferred costs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' (b) the distance between the computed FBNE state trajectory (under the players’ inferred costs) and the ground truth expert data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' An example of such a comparison is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Finally, we evaluate (c) the distance between the inferred FBNE state trajectories and the FBNE state trajectory under the ground truth costs for some randomly sampled initial conditions, which is also visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Collec- tively, the results demonstrate that Algorithm 1 has better robustness and generalization performance than the open-loop baseline when the expert data follows the FBNE assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' To show that Algorithm 1 can infer nonconvex cost functions, we extend the previous 2-vehicle platooning example and assume that the 2-vehicle team encounters a third vehicle and the follower wants to stay close to the leader without colliding with the third vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' We model this scenario as a 3-vehicle game with a 12 dimensional state space and a horizon 𝑇 = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' The dynamics for each vehicle is the same as (12) and the costs are as follows,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' 𝑔1 𝑡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='𝜃 (𝑥𝑡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='𝑢𝑡) =𝜃1 1 ∥𝑝1 𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='𝑡 ∥2 2 + 𝜃1 2 ∥𝑝2 𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='𝑡 − 𝑝∗ 𝑥 ∥2 2 + ∥𝑣1 𝑡 − 2∥2 2 + ∥𝛽1 𝑡 − 𝜋 2 ∥2 2 + ∥𝑢1 𝑡 ∥2 2 𝑔2 𝑡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='𝜃 (𝑥𝑡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='𝑢𝑡) =𝜃2 1 ∥𝑝2 𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='𝑡 ∥2 2 + ∥𝛽2 𝑡 − 𝜋 2 ∥2 2 + 𝜃2 2 ∥𝑝2 𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='𝑡 − 𝑝1 𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='𝑡 ∥2 2 + ∥𝑣2 𝑡 − 2∥2 2 − 1 2 log(∥𝑝2 𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='𝑡 − 𝑝3 𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='𝑡 ∥2 2 + ∥𝑝2 𝑦,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='𝑡 − 𝑝3 𝑦,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='𝑡 ∥2 2) + ∥𝑢2 𝑡 ∥2 2 𝑔3 𝑡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='𝜃 (𝑥𝑡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='𝑢𝑡) =𝜃3 1 ∥𝑝3 𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='𝑡 − 1 2 ∥2 2 + ∥𝑢3 𝑡 ∥2 2 where the ground truth 𝜃∗ ∈ R5 is [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' The ground truth expert state trajectory follows a FBNE strategy from the initial condition 𝑥1 = [0, 1, 𝜋 2 , 2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='3, 0, 𝜋 2 , 2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='5, 𝜋 2 , 2], where the last four elements encode the state of the third vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' The target lane in the expert data is 𝑝∗𝑥 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Similar to the 2-vehicle experiment, we consider two settings, incomplete trajectory data with partial state observation and com- plete trajectory data with full state observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' The partial state observation includes all the states of each vehicle except for the velocity of all the vehicles, and the time indices set of the incom- plete trajectory is T = [𝑇] \\ {11, 12, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' , 19}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' The nonconvex cost of player 2 causes numerical problems in the baseline KKT OLNE Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' 4: Convergence of Algorithm 1 with the Gradient Approxima- tion proposed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' The loss decreases monotonically on the average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' The bold lines and shaded areas represent the mean val- ues and their standard error, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=', the variance divided by the square root of the sample size, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' 5: 2-vehicle platooning scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' The bold lines and shaded ar- eas represent the mean values and their standard error, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=', the vari- ance divided by the square root of the sample size, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' As the noise variance growing, the converged loss value increases, as shown in the red curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' However, Algorithm 1 is still able to learn a more accurate cost and has less generalization error than the base- line, as shown in the blue and yellow curves, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' 6: Full and partial, noisy observation of the expert trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Dashed lines represent predicted trajectories which result from in- ferred costs, and solid lines are ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' The trajectories pre- dicted by Algorithm 1 are closer to the ground truth than the base- line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' solver [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Thus, we add an 𝐿2 regularization 10−4∥𝜃 ∥2 2 to the loss ˆ𝐿(𝜃,𝑥1) and summarize the Monte Carlo study in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' 8, where we see Algorithm 1 is also able to learn better cost functions reflecting the true intentions of each vehicle in feedback games, even with only partial state observations and incomplete trajectory data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' 7: Generalization performance comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' 𝑝∗𝑥 is the target lane position that player 1 wants to guide player 2 toward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' All the costs are inferred from partial observations and incomplete trajectory data, with different noise variance specified in each of the subplot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' The trajectories predicted by Algorithm 1 are closer to the ground truth than the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' 8: 3-vehicle platooning scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' The bold lines and shaded ar- eas represent the mean values and their standard error, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=', the vari- ance divided by the square root of the sample size, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' As the noise variance growing, the converged loss value increases on the average, as shown in the red curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' However, Algorithm 1 is still able to learn a more accurate cost and has less generalization error than the baseline, as shown in the blue and yellow curves, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' 7 CONCLUSION In this work, we propose an efficient cost inference algorithm for inverse feedback nonlinear games, with only partial state observa- tion and incomplete trajectory data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Empirical results show that the proposed solver converges reliably for inverse games with non- convex costs and has superior generalization performance than a state-of-the-art open-loop baseline method when the expert demon- stration reflects a group of agents acting in a dynamic feedback game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' There are many future directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' We can investigate under what conditions the cost can be inferred exactly in feedback games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' The active and online inference are also promising directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' In addition, we are eager to extend this work to settings of closed-loop interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' In such an extension, rather than merely inferring the objectives of observed players, this information would be used to guide the decision-making of an autonomous agent in that scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' APPENDIX Proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Proposition 1 claims that there exists an inverse LQ game, which has isolated global minima and the induced FBNE state trajectories of those solutions match the expert demonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Here, we show such a counterexample, which sup- ports the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Consider a 2-player horizon-3 LQ game with the linear dynamics 𝑥𝑡+1 = 𝑥𝑡 + 𝑢1 𝑡 + 𝑢2 𝑡 , 𝑡 ∈ {1, 2, 3}, (16) and the cost 𝑔1 𝑡 (𝑥𝑡,𝑢𝑡) = 1 2 (𝑄1∥𝑥𝑡 ∥2 2 + ∥𝑢1 𝑡 ∥2 2), 𝑡 ∈ {1, 2}, 𝑔2 𝑡 (𝑥𝑡,𝑢𝑡) = 1 2 (𝑄2∥𝑥𝑡 ∥2 2 + 2∥𝑢2 𝑡 ∥2 2), 𝑡 ∈ {1, 2}, 𝑔1 3(𝑥3,𝑢3) = 1 2𝑄1∥𝑥3∥2 2, 𝑔2 3(𝑥3,𝑢3) = 1 2𝑄2∥𝑥3∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' (17) We assume that the ground truth solutions are 𝑄1 = 1, 𝑄2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' We will show there is also one extra solution ˆ𝑄1 = 1 2 and ˆ𝑄2 = 2, which yields the same FBNE state trajectory as the ground truth for any initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' We follow the same definition of the variable {𝑍𝑖 𝑡 }3,2 𝑡=1,𝑖=1 as in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' By definition, we have𝑍𝑖 𝑡 ≥ 𝑄𝑖 > 0, when𝑄1 ∈ R+ and 𝑄2 ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Following the notations in FBNE condition in Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content='1 of [2], we consider the feedback matrices {𝑃𝑖 𝑡 }2,2 𝑡=1,𝑖=1, �𝑃1 𝑡 𝑃2 𝑡 � = �1 + 𝑍1 𝑡+1 𝑍1 𝑡+1 𝑍2 𝑡+1 2 + 𝑍2 𝑡+1 � ������������������������������������������������ 𝐺𝑖 𝑡 �𝑍1 𝑡+1 𝑍2 𝑡+1 � , ∀𝑡 ∈ {1, 2}, (18) where the matrix 𝐺𝑖 𝑡 is invertible because det(𝐺𝑖 𝑡) = 2 + 𝑍2 𝑡+1 + 2𝑍1 𝑡+1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' The above analysis suggests that the FBNE state tra- jectory for all 𝑄1 > 0 and 𝑄2 > 0 are uniquely determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' We consider the time instant 𝑡 = 2, and observe �𝑃1 2 𝑃2 2 � = �1 + 𝑄1 𝑄1 𝑄2 2 + 𝑄2 �−1�𝑄1 𝑄2 � = 1 2 + 2𝑄1 + 𝑄2 �2𝑄1 𝑄2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' (19) We then have the closed-loop dynamics 𝑥3 = (1 − 𝑃1 2 − 𝑃2 2)𝑥2 = 2 2+2𝑄1+𝑄2 𝑥2, which yields that for two pairs of positive variables (𝑄1,𝑄2) and ( ˆ𝑄1, ˆ𝑄2), a necessary condition for them to have the same FBNE trajectory is that 2𝑄1 + 𝑄2 = 2 ˆ𝑄1 + ˆ𝑄2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' We have 𝑍1 2 = 𝑄1 + 𝑄1+(2𝑄1)2 (2+2𝑄1+𝑄2)2 , 𝑍2 2 = 𝑄2 + 𝑄2+2(𝑄2)2 (2+2𝑄1+𝑄2)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Similarly, for the time instant 𝑡 = 1, we have 𝑥2 = (1 − 𝑃1 1 − 𝑃2 1)𝑥1 = 2 2+2𝑍 1 2+𝑍 2 2 𝑥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' A necessary condition for ( ˆ𝑄1, ˆ𝑄2) to have the same FBNE state trajectory as (𝑄1,𝑄2) is that the following 2 equations are satisfied, 2𝑄1 + 𝑄2 = 2 ˆ𝑄1 + ˆ𝑄2 2�𝑄1 + 𝑄1 + (2𝑄1)2 (2 + 2𝑄1 + 𝑄2)2 � + 𝑄2 + 𝑄2 + 2(𝑄2)2 (2 + 2𝑄1 + 𝑄2)2 = 2� ˆ𝑄1 + ˆ𝑄1 + (2 ˆ𝑄1)2 (2 + 2 ˆ𝑄1 + ˆ𝑄2)2 � + ˆ𝑄2 + ˆ𝑄2 + 2( ˆ𝑄2)2 (2 + 2 ˆ𝑄1 + ˆ𝑄2)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' (20) We substitute 𝑄1 = 1, 𝑄2 = 1 and ˆ𝑄2 = 3−2 ˆ𝑄1 into the second row of (20), which is reduced to a 2-degree polynomial of ˆ𝑄2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' By the fundamental theorem of algebra [4], there exist at most 2 solutions for ˆ𝑄2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' The two pairs of ( ˆ𝑄1, ˆ𝑄2) satisfying (20) are (1, 1) and ( 1 2, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' The two global minima are isolated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' Since the dimension of the state 𝑥𝑡 is 1, for all initial states 𝑥1 ∈ R, the FBNE state trajectories under the costs specified by the two pairs cost parameters (1, 1) and ( 1 2, 2) coincide with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} +page_content=' □ REFERENCES [1] Chaitanya Awasthi and Andrew Lamperski.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfcPwr/content/2301.01398v1.pdf'} diff --git a/t9AyT4oBgHgl3EQfmvi8/content/tmp_files/2301.00478v1.pdf.txt b/t9AyT4oBgHgl3EQfmvi8/content/tmp_files/2301.00478v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7bcc62bf4ac8a951955afccfc83a8299f11dda65 --- /dev/null +++ b/t9AyT4oBgHgl3EQfmvi8/content/tmp_files/2301.00478v1.pdf.txt @@ -0,0 +1,2818 @@ +WEAK QUENCHED LIMIT THEOREMS FOR A RANDOM WALK IN +A SPARSE RANDOM ENVIRONMENT +DARIUSZ BURACZEWSKI, PIOTR DYSZEWSKI AND ALICJA KO�LODZIEJSKA +Abstract. We study the quenched behaviour of a perturbed version of the simple sym- +metric random walk on the set of integers. The random walker moves symmetrically +with an exception of some randomly chosen sites where we impose a random drift. We +show that if the gaps between the marked sites are i.i.d. and regularly varying with a +sufficiently small index, then there is no strong quenched limit laws for the position of +the random walker. As a consequence we study the quenched limit laws in the context +of weak convergence of random measures. +1. Introduction +One of the most classical and well-understood random processes is the simple symmetric +random walk (SRW) on the set of integers, where the particle starting at zero every unit +time moves with probability 1/2 to one of its neighbours. This process is a time and +space homogeneous Markov chain, that is its increments are independent of the past and +the transitions do not depend on time and the current position of the process. In many +cases, the homogeneity of the environment reduces the applicability of the process. In +numerous applied models some kind of obstacles can appear like impurities, fluctuations, +etc. Thus, it is natural to express such irregularities as a random environment and it +is well known that even small perturbations of the environment affect properties of the +random process. In 1981 Harrison and Shepp [15] described the behaviour of the SRW in +a slightly disturbed environment, replacing only the probability of passing from 0 to 1 by +some fixed p ∈ (0, 1). They observed that the scaling limit is not the Brownian motion, +but the skew Brownian motion. +We intend to study random walks in a randomly perturbed environment. Our main +results concern the so-called random walk in a sparse random environment (RWSRE) in- +troduced in [17], in which homogeneity of an environment is perturbed only on a sparse +subset of Z. More precisely, first we choose randomly a subset of integers marked by the +positions of a standard random walk with positive integer jumps and next we impose a +random drift at the chosen sites. The present paper can be viewed as a continuation of +the recent publications [17, 6, 5], where annealed limit theorems were described. These +annelad-type results do not settle however the question if the environment alone is suffi- +cient to determine the distributional behaviour of the process with high certainty. Here +we froze the environment and we are interested in limit behaviour of the random process +in the quenched settings. As we show in the present article, even in a very diluted ran- +dom environment the fluctuations of the random perturbation of the medium affect the +conditional distribution of the random walker. +The model RWSRE we consider here can be viewed as an interpolation between SRW +and the one suggested in the seventies by Solomon [25] called a one dimensional random +walk in random environment (RWRE), where all the sites were associated with random +i.i.d. weights {ωi} describing the probability of passing to the right neighbour. It quickly +2010 Mathematics Subject Classification. Primary: 60K37; secondary 60F05; 60G57. +Key words and phrases. weak convergence, point processes, regular variation, random walk in a random +environment, sparse random environment. +1 +arXiv:2301.00478v1 [math.PR] 1 Jan 2023 + +2 +D. BURACZEWSKI, P. DYSZEWSKI AND A. KO�LODZIEJSKA +became clear that the additional environmental noise in the system has a significant impact +on the behaviour of the model. In fact, the answers to a variety of questions about the +model like limit theorem [16] and large deviations [7, 4] are given only in terms of the +environment marginalizing the impact of the random motion of the process. +1.1. General setting. To define our model let Ω = (0, 1)Z be the set of all possible +configurations of the environment equipped with the corresponding cylindrical σ-algebra +F and a probability measure P. A random element ω = (ωn)n∈Z of (Ω, F) distributed +according to P is called a random environment. Each element ω of Ω and integer x gives +a rise to a probability measure Px +ω on the set X = ZN0 with the cylindrical σ-algebra G +such that Px +ω[X0 = x] = 1 and +Px +ω [Xn+1 = j|Xn = i] = +� +� +� +ωi, +if j = i + 1, +1 − ωi, +if j = i − 1, +0, +otherwise, +where X = (Xn)n∈N0 ∈ X. One sees that under Px +ω, X forms a nearest neighbour random +walk which is a time-homogeneous Markov chain on Z and it is called a random walk +in random environment. The randomness of the environment ω influences significantly +various properties of X. In view of this, it is natural to investigate the behaviour of X +under the annealed measure Px = +� +Px +ωP(dω) which is defined as the unique probability +measure on (Ω × X, F ⊗ G) satisfying +Px[F × G] = +� +F +Px +ω[G] P(dω), +F ∈ F, +G ∈ G. +In the sequel we will write Pω = P0 +ω and P = P0. It turns out that, in general, under the +annealed probability X is no longer a Markov chain, because it usually exhibits a long +range dependence. +We are interested in limit theorems for Xn as → ∞, however in this paper we discuss the +asymptotic behaviour of the corresponding sequence of first passage times T = (Tn)n∈N, +that is +(1.1) +Tn = inf{k ∈ N : Xk = n}. +We will study the distribution of Tn in the quenched setting which means that we will +investigate the behaviour of +µn,ω(·) = Pω [(Tn − bn)/an ∈ · ] +for suitable choices of sequences (an)n∈N and (bn)n∈N possibly depending on ω. In the +present setting µn, defined by µn(ω) = µn,ω, becomes a random element of M1, the space +of probability measures on (R, Bor(R)), where Bor(R) stands for the Borel σ-algebra. +M1 equipped in the Prokhorov distance is a complete, separable metric space. One can +distinguish two types of limiting behaviour of (µn)n∈N. We will say that a strong quenched +limit theorem for T holds if µn → µ almost surely in M1, that is for P a.s. ω the sequence +of measures {µn,ω} converges weakly to µ, and say that a weak quenched limit law for T +holds if µn ⇒ µ in M1. Here and in the sequel ⇒ denotes weak convergence. +We will now discuss different choices of the probability P which is the distribution of +the environment. To keep the introduction brief we will limit the discussion to the i.i.d. +random environment, which is the most classical choice for P, and the sparse random +environment which we will study in depth in the sequel. + +WEAK QUENCHED LIMIT THEOREMS FOR RWSRE +3 +Figure 1.1. Random walk in i.i.d. random environment for ω0 = 1/3 with +probability 1/3, ω0 = 3/4 with probability 2/3 and α ≈ 1, 35 . +1.2. Independent identically distributed environment. One of the simplest and +most studied choices of the environmental distribution P is random walk in i.i.d. random +environment, corresponding to a product measure, under which ω = (ωn)n∈Z forms a +collection of independent, identically distributed (i.i.d.) random variables. In their seminal +work Kesten et al. [16] used the following link between walks and random trees [14]: +the one-dimensional distributions of T are connected to a branching process in random +environment with immigration and a reproduction law with the mean distributed as (1 − +ω0)/ω0. This observation later leads to a conclusion that T lies in the domain of attraction +of an α-stable distribution, where E[ω−α +0 (1 − ω0)α] = 1, provided that such α ∈ (0, 2) +exists (see Figure 1.3). After a close examination of the main results of Kesten et al. [16] it +transpires that the centering and scaling are determined by the distribution of (1−ω0)/ω0, +which means that the behaviour of the walker does not affect the limiting behaviour in a +significant way. In turn, to understand the random motion, one is led to investigate the +behaviour of T under Pω. If α > 2, then a strong quenched limit theorem [24, 13] of the +form +lim +n→∞ Pω +� +(Tn − Eω[Tn])/(σ√n) ∈ dx +� += e−x2/2dx/ +√ +2π +holds almost surely in M1, where σ2 = E[Varω[T1]] < ∞. +As seen from the results +in [18, 20] there is no strong quenched limit theorem for T in the case α < 2. Indeed it +turns out that for α < 2 one can find different strong quenched limits for T along different +sequences. This in turn leads to the analysis of T in the weak quenched setting, that is +weak limits of µn. Consider first the mapping H : Mp → M1 given as follows: for a point +process ζ = � +i≥1 δxi, where {xi}i∈N is an arbitrary enumeration of the points, define +H(ζ)(·) = +� P +� � +i≥1 xi(τi − 1) ∈ · +� +, +� +i≥1 x2 +i < ∞, +δ0(·), +otherwise, +where {τi}i∈N is a sequence of i.i.d. mean one exponential random variables. Then the +main result of [9, 12, 19] states that for α < 2, +Pω +� +n−1/α(Tn − EωTn) ∈ · +� +⇒ H(N) +in M1, where N is a Poisson point process on (0, ∞) with intensity cNx−α−1dx for some +constant cN > 0. +1.3. Sparse random environment. We now specify the object of interest in the present +paper. We will work under a choice of environmental probability P for which the random + +100 +80 +60 +40 +20 +-20 +0 +2000 +4000 +6000 +8000 +100004 +D. BURACZEWSKI, P. DYSZEWSKI AND A. KO�LODZIEJSKA +walk X will move symmetrically except some randomly marked points where we impose +a random drift. The marked sites will be distributed according to a two-sided random +walk. Denote by ((ξk, λk))k∈Z a sequence of independent copies of a random vector (ξ, λ), +where λ ∈ (0, 1) and ξ ∈ N, P-a.s. Considering the aforementioned two-sided random walk +S = (Sn)n∈Z given via +Sn = +� +� +� +�n +k=1 ξk, +if n > 0, +0, +if n = 0, +− �0 +k=n+1 ξk, +if n < 0, +we define a random environment ω = (ωn)n∈Z ∈ Ω given by +(1.2) +ωn = +� +λk, +if n = Sk for some k ∈ Z, +1/2, +otherwise. +The sequence S determines the marked sites in which the random drifts 2λk −1 are placed. +Since for the unmarked sites n (that is, for most of sites) the probabilities of jumping to +the right are deterministic and equal to ωn = 1/2, it is natural to call ω a sparse random +environment. Following [17] we use the term random walk in sparse random environment +(RWSRE) for X as defined above with ω being a sparse random environment. +Example 1.1. In the case when P[ξ = 1] = 1 random walk in sparse random environment +is equivalent to a random walk in i.i.d. environment. +Example 1.2. Suppose that ξ is independent of λ and has a geometric distribution +P[ξ = k] = a(1 − a)k−1, k ≥ 1 for some a ∈ (0, 1). Then the sparse random environment +given in (1.2) is equivalent to an i.i.d. environment with ω0 distributed as P[ω0 ∈ ·] = +aP[λ ∈ ·] + (1 − a)δ1/2(·). +Random walk in a sparse random environment was studied in detail in the annealed +setting in [17, 6, 5]. In [17] the authors address the question of transience and recurrence of +RWSRE and prove a strong law of large numbers and some distributional limit theorems +for X. As in the case of i.i.d. random environment, the fraction +ρ = 1 − λ +λ +appears naturally in the description of the asymptotic behaviour of the random walk. +According to [17, Theorem 3.1], X is P-a.s. transient to +∞ if +(1.3) +E log ρ ∈ [−∞, 0) +and +E log ξ < ∞. +Note that the first condition in (1.3) excludes the degenerate case ρ = 1 a.s. in which X +is a simple random walk. Under (1.3), the RWSRE also satisfies a strong law of large +numbers, that is, +(1.4) +Tn/n → 1/v +P − a.s. +where +v = +� +(1−Eρ)Eξ +(1−Eρ)Eξ2+2EρξEξ +if Eρ < 1, Eρξ < ∞ and Eξ2 < ∞, +0 +otherwise, +see Theorem 3.3 in [17] and Proposition 2.1 in [6]. We note right away that conditions +present in (1.3) are satisfied under the conditions of our main results. Thus, the random +walks in a sparse random environment that we treat here are transient to the right. +The asymptotic behaviour of T is controlled by two ingredients. The first one, similarly +as in the case of i.i.d. environment, is α > 0 such that +(1.5) +E [ρα] = 1. +The parameter α > 0, if it exists, is used to quantify the effect that the random transition +probabilities λk’s have on the asymptotic behaviour of the random walker. The second + +WEAK QUENCHED LIMIT THEOREMS FOR RWSRE +5 +Figure 1.2. RWSRE: β = 1.2 and α ≈ 0.52 (left) and β = 1.2 and +α ≈ 1.85 (right). The grey horizontal lines indicate the marked sites +ingredient is the tail behaviour of ξ, that is the asymptotic of P[ξ > t] as t → ∞. If +E[ξ4] < ∞, then with respect to the annealed probability T is in the domain of attraction +of an α-stable distribution [6, Theorem 2.2] with the exact same behaviour as one observes +in the case of i.i.d. environment. New phenomena appear if ξ has a regularly varying tail +with index −β for β ∈ (0, 4), i.e. as t → ∞, +P[ξ > t] ∼ t−βℓ(t) +for some function ℓ: R → R slowly varying at infinity. Here and in the rest of the article +we write f(t) ∼ g(t) for two functions f, g ∈ R → R whenever f(t)/g(t) → 1 as t → ∞. +Recall that a function ℓ is slowly varying at infinity if ℓ(ct) ∼ ℓ(t) as t → ∞ for any +constant c > 0. It transpires that if the tail of ξ is regularly varying with β ∈ (0, 4) with +E[ξ] < ∞, then with respect to the annealed probability T lies in the domain of attraction +of γ-stable distribution with γ = min{α, β/2}, see [6]. +For small values of β one sees an interplay between the contribution of the sparse random +environment and the random movement of the process in the unmarked sites. To state +this result take ϑ to be a non-negative random variable with the Laplace transform +(1.6) +E +� +e−sϑ� += +1 +cosh(√s), +s > 0. +Note that 2ϑ is equal in distribution to the exit time of the one-dimensional Brownian +motion from the interval [−1, 1], see [23, Proposition II.3.7]. Next consider a measure η +on K = [0, ∞]2 \ {(0, 0)} given via +η({(v, u) ∈ K : u > x1 or v > x2}) = x−β +1 ++ E[ϑβ/2]x−β/2 +2 +− E[min{x−β +1 , ϑβ/2x−β/2 +2 +}] +for x1, x2 > 0. Now let N = � +k δ(tk,jk) be a Poisson point process on [0, ∞) × K with +intensity LEB ⊗ η, where LEB stands for the one-dimensional Lebesgue measure. Under +mild integrability assumptions, see [5, Lemma 6.4], the integral +L(t) = (L1(t), L2(t)) = +� +[0,t]×K +j N(ds, dj), +t ≥ 0 +converges and defines a two-dimensional non-stable L´evy process with L´evy measure η. +Next consider the β-inverse subordinator +L← +1 (t) = inf{s > 0 : L1(s) > t}, +t ≥ 0. + +200 +150 +100 +50 +0 +-50 +0 +2000 +4000 +6000 +8000 +10000800 +600 +400 +200 +0 +-200 +0 +2000 +4000 +6000 +8000 +100006 +D. BURACZEWSKI, P. DYSZEWSKI AND A. KO�LODZIEJSKA +Figure 1.3. RWSRE: β = 0.8 and α ≈ 1.85. The grey horizontal lines +indicate the marked sites. +Finally, if β < 2α and β ∈ (0, 1), then under some additional mild integrability assump- +tions [5, Theorem 21], with respect to the annealed probability +Tn/n2 ⇒ 2L2(L← +1 (1)−) + 2ϑ(1 − L1(L← +1 (1)−))2 +weakly in R. The aim of the present article is to present a quenched version of this result. +As we will see in our main theorem, the terms L2(L← +1 (1)−) and L1(L← +1 (1−)) present +on the right hand side can be viewed as the contribution of the environment, whereas ϑ +represents the contribution of the movement of the random walker in the unmarked sites +that are close to n. For the full treatment of the annealed limit results, in particular the +complementary case β ≥ 2α, we refer the reader to [5]. +The article is organised as follows: in Section 2 we give a precise description of our set- +up and main results. In Section 3 we provide a preliminary analysis of the environment. +The essential parts of the proof of our main results are in Sections 4 and 5 where we prove +an absence of the strong quenched limits and prove weak quenched limits respectively. +2. Weak quenched limit laws +In this section we will present our main results. From this point we will consider only +a sparse random environment given via (1.2). We assume that +(2.1) +P[ξ > t] ∼ t−βℓ(t) +for some β ∈ (0, 4) and slowly varying ℓ. We will focus on the case in which the asymptotic +of the system is not determined solely by the environment and thus we will assume also +that +(2.2) +E[ρ2γ] < 1, +E[ξγρ3γ] < ∞ +for some parameter γ ∈ (β/4, 1 ∧ β). The first condition in (2.2) guarantees that a part of +the fluctuations of Tn will come from the time that the process spends in the unmarked +sites. The second condition is purely technical. Note that we do not assume that there +exists α > 0 for which (1.5) holds. +Our first result states that there is no quenched limit for Tn’s in the strong sense. Take +(an)n∈N to be any sequence of positive real numbers such that +nP[ξ > an] → 1. + +250 +200 +150 +100 +50 +50 +0 +2000 +4000 +6000 +8000 +10000WEAK QUENCHED LIMIT THEOREMS FOR RWSRE +7 +Then, since the tail of ξ is assumed to be regularly varying, the sequence (an)n∈N is also +regularly varying with index 1/β. That is for some slowly varying function ℓ1, +an = n1/βℓ1(n). +The sequence (an)n∈N will play the role of the scaling factor in our results. The first one +shows an absence of strong quenched limit laws for T. +Theorem 2.1. Assume (1.3), (2.1) and (2.2). +Then for P almost every ω there are +no sequences {An(ω)}n∈N and {Cn(ω)}n∈N such that the sequence of normalized random +variables (Tn −Cn(ω))/An(ω) converges in distribution (with respect to Pω) to a nontrivial +random variable. +Therefore, as in the case of i.i.d. environment, the asymptotic quenched behaviour of +Tn’s ought to be expressed in terms of weak quenched convergence. As it is the case for +annealed limit theorem, one needs to distinguish between a moderately (Eξ < ∞) and +strongly (Eξ = ∞) sparse random environment. +To describe the former take {ϑj}j∈N to be a sequence of i.i.d. copies of ϑ distributed +according to (1.6) and let G : Mp → M1 be given via +(2.3) +G(ζ)(·) = +� +P +�� +i≥1 xi(2ϑi − 1) ∈ · +� +, +� +x2ζ(dx) < ∞, +δ0(·) +otherwise, +for ζ = � +i≥1 δxi, where {xi} is an arbitrary enumeration of the point measure. +Theorem 2.2. Assume (1.3), (2.1) and (2.2). If Eξ < ∞, then +Pω +� +(Tn − EωTn)/a2 +n ∈ · +� +⇒ G(N)(·) +in M1, where N is a Poisson point process on (0, ∞) with intensity βx−β/2−1dx/2Eξ. +Before we introduce the notation necessary to state our results in the strongly sparse +random environment, we will first treat the critical case which is relatively simple to state. +Denote +mn = nE +� +ξ1{ξ≤an} +� +. +Note that by Karamata’s theorem [2, Theorem 1.5.11] the sequence {mn}n∈N is regularly +varying with index 1/β. Furthermore an = o(mn) if β = 1 and an ∼ (1 − β)mn if β < 1. +Next let {cn}n∈N be the asymptotic inverse of {mn}n∈N, i.e. any increasing sequence of +natural numbers such that +lim +n→∞ cmn/n = lim +n→∞ mcn/n = 1. +By the properties of an asymptotic inversion of regularly varying sequences [2, Theorem +1.5.12], cn is well defined up to asymptotic equivalence and is regularly varying with index +β. Finally, by the properties of the composition of regularly varying sequences {acn}n∈N +is regularly varying with index 1 and acn = o(n) if β = 1. +Theorem 2.3. Assume (1.3), (2.1) and (2.2). If Eξ = ∞ and β = 1, then +Pω +� +(Tn − EωTn)/a2 +cn ∈ · +� +⇒ G(N)(·) +in M1, where N is a Poisson point process on (0, ∞) with intensity x−3/2dx/2. +The limiting random measures in Theorems 2.2 and 2.3 share some of the properties of +their counterpart in the case of i.i.d. environment [19, Remark 1.5]. Namely, using the +superposition and scaling properties of Poisson point processes, one can directly show that +for each n ∈ N and G, G1, . . . , Gn being i.i.d. copies of the limit random measure G(N) +in Theorem 2.2 or Theorem 2.3, +(2.4) +G1 ∗ G2 ∗ . . . ∗ Gn(·) d= G(·/n2/β). + +8 +D. BURACZEWSKI, P. DYSZEWSKI AND A. KO�LODZIEJSKA +The statement of our results in the strongly sparse case needs some additional notation. +As it is the case for the annealed results, it is most convenient to work in the framework +of non-decreasing c`adl`ag functions rather than point processes. Denote by D↑ the class of +non-decreasing c`adl`ag functions R+ → R+ and for h ∈ D↑ consider +(2.5) +Υ(h) = sup{h(t) : t ∈ R+, h(t) ≤ 1}. +Finally for h ∈ D↑ denote by {xk(h), tk(h)}k an arbitrary enumeration of jumps of h, that +is tk = tk(h) ∈ R+ for k ∈ N are all points on the non-negative half line such that h +has a (left) discontinuity with jump of size xk(h) = h(tk) − h(t− +k ) > 0 at tk. Note that +the random series � +k:h(tk)≤1 xk(h)2(2ϑk − 1) is convergent since it has an expected value +bounded via h(1)E|2ϑ − 1|. Finally let F : D↑ → M1 be given by +F(h)(·) = P +� +�(1 − Υ(h))2(2ϑ0 − 1) + +� +k:h(tk)≤1 +xk(h)2(2ϑk − 1) ∈ · +� +� . +Note that if h(t) = 1 for some t, then necessarily Υ(h) = 1. +Theorem 2.4. Assume (1.3), (2.1) and (2.2). If β ∈ (0, 1), then +Pω +� +(Tn − EωTn)/n2 ∈ · +� +⇒ F(L)(·) +in M1, where L is a β-stable L´evy subordinator with L´evy measure ν(x, +∞) = x−β. +Interestingly the limit measure F(L) does not enjoy a self-similarity property in the +sense of (2.4). Namely, for any a, b ∈ R, b > 0 the laws of +F1 ∗ F2(·) +and +F((· − a)/b) +are different, where F, F1 and F2 are independent copies of the limiting random measure +F(L) in Theorem 2.4. +3. Auxiliary results +We will now present a few lemmas that we will use in our proofs. We will discuss prop- +erties of some random series as well as the asymptotic behaviour of the hitting times (1.1). +3.1. Estimates for the related stochastic processes {Ri}i∈Z and {Wi}i∈Z. We will +frequently make use of the following notation: for integers i ≤ j, +(3.1) +Πi,j = +j� +k=i +ρk, +Ri,j = +j +� +k=i +ξkΠi,k−1, +Wi,j = +j +� +k=i +ξkΠk,j. +We will also make use of the the limits +(3.2) +Ri = lim +j→∞ Ri,j = +∞ +� +k=i +ξkΠi,k−1, +Wj = lim +i→−∞ Wi,j = +j +� +k=−∞ +ξkΠk,j. +Note that if E log ρ < 0 and E log ξ < ∞, both series are convergent as one can see by +a straightforward application of the law of large numbers and the Borel-Cantelli lemma +(see [3, Theorem 2.1.3]). The random variables Ri’s and Wj’s have the same distribution +and obey the recursive formulae +Ri = ξi + ρiRi+1 +and +Wj = ρjξj + ρjWj−1. +We can therefore invoke the proof of [3, Lemma 2.3.1] to infer the following result on the +existence of moments of Ri’s and Wj’s. In what follows we write R (respectively W) for +a generic element of {Ri}i∈Z (respectively {Wj}j∈Z). +Lemma 3.1. Let α > 0. If Eρα < 1, Eραξα < ∞ and Eξα < ∞, then ERα and EW α are +both finite. + +WEAK QUENCHED LIMIT THEOREMS FOR RWSRE +9 +3.2. Hitting times. We describe now some properties of the sequence of stopping times +T = {Tn}n∈N that allow us to better understand the process X and indicate its ingredients +which play an essential role in the proof of our main results. We will first analyse the hitting +times T along the marked sites S, that is +TSi = inf{n : Xn = Si}. +As it turns out, one can use Ri,j’s given in (3.1) to represent the exit probabilities from +interval (Si, Sj). That is, for i < k < j we have +(3.3) +PSk +ω [TSi > TSj] = Ri+1,k +Ri+1,j +, +PSk +ω [TSi < TSj] = Πi+1,k +Rk+1,j +Ri+1,j +. +see the proof of [28, Theorem 2.1.2]. Let +Tk = TSk − TSk−1 +be the time that the particle needs to hit k’th marked point Sk after reaching Sk−1. One +uses Wj’s to describe the expected value of Tk: +(3.4) +EωTk = ESk−1 +ω +TSk = ξ2 +k + 2ξkWk−1, +see the proof of [28, Lemma 2.1.12]. +Observe that the random variable Tk can be decomposed into a sum of two parts: the +time the trajectory, after reaching Sk−1 but before it hits Sk, spends to the left of Sk−1 +and the time it spends to the right of Sk−1. For technical reasons that will become clear +below, we divide the visits exactly at point Sk−1 between these two sets depending on the +direction from which the particle enters Sk−1. To be precise we define +Tl +k = # +� +n ∈ (TSk−1, TSk] : Xn < Sk−1 or (Xn−1, Xn) = (Sk−1 − 1, Sk−1) +� +, +i.e. Tl +k is the sum of the time the particle spends in (−∞, Sk−1 − 1] and the number of +steps from Sk−1 − 1 to Sk−1. Similarly we define +Tr +k = # +� +n ∈ (TSk−1, TSk] : Sk−1 < Xn ≤ Sk or (Xn−1, Xn) = (Sk−1 + 1, Sk−1) +� +. +Thus we can write +Tk = TSk − TSk−1 = Tl +k + Tr +k. +Observe that given ω, the random variables {Tk}k∈N are Pω independent, however for +fixed k, Tl +k and Tr +k mutually depend on each other. Summarizing, we obtain the following +decomposition that will be used repeatedly: +TSk = +k +� +j=1 +Tj = +k +� +j=1 +Tl +j + +k +� +j=1 +Tr +j =: T l +Sk + T r +Sk. +To proceed further we need to analyse Tr +j, Tl +j in details and describe their quenched +expected value and quenched variance. +Below we prove that after hitting any of the +chosen sites (Sk)k the consecutive excursions to the left are negligible. This entails that +behaviour of TSk is determined mainly by T r +Sk. +3.3. The sequence {T r +Sn}. Note that, under Pω, Tr +k equals in distribution to the time +it takes a simple random walk on [0, ξk] with a reflecting barrier placed in 0 to reach ξk +for the first time when starting from 0. This is the reason we include into Tr +k the visits +at Sk−1, but only those from Sk−1 + 1. Indeed, let (Yn)n be a simple random walk on Z +independent of the environment ω. Define +(3.5) +Un = inf{m : |Ym| = n}, +i.e. Un is the first time the reflected random walk hits n. Then for every k > 0, for +fixed environment ω, Tr +k +d= Uξk. +In what follows we investigate how the asymptotic +properties of ξk affect those of Tr +k. To do that, we will utilize the aforementioned equality + +10 +D. BURACZEWSKI, P. DYSZEWSKI AND A. KO�LODZIEJSKA +in distribution and hence we first need to describe the asymptotic properties of Un as n +tends to infinity. The proof of the next lemma is omitted, since it follows from a standard +application of Doob’s optimal stopping theorem to martingales Y 2 +n −n, Y 4 +n −6nY 2 +n +3n2+2n, +Y 6 +n − 15nY 4 +n + (45n2 + 30n)Y 2 +n − (15n3 + 30n2 + 16n) and exp{±tYn}cosh(t)−n. +Lemma 3.2. Let Un, for n ∈ N be given in (3.5). We have +EUn = n2, +EU 2 +n = 5n4/3 − 2n2/3. +Moreover, as n → ∞, +Un/n2 ⇒ 2ϑ, +for ϑ defined in (1.6). Furthermore the family of random variables {n−4U 2 +n}n∈N is uni- +formly integrable. +The sequence T r +Sn = �n +k=1 Tr +k is a sum of Pω independent random variables {Tr +k}. Since, +by Lemma 3.2, +(3.6) +VarωTr +k = 2 +3ξ4 +k − 2 +3ξ2 +k, +the variance VarωT r +Sn behaves asymptotically as (2/3) �n +k=1 ξ4 +k, thus obeys a stable limit +theorem [11, Theorem 3.8.2]. Moreover, we can use precise large deviation results for sums +of i.i.d. regularly varying random variables [8, Theorem 9.1] to describe the deviations of +VarωT r +Sn. That is for any sequence {αn} that tends to infinity, +P[VarωT r +Sn ≥ αna4 +n] ∼ (2/3)β/4nα−β/4 +n +a−β +n ℓ(α1/4 +n an). +We can now use Potter bounds [2, Theorem 1.5.6] to control ℓ(α1/4 +n an) with ℓ(an). This +in turn yields a large deviation result asymptotic on the logarithmic scale. We summarize +this discussion in the following lemma. +Corollary 3.3. The sequence {VarωT r +Sn/a4 +n}n∈N converges in distribution (with respect to +P) to some stable random variable Z. Moreover for any sequence {αn}n∈N that tends to +infinity, +log P[VarωT r +Sn ≥ αna4 +n] ∼ −β log(αn)/4. +3.4. The sequence {T l +Sn}. The structure of Tl +k is more involved. We may express it as a +sum of independent copies of Fk, which denotes the length of a single excursion to the left +from Sk, and thus obtain formulae for its quenched expectation and quenched variance. +Lemma 3.4. The following formulae hold +EωFk = 2(ξk + Wk−1), +VarωFk = 8 +� +j Tξ1) = 1/ξ1. Thus the particle starting at 1 hits the point 0 M1 times +before it reaches ξ1, where M1 is geometrically distributed with parameter 1/ξ1 and mean +ξ1 − 1. This is exactly the number of visits to 0 counted by Tr +1. Between consecutive steps +from 0 to 1, let’s say between mth and (m + 1)’th step, the particle spends some time in +(−∞, 0]. In particular its visits at 0 from the left are exactly those included in Tl +1. Let +us denote such an excursion by G0(m) and denote by G0 its generic copy. That is, G0 +(Gk, resp.) is the time the particle spends in (−∞, 0] ((−∞, Sk], resp.) before visiting 1 +(Sk + 1, resp.). Then G0 +d= T1 − 1 (or more generally Gk +d= TSk+1 − TSk − 1). +The random variable G0 consists of Nm disjoint excursions in (−∞, −1], where Nm is +the number of jumps from 0 to −1 before the next step to 1. Since the particle can jump to +−1 with probability (1 − λ0), Nm has geometric distribution with mean ρ0. Summarizing, +Tl +1 can be decomposed as +(3.9) +Tl +1 = +M1 +� +m=0 +G0(m) = +M1 +� +m=0 +Nm +� +j=1 +F0(j, m), +where F0(j, m) measures the length of a single left excursion from 0. Observe that both +Nm’s and F0(j, m)’s are i.i.d. under Pω. Moreover, the first sum includes m = 0, because +the process starts at 0. +Recall that if SN = �N +k=1 Xi for some random variable N and an i.i.d. sequence {Xn} +independent of N, then +(3.10) +VarSN = EN · VarX + VarN · (EX)2. +The above formula together with (3.9) easily entails +EωTl +1 = ξ1ρ0EωF0, +VarωTl +1 = ξ1ρ0VarωF0 + +� +ξ2 +1ρ2 +0 + ξ1ρ0 +� +(EωF0)2. +(3.11) +Since F0 is the time of a single excursion from 0 that begins with a step left, using the +solution to the classical ruin problem in combination with formula (3.4) we get +EωF0 = 1 + E−1 +ω T0 = 1 + (ξ0 − 1) + 1 +ξ0 +ES−1 +ω +TS0 = 2(ξ0 + W−1), +A formula for quenched variance of crossing times for arbitrary neighbourhood was given +in [13, Lemma 3] and yields (3.7). Inserting these formulae to (3.11), using the fact that +ρ0W−1 = W0 − ξ0ρ0, and finally simplifying the expression leads to (3.8). +□ +Lemma 3.5. For every ε > 0 and θ ≥ 0, +P +� +VarωT l +Sn ≥ εnθa4 +n +� +≤ o(1)/nθγ, +n → ∞, +where γ is a parameter satisfying (2.2). In particular, +1 +a4n +VarωT l +Sn +P→ 0. +Proof. To prove the lemma one needs to deal with the formula for the variance (3.8). To +avoid long and tedious arguments we will explain how to estimate two of the terms, i.e. +we will prove +(3.12) +P +� +n +� +k=1 +ξk · +� +j max{1/(4γ), 1/β}. Notice that one may take e.g. pn = 22n, qn = pn+1/4. +We will need to prove that behaviour of the process in the interval [S2pn, S2qn] is de- +termined by its position after time T2pn and that its previous values up to time 2pn are +negligible when looking at time qn and scale aqn. The trajectory of the random walk X +cannot be divided into independent pieces with respect to P, because the process can have +large excursions to the left and the environment is not homogeneous. To remedy that we +will censor the left excursions of X that become too large. We consider a new process, +say X = {Xk}k∈N. This process essentially behaves as the previous one and evolves in + +WEAK QUENCHED LIMIT THEOREMS FOR RWSRE +13 +the same environment, with a small difference. Namely after X reaches Sqn and before +it reaches S2qn we put a barrier at point Spn, i.e. the process cannot come back below +Spn. However this barrier is removed when X hits S2qn. Of course we can couple both +processes on the same probability space removing all left excursions from Spn after hitting +Sqn and before reaching S2qn. +For any k, we define the random variables T k, Tk, T +r +k, T +l +k in an obvious way, e.g. +T k = inf{j : Xj = k}, +Tk = T Sk − T Sk−1. +Then T +r +k = Tr +k for all k′s and T +l +k = Tl +k for k /∈ � +n(qn, 2qn]. Notice that Tk − Tk is the time +that the process X spends below Spk after hitting Sk−1 and before reaching Sk. The next +lemma ensures that asymptotic properties of the processes X and X are comparable. +Lemma 4.1. For any ε ∈ (0, 1) and P-a.e. ω there is N = N(ω) such that +(4.3) +� +qn N. +Moreover +Tn = Tn a.s. for large (random) n. +Proof. Fix k ∈ (qn, 2qn]. To describe the quenched mean and the quenched variance of +Tk − Tk = Tl +k − T +l +k we need to calculate the time the trajectory X, after it hits Sk−1, +but before reaching Sk, spends below Spn. For this purpose we proceed as in the proof of +Lemma 3.4, that is we decompose +(4.5) +Tk − Tk = +Mk +� +m=1 +Nm +� +j=0 +Fpn(j, m), +where Mk denotes the number of times the walk visits Spn from the right in the time +interval (TSk−1, TSk), Nm is the number of consecutive left excursions from Spn after hitting +it from the right, and Fpn(j, m) is the length of the corresponding excursion. Note that Nm +is geometrically distributed with mean ρpn and variance ρpn(1 + ρpn). Thus, by formulae +(3.10) and (3.7), +Eω +� Nm +� +j=0 +Fpn(j, m) +� += ρpnEωFpn = 2Wpn +Varω +� Nm +� +j=0 +Fpn(j, m) +� += ρpnVarωFpn + ρpn(1 + ρpn) (EωFpn)2 . +(4.6) +Next, observe that for any m > 0, Pω [Mk = m] = rsm−1(1 − s), where +r = PSk−1 +ω +� +TSpn < TSk +� +and, invoking once again the gambler’s ruin problem, +s = PSpn+1 +ω +� +TSpn < TSk +� += 1 − +1 +ξpn+1 +PSpn+1 +ω +� +TSpn > TSk +� +. + +14 +D. BURACZEWSKI, P. DYSZEWSKI AND A. KO�LODZIEJSKA +We may easily calculate the mean and variance of Mk and use the formulae (3.3) to express +them in terms of the environment. We get, after simplifying, +EωMk = +r +1 − s = ξkΠpn+1,k−1, +VarωMk = r(1 + s − r) +(1 − s)2 += ξkΠpn+1,k−1 (2Rpn+1,k−1 + ξkΠpn+1,k−1 − 1) . +(4.7) +Therefore, by (3.1), (4.6) and (4.7), +Eω +� +Tk − Tk +� += 2ξkΠpn+1,k−1Wpn, +Varω +� +Tk − Tk +� += ξkΠpn,k−1VarωFpn ++ ξkΠpn,k−1 (EωFpn)2 +� +�1 + 2 +k−1 +� +j=pn+1 +ξjΠpn,j−1 + ξkΠpn,k−1 +� +� . +(4.8) +Now, we are ready to prove (4.3). We have +P +� +� +qn 0 let +Un(d, D, b, B, ε) = +� +there exists k ∈ (qn, 2qn] such that +Varω(T +r +Sk − T +r +S2pn) +a4 +k+1 +∈ (d, D), +Varω(T +l +Sk − T +l +S2pn) +a4 +k+1 +≤ ε, +VarωGk + (EωGk)2 +ak+1 +≤ ε, +ξ4 +k+1 +a4 +k+1 +∈ (b, B) +� +, +where Gk is the length of the left excursion of X from Sk before hitting Sk + 1. Of course +EωGk ≤ EωGk and VarωGk ≤ EωG2 +k. We want to consider environments which belong to +infinitely many sets Un. However, given ω, we want to have some freedom of choosing all +the parameters. The lemma below justifies that the measure of these environments is one. +Lemma 4.2. Assume that conditions (2.1) and (2.2) are satisfied. Then the event +U = +� � +lim sup +n +Un(d, D, b, B, ε) : d, D, b, B ∈ Q+, d < D, b < B, b > 3 · 24/βD, ε > 0 +� + +WEAK QUENCHED LIMIT THEOREMS FOR RWSRE +15 +has probability one. +Proof. Since in the above formula the intersections are essentially over a countable set of +parameters (one can obviously restrict to the rational parameter ε), it is sufficient to prove +that for fixed parameters d < D, b < B such that b > 3 · 25/β−1D and ε > 0, +P +� +lim sup +n +Un +� += 1, +for Un = Un(d, D, b, B, ε). Observe that the events {Un}n∈N are independent, because +Un depends only on {ωj}j∈[pn,2qn] and thanks to (4.2) the sets {[pn, 2qn]}n∈N are pairwise +disjoint. Thus, invoking the Borel-Cantelli Lemma, it is sufficient to prove that there is +δ0 > 0 such that for large indices n, +(4.9) +P[Un] > δ0. +We need to estimate probabilities of all the events which appear in the definition of Un. +Denote +V 1 +k = +� +Varω(T +r +Sk − T +r +S2pn)/a4 +k+1 ∈ (d, D) +� +, +V 2 +k = +� +Varω(T +l +Sk − T +l +S2pn)/a4 +k+1 ≤ ε +� +, +V 3 +k = +�� +VarωGk + (EωGk)2� +/ak+1 ≤ ε +� +, +V 4 +k+1 = +� +ξ4 +k+1/a4 +k+1 ∈ (b, B) +� +. +(4.10) +To estimate the probability of V 1 +k notice that thanks to (4.2) we have ak−2pn/ak+1 → 1 +for any k ∈ (qn, 2qn], therefore +P[V 1 +k ] = P +� +2/3 � +2pn D. +Summarizing, for some δ′ > 0 and large n +P[Un] = P +� +� +qn 0 VarωTS2pn ≥ a4 +qnε i.o. +� += 0. +Therefore, invoking Lemma 4.2 the set +U ∩ +� +∀ε > 0 VarωTS2pn ≥ a4 +qnε i.o. +�c + +16 +D. BURACZEWSKI, P. DYSZEWSKI AND A. KO�LODZIEJSKA +has probability 1. From now we fix ω from the event above which also satisfies the claim +of Lemma 4.1. Assume that, given ω, +(4.11) +�Tn = Tn − EωTn +√VarωTn +⇒ Yω +n → ∞, +for some random variable Yω. We fix parameters d < D, b < B such that b > 3 · 25/β−1D +and ε > 0. +Take two sequences {nm}m∈N and km ∈ (qnm, q2nm] such that +ω ∈ V 1 +km ∩ V 2 +km ∩ V 3 +km ∩ V 4 +km+1, +where all the sets were defined in (4.10). We can additionally assume (removing a finite +number of elements of the sequence if needed), that for all indices m +(4.12) +VarωTS2pnm < a4 +kmε. +Lemma 4.1 says that, given ω, the difference (Tn − EωTn) − (T n − EωT n) remains a.s. +bounded and VarωTn/VarωT n converges to 1, hence (4.11) yields +(4.13) +T n − EωT n +� +VarωT n +⇒ Yω +n → ∞. +Consider the following decomposition, +(4.14) +T Skm+1 − EωT Skm+1 +� +VarωT Skm+1 += vm · Vm + wm · Wm + Zm, +where +Vm = T Skm − EωT Skm +� +VarωT Skm +, +vm = +� +VarωT Skm +� +VarωT Skm+1 +, +Wm = T +r +km+1 − EωT +r +km+1 +ξ2 +km+1 +, +wm = +ξ2 +km+1 +� +VarωT Skm+1 +, +Zm = T +l +km+1 − EωT +l +km+1 +� +VarωT Skm+1 +. +Random variables Vm and (Wm, Zm) are Pω-independent. By (4.13), Vm converges in +distribution to Yω, whereas Wm, by Lemma 3.2, converges to 2ϑ−1. Therefore we need to +understand the behaviour of both deterministic (given ω) sequences {vm}m∈N, {wm}m∈N +and of the sequence of random variables {Zm}. For this purpose we need to understand +behaviour of the variances which appear in the above formulae. Note first that on the +considered event, recalling (3.10), we have +(4.15) +VarωT +l +km+1 ≤ ξkm+1VarωGkm + ξ2 +km+1(EωGkm)2 ≤ εa3 +km+1B1/2. +Applying the Schwartz inequality and the well-known inequality 2ab ≤ a2/C + Cb2, for +any n and arbitrary large constant C, we can easily prove +−VarωT +r +n/C + (1 − C)VarωT +l +n ≤ VarωTn − VarωT +r +n ≤ VarωT +r +n/C + (1 + C)VarωT +l +n. +Combining the above inequalities with (4.12) and the definition of Un, we have +� +ε(1 − C) + +� +1 − 1 +C +� +d +� +a4 +km+1 ≤ VarωT Skm ≤ +� +ε + ε(1 + C) + +� +1 + 1 +C +� +D +� +a4 +km+1 + +WEAK QUENCHED LIMIT THEOREMS FOR RWSRE +17 +and� +o(1)(1 − C) + +� +1 − 1 +C +� +b +� +a4 +km+1 ≤ VarωTkm+1 ≤ +� +o(1)(1 + C) + +� +1 + 1 +C +� +B +� +a4 +km+1. +The above inequality ensures that VarωT Skm+1 is of the order a4 +km+1. For arbitrary small +δ > 0, choosing appropriate parameters ε, C and large indices km +(4.16) +(1 − δ) +d +D + B ≤ v2 +m ≤ (1 + δ) · +D +d + b +and +(4.17) +(1 − δ) +b +D + B ≤ w2 +m ≤ (1 + δ) · +B +d + b +We can pass with δ to 0 and assume that the parameters satisfy +(4.18) +d +D + B ≤ lim inf +n→∞ v2 +m ≤ lim sup +n→∞ v2 +m ≤ +D +d + b +and +(4.19) +b +D + B ≤ lim inf +n→∞ w2 +m ≤ lim sup +n→∞ w2 +m ≤ +B +d + b. +Observe that for any η > 0, using the Chebyshev inequality and (4.15), we have: +Pω[|Zm| > η] = P +����T +l +km+1 − EωT +l +km+1 +��� > η +� +VarωT Skm+1 +� +≤ +Ca3 +km+1 +η2VarωT Skm+1 +→ 0. +This proves +(4.20) +Zm +Pω +→ 0, +m → ∞. +One can easily see that for any fixed d and D one can construct sequences {bm}, {Bm}, {km} +such that bm, Bm → ∞, bm/Bm → 1 and inequalities (4.18) and (4.19) hold. Then vm → 0 +and wm → 1. Since the sequence {Vm} is tight, we have +T Skm+1 − EωT Skm+1 +� +VarωT Skm+1 += vm · Vm + wm · Wm + Zm ⇒ 2ϑ − 1. +So, if the limits (4.11), (4.13) exist, both must be equal to Yω = 2ϑ − 1. +Next, fixing all the parameters b, B, d, D observe that both sequences {vm}, {wm} are +bounded, therefore we can assume, possibly choosing their subsequences, that they are +convergent to some v and w, respectively. Since the sequences of random variables {Vm} +and {Wm} are independent, we conclude +(4.21) +2ϑ − 1 d= Yω +d= v(2ϑv − 1) + w(2ϑw − 1), +where ϑv, ϑw are independent. However this equation cannot be satisfied e.g. by (1.6). +That leads to a contradiction and proves that the limit (4.11) cannot exist. +Summarizing, we have proved up to now that the sequence { �Tn} defined in (4.1) cannot +converge in distribution. Nevertheless, it still can happen that different normalization +leads to a convergent sequence and we need to exclude this possibility. Let us consider +another normalization �Tn = �Tn/An + Cn for some sequences {An} and {Cn}. Let us also +recall that we already know that if the limit exists, it must be equal to 2ϑ − 1. +Observe first that An must be bounded. Indeed, assume that there exists its subsequence +{Ank} converging to +∞. Then, since { �Tn} is tight, we have �Tnk/Ank +P→ 0 and thus the +sequence {Cnk} must converge to some limit C and finally �Tnk ⇒ C, which is a trivial +limit. + +18 +D. BURACZEWSKI, P. DYSZEWSKI AND A. KO�LODZIEJSKA +Next we fix parameters b, B, d, D and consider the subsequence {km} satisfying all the +above requirements. If the sequence {An} contains a subsequence {Akm} convergent to a +positive constant A, then again the sequence {Ckm} must converge to some C. Repeat- +ing the above calculations we are led once more to equation (4.21), which leads us to a +contradiction. +Thus, the sequence {Akm} must converge to 0. However in this case, since wnWm + +Zm ⇒ w(2ϑ−1), the sequence {(wmWm +Zm)/Akm} cannot be tight and finally, since Vn +is independent, by (4.14) the sequence �Tkm cannot be tight and converge in distribution. +This completes the proof of the theorem. +□ +5. Proofs of the weak quenched limit laws +In this final section we present a complete proof of our main results. We will begin +by presenting a suitable coupling. Then we will treat the moderately sparse and strongly +sparse case separately. +5.1. Coupling. In the first step we will prove our result along the marked sites. That is +we analyse +(5.1) +φn,ω(·) = Pω +� +a−2 +n (TSn − Eω[TSn]) ∈ · +� +. +The main part of the argument concentrates on the limit law of T r +Sn = �n +k=1 Tr +k. Recall +Un defined in (3.5), which is the first time the reflected random walk hits n. For every +k > 0 and for fixed environment ω it holds Tr +k +d= Uξk. By the merit of Lemma 3.2 and +Skorokhod’s representation theorem we may assume that our space holds random variables +U (k) +n +and ϑk such that: +• {U (k) +n }n, ϑk for k ∈ N, are independent copies of {Un}n, ϑ; +• {U (k) +n , ϑk : n, k ∈ N} and {ξk : k ∈ N} are independent; +• U (k) +n /n2 → 2ϑk in L2 as n → ∞; +• for all ω, U (k) +ξk +and Tr +k have the same distribution under Pω. +Observe that the convergence in L2 is secured by the convergence in distribution and +uniform integrability provided in Lemma 3.2. +To simplify the notation we will write Uξk instead of U (k) +ξk . +Proposition 5.1. Assume (2.1). Then as n → ∞, +a−4 +n Varω +� +T r +Sn − EωT r +Sn − +n +� +k=1 +ξ2 +k(2ϑk − 1) +� +P→ 0. +Proof. Firstly note that +Varω +� +T r +Sn − EωT r +Sn − +n +� +k=1 +ξ2 +k(2ϑk − 1) +� += Varω +� n +� +k=1 +� +Uξk − 2ξ2 +kϑk +� +� +. +For ε > 0 let I1 +n = {k ≤ n : ξk > εan} and I2 +n = {k ≤ n : ξk ≤ εan}. Then for any δ > 0, +(5.2) +P +� +Varω +� n +� +k=1 +� +Uξk − 2ξ2 +kϑk +� +� +> δa4 +n +� +≤ +P +� +� � +k∈I1n +ξ4 +kVarω +�Uξk +ξ2 +k +− 2ϑk +� +> δa4 +n +2 +� +� + P +� +� � +k∈I2n +ξ4 +kVarω +�Uξk +ξ2 +k +− 2ϑk +� +> δa4 +n +2 +� +� . + +WEAK QUENCHED LIMIT THEOREMS FOR RWSRE +19 +Since U (k) +n , ϑk are independent copies of Un, ϑ such that Un/n2 → ϑ in L2, there exists +M > 0 such that +Varω +�Uξk +ξ2 +k +− 2ϑk +� +< M +for all k, ω, +and, moreover, for N ∈ N large enough +Varω +� +U (k) +N +N2 − 2ϑk +� +< ε +for all k, ω. +We can hence estimate, for n sufficiently large, +P +� +� � +k∈I1n +ξ4 +kVarω +�Uξk +ξ2 +k +− 2ϑk +� +> δa4 +n +2 +� +� ≤ P +��n +k=1 ξ4 +k +a4n +> δ +2ε +� +. +Since the sequence �n +k=1 ξ4 +k/a4 +n converges weakly (under P) to some β/4-stable variable +Lβ/4, the probability on the right hand side above converges to P[Lβ/4 > δ/(2ε)]. To +estimate the second term in (5.2), note that +P +� +� � +k∈I2n +ξ4 +kVarω +�Uξk +ξ2 +k +− 2ϑk +� +> δa4 +n +2 +� +� ≤ P +� n +� +k=1 +ξ4 +k 1{ξk≤εan} > δa4 +n +2M +� +≤ 2M +δ a−4 +n E +� n +� +k=1 +ξ4 +k 1{ξk≤εan} +� += 2M +δ na−4 +n E +� +ξ4 +1 1{ξ1≤εan} +� +. +By the Fubini theorem, we have +E +� +ξ4 +1 1{ξ1≤εan} +� +≤ +� εan +0 +4t3P[ξ1 > t]dt +and the Karamata theorem [2, Theorem 1.5.11] entails that the expression on the right is +asymptotically equivalent to 4ε4a4 +nP[ξ1 > εan] ∼ 4ε4−βn−1a4 +n. Finally, we can conclude +that for any ε, δ > 0, +lim sup +n +P +� n +� +k=1 +ξ4 +kVarω +�Uξk +ξ2 +k +− ϑk +� +> δa4 +n +� +≤ 8M +δ ε4−β + P +� +Lβ/4 > δ +2ε +� +and passing with ε to 0 we conclude the desired result. +□ +We are now ready to determine the weak limit of the sequence φn(ω) = φn,ω given by +(5.1). Recall the map G defined in (2.3). +Lemma 5.2. The map G is measurable. +Remark 5.3. The proof of Lemma 5.2 is identical to that of Lemma 1.2 in [19] and therefore +will be omitted. Part of the proof is showing that the map +G2 : ℓ2 ∋ (xk)k∈N �→ P +� ∞ +� +k=1 +xk(2ϑk − 1) ∈ · +� +∈ M1(R) +is continuous. +Theorem 5.4. Assume (2.1) and (2.2). Then +φn ⇒ G(N∞), +in M1, where N∞ is a Poisson point process with intensity βx−β/2−1dx/2. +In the proof of this result we will use the following lemma. + +20 +D. BURACZEWSKI, P. DYSZEWSKI AND A. KO�LODZIEJSKA +Lemma 5.5 ([19, Remark 3.4]). Let θn be a sequence of random probability measures +on R2 defined on the same probability space. Let γn and γ′ +n denote the marginals of θn. +Suppose that +Eθn(X − Y ) P→ 0 +and +Varθn(X − Y ) P→ 0, +where X an Y are the coordinate variables in R2. If γn ⇒ γ, then γ′ +n ⇒ γ. +Proof of Theorem 5.4. First, observe that the sequence of random point measures Nn = +�n +k=1 δξ2 +ka−2 +n converges weakly to N∞. Indeed, this follows by an appeal to [21, Proposition +3.21] and checking that +nP[ξ2/a2 +n ∈ ·] → µ(·) +vaguely on (0, ∞] +where µ(dx) = βx−β/2−1dx/2. +Since G is not continuous, we cannot simply apply the continuous mapping theorem +and, similarly as in [19], we are forced to follow a more tedious argument. Define +Gε : Mp((0, ∞]) ∋ +∞ +� +k=1 +δxk �→ P +� ∞ +� +k=1 +xk(2ϑk − 1) 1{xk>ε} ∈ · +� +∈ M1(R). +Then for any ε > 0 the map Gε is continuous on the set Mε +p := {ζ ∈ Mp : ζ({ε, ∞}) = 0}; +indeed, take ζn, ζ ∈ Mε +p such that ζn → ζ vaguely. Then, by [21, Proposition 3.13], since +the set [ε, ∞] is compact in (0, ∞], there exists pε < ∞ and an enumeration of points of ζ +and ζn (for n sufficiently large) such that +ζn(· ∩ [ε, ∞]) = +pε +� +k=1 +δxn +k , +ζ(· ∩ [ε, ∞]) = +pε +� +k=1 +δxk +and +(xn +1, . . . , xn +pε) → (x1, . . . , xpε) +as n → ∞. +Therefore +Gε(ζn)(·) = P +� pε +� +k=1 +xn +k(2ϑk − 1) ∈ · +� +⇒ P +� pε +� +k=1 +xk(2ϑk − 1) ∈ · +� += Gε(ζ)(·). +By [1, Theorem 3.2], to prove that G(Nn) ⇒ G(N∞) it is enough to show +Gε(Nn) ⇒n Gε(N∞) +for all ε > 0, +(5.3) +Gε(N∞) ⇒ε G(N∞), +(5.4) +lim +ε→0 lim sup +n→∞ P [ρ(Gε(Nn), G(Nn)) > δ] = 0 +for all δ > 0, +(5.5) +where ρ is the Prokhorov metric on M1(R). +First, for any ε > 0, N∞ ∈ Mε +p almost surely. Thus (5.3) is satisfied by the continuous +mapping theorem since Gε is continuous. +For any sequence x = (xk)k∈N ∈ ℓ2 and ε > 0 define xε ∈ ℓ2 by xε +k = xk 1{xk>ε}. By +the dominated convergence theorem, xε → x in ℓ2 as ε → 0. Hence, since the map G2 +defined in Remark 5.3 is continuous, also G2(xε) ⇒ G2(x). This means that for any point +process ζ = � +k δxk such that x ∈ ℓ2, +Gε(ζ) = G2(xε) ⇒ G2(x) = G(ζ), +which gives (5.4). + +WEAK QUENCHED LIMIT THEOREMS FOR RWSRE +21 +Recall that if LX, LY are laws of random variables X, Y defined on the same probability +space, then ρ(LX, LY )3 < E|X − Y |2 (c.f. [10, Theorem 11.3.5]). Thus +P [ρ(Gε(Nn), G(Nn)) > δ] ≤ P +� +�Eω +�����a−1 +n +n +� +k=1 +ξk 1{ξk≤εan}(2ϑk − 1) +����� +2 +> δ3 +� +� += P +� +E(2ϑ1 − 1)2a−2 +n +n +� +k=1 +ξ2 +k 1{ξk≤εan} > δ3 +� +, +since (2ϑk − 1)k is a sequence of mean 0 i.i.d. variables independent of the environment. +Denote C = E(2ϑ1 − 1)2, then +lim sup +n→∞ P [ρ(Gε(Nn), G(Nn)) > δ] ≤ lim sup +n→∞ P +� +a−2 +n +n +� +k=1 +ξ2 +k 1{ξk≤εan} > δ3 +C +� +≤ lim sup +n→∞ P +� +a−2 +n +n +� +k=1 +ξkεan 1{ξk≤εan} > δ3 +C +� +≤ lim +n→∞ P +� +a−1 +n +n +� +k=1 +ξk > δ3 +εC +� +. +The sequence a−1 +n +�n +k=1 ξk converges weakly to some β-stable variable Lβ, therefore +lim +n→∞ P +� +a−1 +n +n +� +k=1 +ξk > δ3 +εC +� += P +� +Lβ > δ3 +εC +� +→ 0 +as ε → 0, +which proves (5.5). +Therefore G(Nn) ⇒ G(N∞). Now the claim of the theorem follows from Proposition 5.1 +and Lemmas 5.5 and 3.5. +□ +5.2. Moderately sparse random environment. +Proof of Theorem 2.2. Since Eξ1 < ∞, α = (Eξ1)−1 is well defined. Let N∞ = � +n δxn +be a Poisson point process as in Theorem 5.4 and let Nα +∞ = � +n δα2/βxn. Then Nα +∞ is a +Poisson point process with intensity βαx−β/2−1dx/2. Putting +φα +n(ω)(·) = φα +n,ω(·) = Pω +� +a−2 +n (TSαn − EωTSαn) ∈ · +� += Pω +� +(aαn/an)2a−2 +αn(TSαn − EωTSαn) ∈ · +� +, +where Sαn := S⌊αn⌋, it follows from Lemma 5.5 and Theorem 5.4 that φα +n ⇒ G(Nα +∞). +It remains to show that +a−4 +n Varω [(TSαn − EωTSαn) − (Tn − EωTn)] = a−4 +n Varω [TSαn − Tn] P→ 0, +from which it follows, by Lemma 5.5, that µn ⇒ G(Nα +∞). +Observe that on the event {n ≤ Sαn}, for any k such that Sk ≤ n, +Varω [TSαn − Tn] = +Sαn +� +j=n+1 +Varω [Tj − Tj−1] ≤ +Sαn +� +j=Sk+1 +Varω [Tj − Tj−1] += Varω [TSαn − TSk] +and similarly on {Sαn ≤ n} for any k such that Sk ≥ n, +Varω [TSαn − Tn] ≤ Varω [TSk − TSαn] . + +22 +D. BURACZEWSKI, P. DYSZEWSKI AND A. KO�LODZIEJSKA +Therefore for any δ > 0 and ε > 0, +P +� +a−4 +n Varω [TSαn − Tn] > δ +� +≤ P [|Sαn − n| > εn] ++ P +� +a−4 +n Varω +� +TS⌊αn⌋+⌊εn⌋ − TSαn +� +> δ +� ++ P +� +a−4 +n Varω +� +TSαn − TS⌊αn⌋−⌊εn⌋ +� +> δ +� += P +����� +Sαn +αn − 1 +α +���� > ε +α +� ++ 2P +� +a−4 +n Varω [TSεn] > δ +� +. +The first term tends to 0 by the law of large numbers (recall 1/α = Eξ1). To estimate the +second, note that +Varω +� +T r +Sεn +� += +εn +� +k=1 +VarωT r +ξk +(3.6) += +εn +� +k=1 +2 +3(ξ4 +k − ξ2 +k) ≤ +εn +� +k=1 +ξ4 +k, +and a−4 +n +�εn +k=1 ξ4 +k ⇒ ε−4/βLβ/4 with respect to P, while by Lemma 3.5, a−4 +n Varω +� +T l +Sεn +� P→ +0. Therefore +lim sup +n→∞ P +� +a−4 +n Varω [TSεn] > δ +� +≤ P +� +Lβ/4 > +δ +ε4/β +� +. +The last expression can be made arbitrary small by taking sufficiently small ε. +□ +5.3. Strong sparsity: preliminaries. From now we assume that Eξ = ∞. This case +is technically more involved, however the underlying principle remains the same. Denote +the first passage time of S via +νn = inf {k > 0 : Sk > n} . +Recall that we write +mn = nE +� +ξ1{ξ≤an} +� +and +dn = 1/P[ξ > n]. +and we denote by {cn}n∈N for the asymptotic inverse of {mn}n∈N, i.e. any increasing +sequence of real numbers such that +lim +n→∞ cmn/n = lim +n→∞ mcn/n = 1. +As one may expect Sn grows at a scale mn and thus νn must grow at a scale cn (in the sense +of limit theorem which we will soon make precise). For our purposes we need to justify +that Sn/mn and νn/cn converge jointly with some other characteristics of the trajectory +of S. For this reason we will need to use the setting of c`adl`ag functions. Recall that D +stands for the space of right continuous functions that have a left limit at each point. For +h ∈ D we define h← ∈ D via +h←(t) = inf{s : h(s) > t}. +Consider D↑ ⊆ D consisting of non-decreasing functions and take M : D↑ → M given via +M(h) = +� +k +δxk ⊗ δtk, +where for h ∈ D↑, {tk} are the discontinuity points of h and xk = h(tk) − h(t− +k ) is the size +of the jump at tk. +Lemma 5.6. The function M : D↑ → M is continuous with respect to J1 topology. +Proof. Let fn, f ∈ D↑ be such that fn → f in J1 topology. For any nonnegative, continuous +function ϕ: (0, +∞] × [0, +∞) → R with compact support we can find ε > 0 and T > 0 + +WEAK QUENCHED LIMIT THEOREMS FOR RWSRE +23 +such that ϕ(x, t) = 0 if x ≤ ε or t ≥ T. Since f ∈ D↑, it has only finitely many jumps on +the interval [0, T] that are greater than ε, +� +ϕ(x, t) Mf(dx, dt) = +N +� +k=1 +ϕ(xk, tk) +for some N, t1 < · · · < tN < T and xk > ε. +By the definition of J1 topology, there exists a sequence of continuous increasing func- +tions λn : (0, ∞) → (0, ∞) such that +(5.6) +sup +t∈[0,T] +|λn(t) − t| → 0, +sup +t∈[0,T] +|fn(t) − f(λn(t))| → 0. +For n sufficiently large, supt∈[0,T] |λn(t) − t| < T − tN, which means that f ◦ λn has +exactly N jumps on the interval [0, T), at times λ−1 +n (tk). Moreover, for large enough n, +supt∈[0,T] |fn(t) − f(λn(t))| < ε/3, from which it follows that fn cannot have jumps bigger +than ε apart from the discontinuity points of f ◦ λn. +Fix k ∈ {1, . . . N}. It follows from (5.6) that for n large enough fn does have a jump +at λ−1 +n (tk), denote it by xn +k, and observe that xn +k → xk as n → ∞; in particular xn +k > ε for +large n. It also follows that λ−1 +n (tk) → tk as n → ∞. This means that for n sufficiently +large +� +ϕ(x, t) Mfn(dx, dt) = +N +� +k=1 +ϕ(xn +k, λ−1 +n (tk)) +and the last expression tends to +� +ϕ(x, t) Mf(dx, dt) as n → ∞, which gives Mfn → +Mf. +□ +Consider a random element of M1((0, +∞] × [0, +∞)) given by +Λn = +∞ +� +j=1 +δξj/an ⊗ δj/n +and random elements of D↑ defined via +(5.7) +Ln(t) = S⌊nt⌋/an for β < 1, +and +�Ln(t) = S⌊nt⌋/mn for β = 1. +Recall Υ: D↑ → R defined in (2.5). +Lemma 5.7. If β < 1, then +(5.8) +� +Ln, Λn, νn +dn +, Sνn−1 +n +� +⇒ (L, M(L), L←(1), Υ(L)) +in (D, J1)×M×R×R, where L = (Lt)t≥0 is strictly increasing β-stable subordinator with +L´evy measure given by ν(x, +∞) = x−β. +If β = 1, then +(5.9) +� +�Ln, νn +cn +, Sνn−1 +n +� +⇒ (id, 1, 1) +in (D, J1) × R × R, where id: R+ → R+ is the identity function. +Proof. Consider first β < 1. By an appeal to standard functional weak convergence to +stable L´evy motion [22, Corollary 7.1], +Ln ⇒ L +in (D, J1). +Note that +Λn = M(Ln) + +24 +D. BURACZEWSKI, P. DYSZEWSKI AND A. KO�LODZIEJSKA +and the function M is J1-continuous by Lemma 5.6. Moreover, +νn +dn += L← +dn +� n +adn +� +and the map h �→ h← is continuous in M1 topology by [27]. In what follows, we will use +notation introduced in [26]. For h ∈ D let h− be the lcrl (left-continuous, having right- +hands limits) version of h, that is, h−(t) = limε→0+ h(t − ε) and h−(0) = 0. Similarly, let +h+ denote rcll version of a lcrl path. Let Φ : D↑ → D be given by +Φ(h) = (h− ◦ (h←)−)+. +Finally, observe that for any k ∈ N, Φ(Ldn) on the set [Sk/adn, Sk+1/adn) is constant and +equal to Sk/adn, therefore +Sνn−1 +adn += Φ (Ldn) +� n +adn +� +. +By [26], Φ is J1-continuous on D↑↑ ⊂ D, the set of strictly increasing, unbounded functions. +Since L ∈ D↑↑ almost surely, by the continuous mapping theorem we have joint convergence +in distribution +(Ln, M(Ln), L← +n , Φ(Ln)) → (L, M(L), L←, Φ(L)) +in (D, J1)×Mp((0, ∞]×[0, ∞))×(D, M1)×(D, J1). By Skorokhod’s representation theorem +we may assume the above convergence holds almost surely. +Since the limiting processes admit no fixed discontinuities, Proposition 2.4 in [26] gives +νn +dn +→ L←(1) +and +Sνn−1 +adn +→ Φ(L)(1) = Υ(L) +almost surely. +The case β = 1 is similar and follows from the fact that by [22, Corollary 7.1] and +properties of J1 topology, +�Ln ⇒ id +in (D, J1). +One can combine this with +νn +cn += �L← +cn +� n +mcn +� +, +Sνn−1 +mcn += Φ +��Lmn +� � n +mcn +� +and the arguments presented in the case β < 1 to get the desired claim. +□ +Remark 5.8. Observe that all information on the sequence (ξk)k is carried by the process +Λn and therefore by Ln or, equivalently, �Ln. We may thus assume that our space holds +random variables U (k) +n , ϑk as described in Section 5.1 and at the same time the convergence +given in Lemma 5.7 holds almost surely. +Lemma 5.9. Assume that (2.1) holds true. If β < 1, then +n−4Varω +� +T r +Sνn−1 − EωT r +Sνn−1 − +νn−1 +� +k=1 +ξ2 +k(2ϑk − 1) +� +P→ 0. +If β = 1 and Eξ = ∞, then +a−4 +cn Varω +� +T r +Sνn−1 − EωT r +Sνn−1 − +νn−1 +� +k=1 +ξ2 +k(2ϑk − 1) +� +P→ 0. +Proof. One can use the same arguments as in the proof of Proposition 5.1. First consider +β ∈ (0, 1). By tightness of {νn/dn}n∈N we can choose C > 0 to make the probability +P[νn > Cdn] arbitrarily small. Next, on the event {νn ≤ Cdn}, +Varω +� +T r +Sνn−1 − EωT r +Sνn−1 − +νn−1 +� +k=1 +ξ2 +k(2ϑk − 1) +� +≤ Varω +�Cdn +� +k=1 +� +Uξk − 2ξ2 +kϑk +� +� +. + +WEAK QUENCHED LIMIT THEOREMS FOR RWSRE +25 +From here, since aCdn ∼ C1/βn, one argues as in the proof of Proposition 5.1 to show that +� +k∈I1n +ξ4 +k +n4 Varω +�Uξk +ξ2 +k +− 2ϑk +� +P→ 0 +and +� +k∈I2n +ξ4 +k +n4 Varω +�Uξk +ξ2 +k +− 2ϑk +� +P→ 0, +where I1 +n = {k ≤ Cdn : ξk > εn}, I2 +n = {k ≤ Cdn : ξk ≤ εn} with fixed ε > 0. In the +case β = 1 and Eξ = ∞ one can invoke that same arguments combined with the tightness +of {νn/cn}n∈N. +□ +Lemma 5.10. Assume that (2.1) holds true. If β ∈ (0, 1), then +n−4VarωT l +Sνn +P→ 0. +If β = 1 and Eξ = ∞, then +a−4 +cn VarωT l +Sνn +P→ 0. +Proof. Consider the case β < 1. Take any C > 0 and write +P +� +n−4VarωT l +Sνn ≥ ε +� +≤ P [νn ≥ Cdn] + P +� +VarωT l +S[Cdn] ≥ εn4� +. +Since aCdn ∼ C1/βn, an appeal to Lemma 3.5 shows that the second term tends to 0 as +n → ∞. The first term can be made arbitrary small by taking C > 0 sufficiently big. In +the case β = 1 we can use an analogous argument with dn replaced with cn. +□ +For the purpose of the next lemma. let ({U 0 +n}n∈N, ϑ0) be, as before, a copy of ({Un}n∈N, ϑ) +given by the claim of Lemma 3.2 independent of the environment. +Lemma 5.11. Assume that (2.1) and (2.2) hold true for β ≤ 1 and Eξ = ∞. Then +U 0 +n−Sνn−1 − EωU 0 +n−Sνn−1 +n2 +− (1 − Ξ)2(2ϑ0 − 1) P→ 0, +where Ξ = Υ(L) for β < 1 and Ξ = 1 for β = 1. +Proof. By the merit of Remark 5.8, Sνn−1/n → Ξ, P-almost surely. Secondly, by a stand- +ard application of the key renewal theorem [11, Theorem 2.6.12], the condition Eξ = ∞ +implies that n − Sνn−1 +P→ ∞. The claim of the lemma follows from the fact that +U 0 +n−Sνn−1 − EωU 0 +n−Sνn−1 +n2 +− (1 − Ξ)2(2ϑ0 − 1) = +− +� +1 − Sνn−1 +n +�2 ++ (1 − Ξ)2 + +� +1 − Sνn−1 +n +�2 +U 0 +n−Sνn−1 +(n − Sνn−1)2 − (1 − Ξ)22ϑ0 +and Lemma 3.2. +□ +5.4. Strong sparsity: β = 1. We will now focus on the case when β = 1 and Eξ = ∞. +By Lemmas 5.5, 5.9, 5.10 and 5.11, it is sufficient to study the quenched behaviour of +�νn−1 +k=1 ξ2 +k(2ϑk − 1). +Proof of Theorem 2.3. Fix ε > 0. On the set {|νn − cn| ≤ εcn}, +Eω +� +νn +� +k=cn+1 +ξ2 +k(2ϑk − 1) +�2 += +νn +� +k=cn+1 +ξ4 +kE(2ϑ − 1)2 ≤st C +εcn +� +k=1 +ξ4 +k. +Observe that +a−4 +cn +εcn +� +k=1 +ξ4 +k = ε4(1 + o(1)) +εcn +� +k=1 +ξ4 +k/a4 +εcn. + +26 +D. BURACZEWSKI, P. DYSZEWSKI AND A. KO�LODZIEJSKA +Since the sequence on the right hand side is tight in n, it follows that +a−4 +cn Eω +� +νn +� +k=cn+1 +ξ2 +k(2ϑk − 1) +�2 +P→ 0. +In a similar fashion, +a−4 +cn Eω +� +cn +� +k=νn+1 +ξ2 +k(2ϑk − 1) +�2 +P→ 0. +Therefore the weak limit of the quenched law of (Tn − EωTn)/a2 +cn will coincide with the +limit of +Pω +� cn +� +k=1 +ξ2 +k(2ϑk − 1)/a2 +cn ∈ · +� +. +The weak limit of the latter is G(N), which follows from the proof of Theorem 2.2. +□ +5.5. Strong sparsity: β < 1. +Proof of Theorem 2.4. Let µn,ω denote the quenched law of (Tn − EωTn)/n2. Then +µn,ω(·) = Pω +�Tn − TSνn−1 − Eω[Tn − TSνn−1] +n2 ++ TSνn−1 − Eω[TSνn−1] +n2 +∈ · +� +To treat the second term under the probability we can, similarly as previously, decouple +the times that the random walker spends between consecutive Sk’s for k ≤ n. The first +part will be controlled with the help of Lemma 5.11. Let (U 0 +n, ϑ0) be, as before, a copy of +(Un, ϑ) given by the claim of Lemma 3.2 independent of the environment. Then Un−Sνn−1 +has, under Pω, the same distribution as the time the walk spends in [Sνn−1, n) after +reaching Sνn−1 and before reaching n. By Lemma 5.10 and Lemma 5.5 the weak limit of +µn,ω is the same as that of +¯µn,ω(·) = Pω +� +U 0 +n−Sνn−1 − EωU 0 +n−Sνn−1 +n2 ++ +T r +Sνn−1 − Eω[T r +Sνn−1] +n2 +∈ · +� +. +Recall the random functions Ln given in (5.7) and that for a c`adl`ag function h we +denote by {xk(h), tk(h)} an arbitrary enumeration of its discontinuities, i.e. +xk(h) = +h(tk) − h(t− +k ) > 0, where tk(h) = tk. Note that, with Υ given in (2.5), one has by the +merit of Lemmas 5.5, 5.9, 5.10 and 5.11 that the limit of ¯µn,ω will coincide with the limit +of +F n(·) = Pω +� +a2 +dn +n2 (1 − Υ(Ldn))2(2ϑ0 − 1) + a2 +dn +n2 +� +k +xk(Ldn)2(2ϑk − 1)1{Ldn(tk) 0 +F n +ε (·) = Pω +�a2 +dn +n2 (1 − Υ(Ldn))2(2ϑ0 − 1) ++ a2 +dn +n2 +� +k +xk(Ldn)2(2ϑk − 1)1{xk(Ldn)>ε}1{Ldn(tk) 0, F n +ε → F ∞ +ε , where +F ∞ +ε (·) = Pω +� +(1 − Υ(L))2(2ϑ0 − 1) + +� +k +xk(L)2(2ϑk − 1)1{xk(L)>ε}1{L(tk)≤1} ∈ · +� +since associated point processes converge and adn/n → 1. Then we show that F ∞ +ε +⇒ F(L) +as ε → 0. We finally prove that (5.5) also holds in this context and conclude the result. +□ + +WEAK QUENCHED LIMIT THEOREMS FOR RWSRE +27 +Acknowledgement. DB nad PD were supported by the National Science Center, Poland +(Opus, grant number 2020/39/B/ST1/00209). AK was supported by the National Science +Center, Poland (Opus, grant number 2019/33/B/ST1/00207). +References +1. P. 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Roitershtein, Random +walks in a moderately sparse random environment, Electronic Journal of Probability +24 (2019). +7. A. Dembo, Y. Peres, and O. Zeitouni, Tail estimates for one-dimensional random walk +in random environment, Comm. Math. Phys. 181 (1996), no. 3, 667–683. +8. D. Denisov, A. B. Dieker, and V. Shneer, Large deviations for random walks under +subexponentiality: The big-jump domain, Annals of Probability 36 (2008), 1946–1991. +9. D. Dolgopyat and I. Goldsheid, Quenched limit theorems for nearest neighbour random +walks in 1D random environment, Comm. Math. Phys. 315 (2012), no. 1, 241–277. +10. R. M. Dudley, Real analysis and probability, CRC Press, 2018. +11. R. Durrett, Probability—theory and examples, Cambridge Series in Statistical and +Probabilistic Mathematics, vol. 49, Cambridge University Press, Cambridge, 2019. +12. N. Enriquez, C. Sabot, L. Tournier, and O. Zindy, Quenched limits for the fluctuations +of transient random walks in random environment on Z1, Ann. 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Samorodnitsky, Weak quenched limiting distributions for transient +one-dimensional random walk in a random environment, Ann. Inst. Henri Poincar´e +Probab. Stat. 49 (2013), no. 3, 722–752. + +28 +D. BURACZEWSKI, P. DYSZEWSKI AND A. KO�LODZIEJSKA +20. J. Peterson and O. Zeitouni, Quenched limits for transient, zero speed one-dimensional +random walk in random environment, The Annals of Probability 37 (2009), no. 1, 143– +188. +21. S. I. Resnick, Extreme values, regular variation and point processes, Springer New +York, 1987. +22. +, Heavy-tail phenomena: probabilistic and statistical modeling, Springer Science +& Business Media, 2007. +23. D. Revuz and M. Yor, Continuous martingales and brownian motion, 2004. +24. A. Sma¨ıl, Asymptotic behaviour for random walks in random environments, Journal +of applied probability 36 (1999), no. 2, 334–349. +25. F. Solomon, Random walks in a random environment, The Annals of Probability 3 +(1975), no. 1, 1–31. +26. P. Straka and B. I. Henry, Lagging and leading coupled continuous time random walks, +renewal times and their joint limits, Stochastic Processes and their Applications 121 +(2011), 324–336. +27. W. Whitt, Weak convergence of first passage time processes, 1971, pp. 417–422. +28. O. Zeitouni, Random walks in random environment, 2004. +Dariusz Buraczewski, Piotr Dyszewski and Alicja Kolodziejska: Mathematical Institute, Uni- +versity of Wroclaw, 50-384 Wroclaw, Poland +E-mail: dariusz.buraczewski@math.uni.wroc.pl, piotr.dyszewski@math.uni.wroc.pl, +alicja.kolodziejska@math.uni.wroc.pl + diff --git a/t9AyT4oBgHgl3EQfmvi8/content/tmp_files/load_file.txt b/t9AyT4oBgHgl3EQfmvi8/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c9c05dade5d8c35f8226467e3d179490fcbcf819 --- /dev/null +++ b/t9AyT4oBgHgl3EQfmvi8/content/tmp_files/load_file.txt @@ -0,0 +1,1192 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf,len=1191 +page_content='WEAK QUENCHED LIMIT THEOREMS FOR A RANDOM WALK IN A SPARSE RANDOM ENVIRONMENT DARIUSZ BURACZEWSKI, PIOTR DYSZEWSKI AND ALICJA KO�LODZIEJSKA Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' We study the quenched behaviour of a perturbed version of the simple sym- metric random walk on the set of integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' The random walker moves symmetrically with an exception of some randomly chosen sites where we impose a random drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' We show that if the gaps between the marked sites are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' and regularly varying with a sufficiently small index, then there is no strong quenched limit laws for the position of the random walker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' As a consequence we study the quenched limit laws in the context of weak convergence of random measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Introduction One of the most classical and well-understood random processes is the simple symmetric random walk (SRW) on the set of integers, where the particle starting at zero every unit time moves with probability 1/2 to one of its neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' This process is a time and space homogeneous Markov chain, that is its increments are independent of the past and the transitions do not depend on time and the current position of the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' In many cases, the homogeneity of the environment reduces the applicability of the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' In numerous applied models some kind of obstacles can appear like impurities, fluctuations, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Thus, it is natural to express such irregularities as a random environment and it is well known that even small perturbations of the environment affect properties of the random process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' In 1981 Harrison and Shepp [15] described the behaviour of the SRW in a slightly disturbed environment, replacing only the probability of passing from 0 to 1 by some fixed p ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' They observed that the scaling limit is not the Brownian motion, but the skew Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' We intend to study random walks in a randomly perturbed environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Our main results concern the so-called random walk in a sparse random environment (RWSRE) in- troduced in [17], in which homogeneity of an environment is perturbed only on a sparse subset of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' More precisely, first we choose randomly a subset of integers marked by the positions of a standard random walk with positive integer jumps and next we impose a random drift at the chosen sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' The present paper can be viewed as a continuation of the recent publications [17, 6, 5], where annealed limit theorems were described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' These annelad-type results do not settle however the question if the environment alone is suffi- cient to determine the distributional behaviour of the process with high certainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Here we froze the environment and we are interested in limit behaviour of the random process in the quenched settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' As we show in the present article, even in a very diluted ran- dom environment the fluctuations of the random perturbation of the medium affect the conditional distribution of the random walker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' The model RWSRE we consider here can be viewed as an interpolation between SRW and the one suggested in the seventies by Solomon [25] called a one dimensional random walk in random environment (RWRE), where all the sites were associated with random i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' weights {ωi} describing the probability of passing to the right neighbour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' It quickly 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Primary: 60K37;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' secondary 60F05;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' 60G57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' weak convergence, point processes, regular variation, random walk in a random environment, sparse random environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='00478v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='PR] 1 Jan 2023 2 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' BURACZEWSKI, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' DYSZEWSKI AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' KO�LODZIEJSKA became clear that the additional environmental noise in the system has a significant impact on the behaviour of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' In fact, the answers to a variety of questions about the model like limit theorem [16] and large deviations [7, 4] are given only in terms of the environment marginalizing the impact of the random motion of the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' General setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' To define our model let Ω = (0, 1)Z be the set of all possible configurations of the environment equipped with the corresponding cylindrical σ-algebra F and a probability measure P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' A random element ω = (ωn)n∈Z of (Ω, F) distributed according to P is called a random environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Each element ω of Ω and integer x gives a rise to a probability measure Px ω on the set X = ZN0 with the cylindrical σ-algebra G such that Px ω[X0 = x] = 1 and Px ω [Xn+1 = j|Xn = i] = � � � ωi, if j = i + 1, 1 − ωi, if j = i − 1, 0, otherwise, where X = (Xn)n∈N0 ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' One sees that under Px ω, X forms a nearest neighbour random walk which is a time-homogeneous Markov chain on Z and it is called a random walk in random environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' The randomness of the environment ω influences significantly various properties of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' In view of this, it is natural to investigate the behaviour of X under the annealed measure Px = � Px ωP(dω) which is defined as the unique probability measure on (Ω × X, F ⊗ G) satisfying Px[F × G] = � F Px ω[G] P(dω), F ∈ F, G ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' In the sequel we will write Pω = P0 ω and P = P0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' It turns out that, in general, under the annealed probability X is no longer a Markov chain, because it usually exhibits a long range dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' We are interested in limit theorems for Xn as → ∞, however in this paper we discuss the asymptotic behaviour of the corresponding sequence of first passage times T = (Tn)n∈N, that is (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='1) Tn = inf{k ∈ N : Xk = n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' We will study the distribution of Tn in the quenched setting which means that we will investigate the behaviour of µn,ω(·) = Pω [(Tn − bn)/an ∈ · ] for suitable choices of sequences (an)n∈N and (bn)n∈N possibly depending on ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' In the present setting µn, defined by µn(ω) = µn,ω, becomes a random element of M1, the space of probability measures on (R, Bor(R)), where Bor(R) stands for the Borel σ-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' M1 equipped in the Prokhorov distance is a complete, separable metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' One can distinguish two types of limiting behaviour of (µn)n∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' We will say that a strong quenched limit theorem for T holds if µn → µ almost surely in M1, that is for P a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' ω the sequence of measures {µn,ω} converges weakly to µ, and say that a weak quenched limit law for T holds if µn ⇒ µ in M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Here and in the sequel ⇒ denotes weak convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' We will now discuss different choices of the probability P which is the distribution of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' To keep the introduction brief we will limit the discussion to the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' random environment, which is the most classical choice for P, and the sparse random environment which we will study in depth in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' WEAK QUENCHED LIMIT THEOREMS FOR RWSRE 3 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Random walk in i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' random environment for ω0 = 1/3 with probability 1/3, ω0 = 3/4 with probability 2/3 and α ≈ 1, 35 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Independent identically distributed environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' One of the simplest and most studied choices of the environmental distribution P is random walk in i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' random environment, corresponding to a product measure, under which ω = (ωn)n∈Z forms a collection of independent, identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=') random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' In their seminal work Kesten et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' [16] used the following link between walks and random trees [14]: the one-dimensional distributions of T are connected to a branching process in random environment with immigration and a reproduction law with the mean distributed as (1 − ω0)/ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' This observation later leads to a conclusion that T lies in the domain of attraction of an α-stable distribution, where E[ω−α 0 (1 − ω0)α] = 1, provided that such α ∈ (0, 2) exists (see Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' After a close examination of the main results of Kesten et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' [16] it transpires that the centering and scaling are determined by the distribution of (1−ω0)/ω0, which means that the behaviour of the walker does not affect the limiting behaviour in a significant way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' In turn, to understand the random motion, one is led to investigate the behaviour of T under Pω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' If α > 2, then a strong quenched limit theorem [24, 13] of the form lim n→∞ Pω � (Tn − Eω[Tn])/(σ√n) ∈ dx � = e−x2/2dx/ √ 2π holds almost surely in M1, where σ2 = E[Varω[T1]] < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' As seen from the results in [18, 20] there is no strong quenched limit theorem for T in the case α < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Indeed it turns out that for α < 2 one can find different strong quenched limits for T along different sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' This in turn leads to the analysis of T in the weak quenched setting, that is weak limits of µn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Consider first the mapping H : Mp → M1 given as follows: for a point process ζ = � i≥1 δxi, where {xi}i∈N is an arbitrary enumeration of the points, define H(ζ)(·) = � P � � i≥1 xi(τi − 1) ∈ · � , � i≥1 x2 i < ∞, δ0(·), otherwise, where {τi}i∈N is a sequence of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' mean one exponential random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Then the main result of [9, 12, 19] states that for α < 2, Pω � n−1/α(Tn − EωTn) ∈ · � ⇒ H(N) in M1, where N is a Poisson point process on (0, ∞) with intensity cNx−α−1dx for some constant cN > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Sparse random environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' We now specify the object of interest in the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' We will work under a choice of environmental probability P for which the random 100 80 60 40 20 20 0 2000 4000 6000 8000 100004 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' BURACZEWSKI, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' DYSZEWSKI AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' KO�LODZIEJSKA walk X will move symmetrically except some randomly marked points where we impose a random drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' The marked sites will be distributed according to a two-sided random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Denote by ((ξk, λk))k∈Z a sequence of independent copies of a random vector (ξ, λ), where λ ∈ (0, 1) and ξ ∈ N, P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Considering the aforementioned two-sided random walk S = (Sn)n∈Z given via Sn = � � � �n k=1 ξk, if n > 0, 0, if n = 0, − �0 k=n+1 ξk, if n < 0, we define a random environment ω = (ωn)n∈Z ∈ Ω given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='2) ωn = � λk, if n = Sk for some k ∈ Z, 1/2, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' The sequence S determines the marked sites in which the random drifts 2λk −1 are placed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Since for the unmarked sites n (that is, for most of sites) the probabilities of jumping to the right are deterministic and equal to ωn = 1/2, it is natural to call ω a sparse random environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Following [17] we use the term random walk in sparse random environment (RWSRE) for X as defined above with ω being a sparse random environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' In the case when P[ξ = 1] = 1 random walk in sparse random environment is equivalent to a random walk in i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Suppose that ξ is independent of λ and has a geometric distribution P[ξ = k] = a(1 − a)k−1, k ≥ 1 for some a ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Then the sparse random environment given in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='2) is equivalent to an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' environment with ω0 distributed as P[ω0 ∈ ·] = aP[λ ∈ ·] + (1 − a)δ1/2(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Random walk in a sparse random environment was studied in detail in the annealed setting in [17, 6, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' In [17] the authors address the question of transience and recurrence of RWSRE and prove a strong law of large numbers and some distributional limit theorems for X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' As in the case of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' random environment, the fraction ρ = 1 − λ λ appears naturally in the description of the asymptotic behaviour of the random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' According to [17, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='1], X is P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' transient to +∞ if (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='3) E log ρ ∈ [−∞, 0) and E log ξ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Note that the first condition in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='3) excludes the degenerate case ρ = 1 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' in which X is a simple random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Under (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='3), the RWSRE also satisfies a strong law of large numbers, that is, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='4) Tn/n → 1/v P − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' where v = � (1−Eρ)Eξ (1−Eρ)Eξ2+2EρξEξ if Eρ < 1, Eρξ < ∞ and Eξ2 < ∞, 0 otherwise, see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='3 in [17] and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='1 in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' We note right away that conditions present in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='3) are satisfied under the conditions of our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Thus, the random walks in a sparse random environment that we treat here are transient to the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' The asymptotic behaviour of T is controlled by two ingredients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' The first one, similarly as in the case of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' environment, is α > 0 such that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='5) E [ρα] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' The parameter α > 0, if it exists, is used to quantify the effect that the random transition probabilities λk’s have on the asymptotic behaviour of the random walker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' The second WEAK QUENCHED LIMIT THEOREMS FOR RWSRE 5 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' RWSRE: β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='2 and α ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='52 (left) and β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='2 and α ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='85 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' The grey horizontal lines indicate the marked sites ingredient is the tail behaviour of ξ, that is the asymptotic of P[ξ > t] as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' If E[ξ4] < ∞, then with respect to the annealed probability T is in the domain of attraction of an α-stable distribution [6, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='2] with the exact same behaviour as one observes in the case of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' New phenomena appear if ξ has a regularly varying tail with index −β for β ∈ (0, 4), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' as t → ∞, P[ξ > t] ∼ t−βℓ(t) for some function ℓ: R → R slowly varying at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Here and in the rest of the article we write f(t) ∼ g(t) for two functions f, g ∈ R → R whenever f(t)/g(t) → 1 as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Recall that a function ℓ is slowly varying at infinity if ℓ(ct) ∼ ℓ(t) as t → ∞ for any constant c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' It transpires that if the tail of ξ is regularly varying with β ∈ (0, 4) with E[ξ] < ∞, then with respect to the annealed probability T lies in the domain of attraction of γ-stable distribution with γ = min{α, β/2}, see [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' For small values of β one sees an interplay between the contribution of the sparse random environment and the random movement of the process in the unmarked sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' To state this result take ϑ to be a non-negative random variable with the Laplace transform (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='6) E � e−sϑ� = 1 cosh(√s), s > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Note that 2ϑ is equal in distribution to the exit time of the one-dimensional Brownian motion from the interval [−1, 1], see [23, Proposition II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Next consider a measure η on K = [0, ∞]2 \\ {(0, 0)} given via η({(v, u) ∈ K : u > x1 or v > x2}) = x−β 1 + E[ϑβ/2]x−β/2 2 − E[min{x−β 1 , ϑβ/2x−β/2 2 }] for x1, x2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Now let N = � k δ(tk,jk) be a Poisson point process on [0, ∞) × K with intensity LEB ⊗ η, where LEB stands for the one-dimensional Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Under mild integrability assumptions, see [5, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='4], the integral L(t) = (L1(t), L2(t)) = � [0,t]×K j N(ds, dj), t ≥ 0 converges and defines a two-dimensional non-stable L´evy process with L´evy measure η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Next consider the β-inverse subordinator L← 1 (t) = inf{s > 0 : L1(s) > t}, t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' 200 150 100 50 0 50 0 2000 4000 6000 8000 10000800 600 400 200 0 200 0 2000 4000 6000 8000 100006 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' BURACZEWSKI, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' DYSZEWSKI AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' KO�LODZIEJSKA Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' RWSRE: β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='8 and α ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' The grey horizontal lines indicate the marked sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Finally, if β < 2α and β ∈ (0, 1), then under some additional mild integrability assump- tions [5, Theorem 21], with respect to the annealed probability Tn/n2 ⇒ 2L2(L← 1 (1)−) + 2ϑ(1 − L1(L← 1 (1)−))2 weakly in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' The aim of the present article is to present a quenched version of this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' As we will see in our main theorem, the terms L2(L← 1 (1)−) and L1(L← 1 (1−)) present on the right hand side can be viewed as the contribution of the environment, whereas ϑ represents the contribution of the movement of the random walker in the unmarked sites that are close to n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' For the full treatment of the annealed limit results, in particular the complementary case β ≥ 2α, we refer the reader to [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' The article is organised as follows: in Section 2 we give a precise description of our set- up and main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' In Section 3 we provide a preliminary analysis of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' The essential parts of the proof of our main results are in Sections 4 and 5 where we prove an absence of the strong quenched limits and prove weak quenched limits respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Weak quenched limit laws In this section we will present our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' From this point we will consider only a sparse random environment given via (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' We assume that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='1) P[ξ > t] ∼ t−βℓ(t) for some β ∈ (0, 4) and slowly varying ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' We will focus on the case in which the asymptotic of the system is not determined solely by the environment and thus we will assume also that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='2) E[ρ2γ] < 1, E[ξγρ3γ] < ∞ for some parameter γ ∈ (β/4, 1 ∧ β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' The first condition in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='2) guarantees that a part of the fluctuations of Tn will come from the time that the process spends in the unmarked sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' The second condition is purely technical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Note that we do not assume that there exists α > 0 for which (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='5) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Our first result states that there is no quenched limit for Tn’s in the strong sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Take (an)n∈N to be any sequence of positive real numbers such that nP[ξ > an] → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' 250 200 150 100 50 50 0 2000 4000 6000 8000 10000WEAK QUENCHED LIMIT THEOREMS FOR RWSRE 7 Then, since the tail of ξ is assumed to be regularly varying, the sequence (an)n∈N is also regularly varying with index 1/β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' That is for some slowly varying function ℓ1, an = n1/βℓ1(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' The sequence (an)n∈N will play the role of the scaling factor in our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' The first one shows an absence of strong quenched limit laws for T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Assume (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='3), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Then for P almost every ω there are no sequences {An(ω)}n∈N and {Cn(ω)}n∈N such that the sequence of normalized random variables (Tn −Cn(ω))/An(ω) converges in distribution (with respect to Pω) to a nontrivial random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Therefore, as in the case of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' environment, the asymptotic quenched behaviour of Tn’s ought to be expressed in terms of weak quenched convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' As it is the case for annealed limit theorem, one needs to distinguish between a moderately (Eξ < ∞) and strongly (Eξ = ∞) sparse random environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' To describe the former take {ϑj}j∈N to be a sequence of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' copies of ϑ distributed according to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='6) and let G : Mp → M1 be given via (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='3) G(ζ)(·) = � P �� i≥1 xi(2ϑi − 1) ∈ · � , � x2ζ(dx) < ∞, δ0(·) otherwise, for ζ = � i≥1 δxi, where {xi} is an arbitrary enumeration of the point measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Assume (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='3), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' If Eξ < ∞, then Pω � (Tn − EωTn)/a2 n ∈ · � ⇒ G(N)(·) in M1, where N is a Poisson point process on (0, ∞) with intensity βx−β/2−1dx/2Eξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Before we introduce the notation necessary to state our results in the strongly sparse random environment, we will first treat the critical case which is relatively simple to state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Denote mn = nE � ξ1{ξ≤an} � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Note that by Karamata’s theorem [2, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='11] the sequence {mn}n∈N is regularly varying with index 1/β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Furthermore an = o(mn) if β = 1 and an ∼ (1 − β)mn if β < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Next let {cn}n∈N be the asymptotic inverse of {mn}n∈N, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' any increasing sequence of natural numbers such that lim n→∞ cmn/n = lim n→∞ mcn/n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' By the properties of an asymptotic inversion of regularly varying sequences [2, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='12], cn is well defined up to asymptotic equivalence and is regularly varying with index β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Finally, by the properties of the composition of regularly varying sequences {acn}n∈N is regularly varying with index 1 and acn = o(n) if β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Assume (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='3), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' If Eξ = ∞ and β = 1, then Pω � (Tn − EωTn)/a2 cn ∈ · � ⇒ G(N)(·) in M1, where N is a Poisson point process on (0, ∞) with intensity x−3/2dx/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' The limiting random measures in Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='3 share some of the properties of their counterpart in the case of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' environment [19, Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Namely, using the superposition and scaling properties of Poisson point processes, one can directly show that for each n ∈ N and G, G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' , Gn being i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' copies of the limit random measure G(N) in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='2 or Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='3, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='4) G1 ∗ G2 ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' ∗ Gn(·) d= G(·/n2/β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' 8 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' BURACZEWSKI, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' DYSZEWSKI AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' KO�LODZIEJSKA The statement of our results in the strongly sparse case needs some additional notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' As it is the case for the annealed results, it is most convenient to work in the framework of non-decreasing c`adl`ag functions rather than point processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Denote by D↑ the class of non-decreasing c`adl`ag functions R+ → R+ and for h ∈ D↑ consider (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='5) Υ(h) = sup{h(t) : t ∈ R+, h(t) ≤ 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Finally for h ∈ D↑ denote by {xk(h), tk(h)}k an arbitrary enumeration of jumps of h, that is tk = tk(h) ∈ R+ for k ∈ N are all points on the non-negative half line such that h has a (left) discontinuity with jump of size xk(h) = h(tk) − h(t− k ) > 0 at tk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Note that the random series � k:h(tk)≤1 xk(h)2(2ϑk − 1) is convergent since it has an expected value bounded via h(1)E|2ϑ − 1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Finally let F : D↑ → M1 be given by F(h)(·) = P � �(1 − Υ(h))2(2ϑ0 − 1) + � k:h(tk)≤1 xk(h)2(2ϑk − 1) ∈ · � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Note that if h(t) = 1 for some t, then necessarily Υ(h) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Assume (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='3), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' If β ∈ (0, 1), then Pω � (Tn − EωTn)/n2 ∈ · � ⇒ F(L)(·) in M1, where L is a β-stable L´evy subordinator with L´evy measure ν(x, +∞) = x−β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Interestingly the limit measure F(L) does not enjoy a self-similarity property in the sense of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Namely, for any a, b ∈ R, b > 0 the laws of F1 ∗ F2(·) and F((· − a)/b) are different, where F, F1 and F2 are independent copies of the limiting random measure F(L) in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Auxiliary results We will now present a few lemmas that we will use in our proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' We will discuss prop- erties of some random series as well as the asymptotic behaviour of the hitting times (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Estimates for the related stochastic processes {Ri}i∈Z and {Wi}i∈Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' We will frequently make use of the following notation: for integers i ≤ j, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='1) Πi,j = j� k=i ρk, Ri,j = j � k=i ξkΠi,k−1, Wi,j = j � k=i ξkΠk,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' We will also make use of the the limits (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='2) Ri = lim j→∞ Ri,j = ∞ � k=i ξkΠi,k−1, Wj = lim i→−∞ Wi,j = j � k=−∞ ξkΠk,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Note that if E log ρ < 0 and E log ξ < ∞, both series are convergent as one can see by a straightforward application of the law of large numbers and the Borel-Cantelli lemma (see [3, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' The random variables Ri’s and Wj’s have the same distribution and obey the recursive formulae Ri = ξi + ρiRi+1 and Wj = ρjξj + ρjWj−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' We can therefore invoke the proof of [3, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='1] to infer the following result on the existence of moments of Ri’s and Wj’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' In what follows we write R (respectively W) for a generic element of {Ri}i∈Z (respectively {Wj}j∈Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Let α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' If Eρα < 1, Eραξα < ∞ and Eξα < ∞, then ERα and EW α are both finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' WEAK QUENCHED LIMIT THEOREMS FOR RWSRE 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Hitting times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' We describe now some properties of the sequence of stopping times T = {Tn}n∈N that allow us to better understand the process X and indicate its ingredients which play an essential role in the proof of our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' We will first analyse the hitting times T along the marked sites S, that is TSi = inf{n : Xn = Si}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' As it turns out, one can use Ri,j’s given in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='1) to represent the exit probabilities from interval (Si, Sj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' That is, for i < k < j we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='3) PSk ω [TSi > TSj] = Ri+1,k Ri+1,j , PSk ω [TSi < TSj] = Πi+1,k Rk+1,j Ri+1,j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' see the proof of [28, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Let Tk = TSk − TSk−1 be the time that the particle needs to hit k’th marked point Sk after reaching Sk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' One uses Wj’s to describe the expected value of Tk: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='4) EωTk = ESk−1 ω TSk = ξ2 k + 2ξkWk−1, see the proof of [28, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Observe that the random variable Tk can be decomposed into a sum of two parts: the time the trajectory, after reaching Sk−1 but before it hits Sk, spends to the left of Sk−1 and the time it spends to the right of Sk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' For technical reasons that will become clear below, we divide the visits exactly at point Sk−1 between these two sets depending on the direction from which the particle enters Sk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' To be precise we define Tl k = # � n ∈ (TSk−1, TSk] : Xn < Sk−1 or (Xn−1, Xn) = (Sk−1 − 1, Sk−1) � , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Tl k is the sum of the time the particle spends in (−∞, Sk−1 − 1] and the number of steps from Sk−1 − 1 to Sk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Similarly we define Tr k = # � n ∈ (TSk−1, TSk] : Sk−1 < Xn ≤ Sk or (Xn−1, Xn) = (Sk−1 + 1, Sk−1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Thus we can write Tk = TSk − TSk−1 = Tl k + Tr k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Observe that given ω, the random variables {Tk}k∈N are Pω independent, however for fixed k, Tl k and Tr k mutually depend on each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Summarizing, we obtain the following decomposition that will be used repeatedly: TSk = k � j=1 Tj = k � j=1 Tl j + k � j=1 Tr j =: T l Sk + T r Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' To proceed further we need to analyse Tr j, Tl j in details and describe their quenched expected value and quenched variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Below we prove that after hitting any of the chosen sites (Sk)k the consecutive excursions to the left are negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' This entails that behaviour of TSk is determined mainly by T r Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' The sequence {T r Sn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Note that, under Pω, Tr k equals in distribution to the time it takes a simple random walk on [0, ξk] with a reflecting barrier placed in 0 to reach ξk for the first time when starting from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' This is the reason we include into Tr k the visits at Sk−1, but only those from Sk−1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Indeed, let (Yn)n be a simple random walk on Z independent of the environment ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Define (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='5) Un = inf{m : |Ym| = n}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Un is the first time the reflected random walk hits n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Then for every k > 0, for fixed environment ω, Tr k d= Uξk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' In what follows we investigate how the asymptotic properties of ξk affect those of Tr k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' To do that, we will utilize the aforementioned equality 10 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' BURACZEWSKI, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' DYSZEWSKI AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' KO�LODZIEJSKA in distribution and hence we first need to describe the asymptotic properties of Un as n tends to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' The proof of the next lemma is omitted, since it follows from a standard application of Doob’s optimal stopping theorem to martingales Y 2 n −n, Y 4 n −6nY 2 n +3n2+2n, Y 6 n − 15nY 4 n + (45n2 + 30n)Y 2 n − (15n3 + 30n2 + 16n) and exp{±tYn}cosh(t)−n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Let Un, for n ∈ N be given in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' We have EUn = n2, EU 2 n = 5n4/3 − 2n2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Moreover, as n → ∞, Un/n2 ⇒ 2ϑ, for ϑ defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Furthermore the family of random variables {n−4U 2 n}n∈N is uni- formly integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' The sequence T r Sn = �n k=1 Tr k is a sum of Pω independent random variables {Tr k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Since, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='6) VarωTr k = 2 3ξ4 k − 2 3ξ2 k, the variance VarωT r Sn behaves asymptotically as (2/3) �n k=1 ξ4 k, thus obeys a stable limit theorem [11, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Moreover, we can use precise large deviation results for sums of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' regularly varying random variables [8, Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='1] to describe the deviations of VarωT r Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' That is for any sequence {αn} that tends to infinity, P[VarωT r Sn ≥ αna4 n] ∼ (2/3)β/4nα−β/4 n a−β n ℓ(α1/4 n an).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' We can now use Potter bounds [2, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='6] to control ℓ(α1/4 n an) with ℓ(an).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' This in turn yields a large deviation result asymptotic on the logarithmic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' We summarize this discussion in the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' The sequence {VarωT r Sn/a4 n}n∈N converges in distribution (with respect to P) to some stable random variable Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Moreover for any sequence {αn}n∈N that tends to infinity, log P[VarωT r Sn ≥ αna4 n] ∼ −β log(αn)/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' The sequence {T l Sn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' The structure of Tl k is more involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' We may express it as a sum of independent copies of Fk, which denotes the length of a single excursion to the left from Sk, and thus obtain formulae for its quenched expectation and quenched variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AyT4oBgHgl3EQfmvi8/content/2301.00478v1.pdf'} +page_content=' The following formulae hold EωFk = 2(ξk + Wk−1), VarωFk = 8 � j 0, and for every pair of distinct vertices 푖, 푗 ∈ [푛] the edge {푖, 푗} +is independently added to the graph G with probability (1 + 훾 · 푥푖 · 푥푗) 푑 +푛 . +Note that for distinct vertices 푖, 푗 ∈ [푛], the edge {푖, 푗} is present in G with probability +(1 + 훾) 푑 +푛 if the vertices have the same label 푥푖 = 푥푗 and with probability (1 − 훾) 푑 +푛 if the +vertices have different labels 푥푖 ≠ 푥푗.3 +Given a graph G sampled according to this model, the goal is to recover the (unknown) +underlying vector of labels as well as possible. In particular, for a chosen algorithm return- +ing a partition ˆ푥(G) ∈ {±1}푛, there are two main objective of interest: weak recovery and exact +recovery. The former amounts to finding a partition ˆ푥(G) correlated with the true partition. +The latter instead corresponds to actually recovering the true partition with high probabil- +ity. As shown in the following table, by now the statistical and computational landscape +of these problems is well understood [DKMZ11, Mas14, MNS15b, MNS18, GV16]: +Objective +can be achieved (and +efficiently so) iff +weak recovery +ℙG∼SBM푛(푑,훾,푥) +� +1 +푛 |⟨푥, ˆ푥(G)⟩| ⩾ Ω푑,훾(1) +� +⩾ 1 − 표(1) +훾2 · 푑 ⩾ 1 +exact recovery +ℙG∼SBM푛(푑,훾,푥) +� +ˆ푥(G) ∈ {푥, −푥} +� +⩾ 1 − 표(1) +푑 +log 푛 +� +1 − +� +1 − 훾2 +� +⩾ 1 +Learning mixtures of spherical Gaussians. +The Gaussian Mixture Model we consider +is the following. +Model 1.2 (Mixtures of spherical Gaussians). Let 퐷1, . . . , 퐷푘 be Gaussian distributions +on ℝ푑 with covariance Id and means 휇1, . . . , 휇푘 satisfying ∀푖 ≠ 푗 , +��휇푖 − 휇푗 +�� ⩾ Δ . Given +a set Y = {y1, . . . , y푛} of 푛 samples from the uniform mixture over 퐷1, . . . , 퐷푘, estimate +휇1, . . . , 휇푘. +1We use bold characters to denote random variables. +2More general versions of the stochastic block model allow for more than two labels and general edge +probabilities depending on the label assignment. However, many of the algorithmic phenomena of the +general version can in their essence already be observed for the basic version that we consider in this work. +3At times we may write 푑푛 , 훾푛 to emphasize that these may be functions of 푛. We write 표(1), 휔(1) for +functions tending to zero (resp. infinity) as 푛 grows. +3 + +It is known that when the minimum separation is Δ = 표( +� +log 푘), superpolynomially +many samples are required to estimate the means up to small constant error [RV17]. +Just above this threshold, at separation 푘푂(1/훾) for any constant 훾, there exist efficient +algorithms based on the sum-of-squares hierarchy recovering the means up to accuracy +1/poly(푘) [HL18, KSS18, ST21]. In the regime where Δ = 푂( +� +log 푘) these algorithms yield +the same guarantees but require quasipolynomial time. Recently, [LL22] showed how to +efficiently recover the means as long as Δ = 푂 +� +log(푘) +1 +2 +푐 +� +for any constant 푐 > 0. +1.1 +Results +Stochastic block model. +We present here the first (휀, 훿)-differentially private efficient +algorithms for exact recovery. +Theorem 1.3 (Private exact recovery of SBM). Let 푥 ∈ {±1}푛 be balanced4. For any 훾, 푑, 휀, 훿 > +0 satisfying +푑 +log 푛 +� +1 − +� +1 − 훾2 +� +⩾ 4 +and +훾푑 +log 푛 ⩾ Ω +� +1 +휀2 · log(1/훿) +log 푛 ++ 1 +휀 +� +, +there exists an algorithm that, on input G ∼ SBM푛(푑, 훾, 푥), returns ˆ푥(G) ∈ {푥, −푥} with +probability 1 − 표(1). Moreover, the algorithm is (휀, 훿)-differentially private5 for any input graph, +and runs in polynomial time. +For any constant 휀 > 0, Theorem 1.3 states that (휀, 훿)-differentially private exact re- +covery is possible, in polynomial time, already a constant factor close to the non-private +threshold. Previous results [SNVT22] could only achieve comparable guarantees in time +푂� +푛푂(log 푛)� +. It is also important to observe that the theorem provides a trade-off between +signal-to-noise ratio of the instance (captured by the expression on the left-hand side +with 훾, 푑) and the privacy parameter 휀 . In particular, we distinguish two regimes: for +푑 = 퐶∗ ·log 푛 one can achieve exact recovery with high probability and privacy parameters +훿 = 푛−푂(퐶∗) , 휀 > 퐶−푂(1) +∗ +/훾. For +√ +푑 ⩾ 휔(log 푛) one can achieve exact recovery with high +probability and privacy parameters 휀 = 표(1), 훿 = 푛−휔(1) . +Further, we present a second, exponential-time, algorithm based on the exponential +mechanism [MT07] which improves over the above in two regards: First, it gives pure +privacy guarantees, i.e., 훿 = 0, and second, provides strong privacy guarantees for a larger +range of graph parameters. In fact, we will also prove a lower bound which shows that +its privacy guarantees are information theoretically optimal.6 All hidden constants are +absolute and in particular, do not depend on any graph or privacy parameters unless +4A vector 푥 ∈ {±1}푛 is said to be balanced if �푛 +푖=1 푥푖 = 0. +5See Definition 5.1 for a precise definition of adjacent graphs. +6Optimality only holds in the "small error" regime, otherwise it is almost optimal. See the lower bound +for more detail. +4 + +stated otherwise. In what follows we denote by err( ˆ푥, 푥) the minimum of the hamming +distance of ˆ푥 and 푥, and the one of − ˆ푥 and 푥, divided by 푛. +Theorem 1.4 (Slightly informal, see Theorem 5.17 for full version). Let 훾 +√ +푑 ⩾ Ω(1), 푥 ∈ +{±1}푛 be balanced, and 휁 ⩾ exp� +−Ω� +훾2푑�� +. For any 휀 ⩾ Ω +� +log(1/휁) +훾푑 +� +, there exists an algorithm +which on input G ∼ SBM푛(훾, 푑, 푥) outputs an estimate ˆ푥(G) ∈ {±1}푛 satisfying +err( ˆ푥(G), 푥) ⩽ 휁 +with probability at least 1 − 휁. In addition, the algorithm is 휀-private. Further, we can achieve error +Θ +� +1/ +� +log(1/휁) +� +with the increased success probability 1 − 푒−푛. +A couple of remarks are in order. First, our algorithm works across all degree-regimes +in the literature and matches known non-private thresholds and rates up to constants.7 +In particular, for 훾2푑 = Θ(1), we achieve weak/partial recovery with either constant or +exponentially high success probability. Recall that the optimal non-private threshold is +훾2푑 > 1. For the regime, where 훾2푑 = 휔(1), it is known that the optimal error rate is +exp� +−(1 − 표(1))훾2푑� +[ZZ16] even non-privately which we match up to constants - here +표(1) denotes a function that tends to zero as 훾2푑 tends to infinity. Moreover, our algo- +rithm achieves exact recovery as soon as 훾2푑 = Ω� +log 푛� +since then 휁 < +1 +푛. This also +matches known non-private threshholds up to constants [ABH15, MNS15a]. We remark +that [SNVT22] gave an 휀-DP exponential time algorithm which achieved exact recovery +and has inverse polynomial success probability in the utility case as long as 휀 ⩾ Ω +� +log 푛 +훾푑 +� +. +We recover this result as a special case.8 In fact, their algorithm is also based on the expo- +nential mechanism, but their analysis only applies to the setting of exact recovery, while +our result holds much more generally. Another crucial difference is that we show how to +privatize a known boosting technique frequently used in the non-private setting, allowing +us to achieve error guarantees which are optimal up to constant factors. +It is natural to ask whether, for a given set of parameters 훾, 푑, 휁 one could hope to obtain +better privacy guarantees than Theorem 1.4. Our next result is an information theoretic +lower bound which shows that our guarantees are almost tight. +Theorem 1.5 (Slightly informal, see Theorem 5.21 for full version). Suppose there exists an 휀- +differentiallyprivate algorithm such that for any balanced 푥 ∈ {±1}푛, on input G ∼ SBM푛(푑, 훾, 푥), +outputs ˆ푥(G) ∈ {±1}푛 satisfying +ℙ(err( ˆ푥(G), 푥) < 휁) ⩾ 1 − 휂 . +Then, +휀 ⩾ Ω +�log(1/휁) +훾푑 ++ log(1/휂) +휁푛훾푑 +� +. +(1.1) +7For ease of exposition we did not try to optimize these constants. +8With slightly worse constants. +5 + +Notice that this is a lower bound for all desired error rates (weak to exact recovery). +For failure probability 휂 = 휁, the lower bound simplifies to 휀 ⩾ Ω +� +log(1/휁) +훾푑 +� +and hence +matches Theorem 1.4 up to constants. For exponentially small failure probability, 휂 = 푒−푛, +it becomes 휀 ⩾ Ω +� +1 +휁훾푑 +� +. To compare, using the substitution +� +log(1/휁) → 휁, Theorem 1.4 +requires 휀 ⩾ Ω +� +1 +휁2훾푑 +� +in this regime. +Further, while formally incomparable, this lower bound also suggests that the guar- +antees obtained by our efficient algorithm in Theorem 1.3 might be close to optimal. In +particular, setting 훿 = 푛−Θ(1), implies that Theorem 1.3 achieves (휀, 푛−Θ(1))-private exact +recover, i.e., 휁 < 1/푛, whenever9 휀 ⩾ Ω +�� +log 푛 +훾푑 +� +. Theorem 1.5 states that in this exact +recovery setting 휀 ⩾ Ω +� +log 푛 +훾푑 +� +is necessary. We leave it as fascinating open questions to +bridge the gaps between upper and lower bounds in some cases. +Learning mixtures of spherical Gaussians. +Our algorithm for privately learning mix- +tures of 푘 spherical Gaussians is the first to break the +√ +푘 separation barrier. +Theorem 1.6 (Privately learning mixtures of spherical Gaussians). Consider an instance of +Model 1.2. Let 푡 > 0 be such that Δ ⩾ 푂 +�√ +푡푘1/푡� +. For 푛 ⩾ Ω� +푘푂(1) · 푑푂(푡)� +, 푘 ⩾ (log 푛)1/5 , +there exists an algorithm, running in time (푛푑)푂(푡), that outputs vectors ˆ흁1, . . . , ˆ흁푘 satisfying +max +ℓ∈[푘] +�� ˆ흁ℓ − 휇휋(ℓ) +�� +2 ⩽ 푂(푘−12) , +with high probability, for some permutation 휋 : [푘] → [푘] . Moreover, for 휀 ⩾ 푘−10 , 훿 ⩾ 푛−10 , +the algorithm is (휀, 훿)-differentially private10 for any input 푌. +Prior to this work, known differentially private algorithms [KSSU19] could learn a +mixture of 푘-spherical Gaussian only if: (1) they were given a ball of radius 푅 containing all +centers;11 and (2) the minimum separation between centers satisfied Δ ⩾ +√ +푘. Theorem 1.6 +overcomes both obstacles. First, no explicit upper bounds on the means is required (this +also means the sample complexity does not depend on 푅). Second, the theorem drastically +improves over the separation requirements of [KSSU19]. Further, our algorithms not only +work for mixtures of Gaussians but for the significantly more general class of mixtures +of Poincaré distributions, for which previous private algorithms are not known to work. +Concretely, in the high dimensional regime 푘 ⩾ +� +log 푑, our algorithm recovers the state- +of-the-art guarantees provided by non-private algorithms which are based on the sum-of- +squares hierarchy [KSS18, HL18, ST21]:12 +9in addition to the condition independent of 휀 +10Our notion of adjacent databases here is the obvious one. See Definition 6.1. +11We remark that in [KSSU19] the sample complexity of the algorithm depends on this radius 푅. +12We remark that [LL22] give a polynomial time algorithm for separation Ω(log(푘)1/2+푐) for constant 푐 > 0 +in the non-private setting but for a less general class of mixture distributions. +6 + +• If Δ ⩾ 푘1/푡∗ for some 푡∗ ∈ ℕ, then by choosing 푡 ⩾ Ω(푡∗) the algorithm recovers the +centers, up to a 1/poly(푘) error, in time poly(푘, 푑) and using only poly(푘, 푑) samples. +• If Δ ⩾ Ω( +� +log 푘) then choosing 푡 = 푂(log 푘) the algorithm recovers the centers, up +to a 1/poly(푘) error, in quasi-polynomial time poly(푘푂(푡), 푑푂(푡2)) and using a quasi- +polynomial number of samples poly(푘, 푑푂(푡)) . +For simplicity of exposition we will limit the presentation to mixtures of spherical Gaus- +sians. We reiterate that separation Ω( +� +log 푘) is information-theoretically necessary for +algorithms with polynomial sample complexity [RV17]. +2 +Techniques +We present here our general tools for designing efficient private estimation algorithms in +the high-dimensional setting whose statistical guarantees almost match those of the best +know non-private algorithms. The algorithms we design have the following structure in +common: First, we solve a convex optimization problem with constraints and objective +function depending on our input 푌. Second, we round the optimal solution computed in +the first step to a solution 푋 for the statistical estimation problem at hand. +We organize our privacy analyses according to this structure. In order to analyze the +first step, we prove a simple sensitivity bound for strongly convex optimization problems, +which bounds the ℓ2-sensitivity of the optimal solution in terms of a uniform sensitivity +bound for the objective function and the feasible region of the optimization problem. +For bounded problems –such as recovery of stochastic block models– we use this +sensitivity bound, in the second step, to show that introducing small additive noise to +standard rounding algorithms is enough to achieve privacy. +For unbounded problems –such as learning GMMs– we use this sensitivity bound +to show that on adjacent inputs, either most entries of 푋 only change slightly, as in +the bounded case, or few entries vary significantly. We then combine different privacy +techniques to hide both type of changes. +Privacy from sensitivity of strongly convex optimization problems. +Before illustrating +our techniques with some example, it is instructive to explicit our framework. Here we +have a set of inputs 풴 and a family of strongly convex functions ℱ (풴) and convex sets +풦(풴) parametrized by these inputs. The generic non-private algorithm based on convex +optimization we consider works as follows: +1. Compute ˆ푋 := argmin푋∈풦(푌) 푓푌(푋) ; +2. Round ˆ푋 into an integral solution. +7 + +For an estimation problem, a distributional assumption on 풴 is made. Then one shows +how, for typical inputs Y sampled according to that distribution, the above scheme recovers +the desired structured information. +We can provide a privatized version of this scheme by arguing that, under reasonable +assumptions on ℱ (푌) and 풦(풴), the output of the function argmin푋∈풦(푌) 푓푌(푋) has low +ℓ2-sensitivity. The consequence of this crucial observation is that one can combine the +rounding step 2 with some standard privacy mechanism and achieve differential privacy. +That is, the second step becomes: +2. Add random noise N and round ˆ푋 + N into an integral solution. +Our sensitivity bound is simple, yet it generalizes previously known bounds for +strongly convex optimization problems (we provide a detailed comparison later in the +section). For adjacent 푌, 푌′ ∈ 풴 , it requires the following ingredients: +(i) For each 푋 ∈ 풦(푌) ∩ 풦(푌′) it holds | 푓푌(푋) − 푓푌′(푋)| ⩽ 훼; +(ii) For each 푋 ∈ 풦(푌) its projection 푋′ onto 풦(푌′)∩풦(푌) satisfies | 푓푌(푋)− 푓푌(푋′)| ⩽ 훼 . +Here we think of 훼 as some small quantity (relatively to the problem parameters). Notice, +we may think of (i) as Lipschitz-continuity of the function 푔(푌, 푋) = 푓푌(푋) with respect +to 푌 and of (ii) as a bound on the change of the constrained set on adjacent inputs. In fact, +these assumptions are enough to conclude low ℓ2 sensitivity. If ˆ푋 and ˆ푋′ are the outputs +of the first step on inputs 푌, 푌′, then there exists 푋 ∈ 풦(푌) ∩ 풦(푌′) such that +| 푓푌( ˆ푋) − 푓푌(푋)| + | 푓푌′( ˆ푋′) − 푓푌′(푋)| ⩽ 푂(훼) . +By strong convexity of 푓푌 , 푓푌′ this implies +��� ˆ푋 − 푋 +��� +2 +2 + +��� ˆ푋′ − 푋 +��� +2 +2 ⩽ 푂(훼) +which ultimately means ∥ ˆ푋 − ˆ푋′∥2 +2 ⩽ 푂(훼). Thus, starting from our assumptions on the +point-wise distance of 푓푌 , 푓푌′ we were able to conclude low ℓ2-sensitivity of our output! +A simple application: weak recovery of stochastic block models. +The ideas introduced +above, combined with existing algorithms for weak recovery of stochastic block mod- +els, immediately imply a private algorithm for the problem. To illustrate this, consider +Model 1.1 with parameters 훾2푑 ⩾ 퐶, for some large enough constant 퐶 > 1. Let 푥 ∈ {±1}푛 +be balanced. Here the input 푌 is an 푛-by-푛 matrix corresponding to the rescaled centered +adjacency matrix of the graph: +푌푖푗 = +� 1 +훾푑 +� +1 − 푑 +푛 +� +if 푖푗 ∈ 퐸(퐺) +− 1 +훾푛 +otherwise. +8 + +The basic semidefinite program [GV16, MS16] can be recast as the strong constrained +optimization question of finding the orthogonal projection of the matrix 푌 onto the set +풦 := {푋 ∈ ℝ푛×푛 | 푋 ⪰ 0 , ∥푋∥∞ ⩽ 1/푛} . That is: +ˆ푋 := argmin푋∈풦 ∥푌 − 푋∥2 +F . +It is a standard fact that, if our input was G ∼ SBM푛(푑, 훾, 푥), then with high probability +푋(G) = argmin푋∈풦 푓푌(G)(푋) would have leading eigenvalue, eigenvector pair satisfying +휆1(G) ⩾ 1 − 푂(1/훾2푑) , +⟨푣1(G), 푥/∥푥∥⟩2 ⩾ 1 − 푂� +1/훾2푑� +. +This problem fits perfectly the description of the previous paragraph. In fact, it stands to +reason that the closeness of the projections ˆ푋, ˆ푋′ of inputs 푌, 푌′ should be proportional +to the distance between 푌 and 푌′. Our sensitivity argument above formalizes this simple +intuition. Concretely, observe that the constrained set 풦 is fixed and that for each 푋 ∈ 풦 +it holds | 푓푌(푋) − 푓푌′(푋)| ⩽ 푂� +∥푌 − 푌′∥2 +F + ∥푌 − 푌′∥1 +� +. It is easy to see that on adjacent +input we have ∥푌 − 푌′∥2 +F + ∥푌 − 푌′∥1 ⩽ 푂(1/푛훾푑) and thus this immediately yields +∥ ˆ푋 − ˆ푋′∥2 +F ⩽ 푂(1/푛훾푑). +The rounding step is now straightforward. Using the Gaussian mechanism we return +the leading eigenvector of ˆ푋 + N where N ∼ 푁 +� +0, +1 +푛훾푑 · log(1/훿) +휀2 +�푛×푛 +. This matrix has Frobei- +nus norm significantly larger than ˆ푋 but its spectral norm is only +∥N∥ ⩽ +� +푛 log(1/훿) +휀 +· +� +1 +푛훾푑 ⩽ 1 +휀 · +� +log(1/훿) +훾푑 +. +Thus by standard linear algebra, for typical instances G ∼ SBM푛(푑, 훾, 푥), the leading +eigenvector of ˆ푋(G) + N will be highly correlated with the true community vector 푥 +whenever the average degree 푑 is large enough. In conclusion, a simple randomized +rounding step is enough! +Remark 2.1 (From weak recovery to exact recovery). In the non-private setting, given a +weak recovery algorithm for the stochastic block model, one can use this as an initial +estimate for a boosting procedure based on majority voting to achieve exact recovery. We +show that this can be done privately. See Section 5.2. +An advanced application: learning mixtures of Gaussians. +In the context of stochastic +block models our argument greatly benefited from two key properties: first, on adjacent +inputs the difference ∥푌 − 푌′∥F was bounded; and second, the convex set 풦 was fixed. In +the context of learning mixtures of spherical Gaussians as in Model 1.2, both this properties +are not satisfied (notice how one of this second properties would be satisfied assuming +bounded centers!). So additional ingredients are required. +9 + +The first observation, useful to overcome the first obstacle, is that before finding the +centers, one can first find the 푛-by-푛 membership matrix 푊(푌) where 푊(푌)푖푗 = 1 if +푖, 푗 where sampled from the same mixture component and 0 otherwise. The advantage +here is that, on adjacent inputs, ∥푊(푌) − 푊(푌′)∥2 +F ⩽ 2푛/푘 and thus one recovers the first +property.13 Here early sum-of-squares algorithms for the problem [HL18, KSS18] turns out +to be convenient as they rely on minimizing the function ∥푊 ∥2 +F subject to the following +system of polynomial inequalities in variables 푧11 , . . . , , 푧1푘 , . . . , 푧푛푘, with 푊푖푗 = � +ℓ 푧푖ℓ 푧푗ℓ +for all 푖, 푗 ∈ [푛] and a parameter 푡 > 0. + + +푧2 +푖ℓ = 푧푖ℓ +∀푖 ∈ [푛] , ℓ ∈ [푘] +(indicators) +� +ℓ∈[푘] +푧푖ℓ ⩽ 1 +∀푖 ∈ [푛] +(cluster membership) +푧푖ℓ · 푧푖ℓ′ = 0 +∀푖 ∈ [푛] , ℓ ∈ [푘] +(unique membership) +� +푖 +푧푖ℓ = 푛/푘 +∀ℓ ∈ [푘] +(size of clusters) +휇′ +ℓ = 푘 +푛 +� +푖 +푧푖ℓ · 푦푖 +∀ℓ ∈ [푘] +(means of clusters) +푘 +푛 +� +푖 +푧푖ℓ ⟨푦푖 − 휇′ +ℓ , 푢⟩2푡 ⩽ (2푡)푡 · ∥푢∥푡 +2 +∀푢 ∈ ℝ푑 , ℓ ∈ [푘] +(subgaussianity of 푡-moment) + + +(풫(푌)) +For the scope of this discussion,14 we may disregard computational issues and assume we +have access to an algorithm returning a point from the convex hull 풦(푌) of all solutions to +our system of inequalities.15 Here each indicator variable 푧푖ℓ ∈ {0, 1} is meant to indicate +whether sample 푦푖 is believed to be in cluster 퐶ℓ. In the non-private settings, the idea +behind the program is that –for typical Y sampled according to Model 1.2 with minimum +separation Δ ⩾ 푘1/푡√ +푡– any solution푊(Y) ∈ 풦(Y) is close to the ground truth matrix +푊∗(Y) in Frobenius norm: ∥푊(Y) − 푊∗(Y)∥2 +F ⩽ 1/poly(푘) . Each row 푊(Y)푖 may be seen as +inducing a uniform distribution over a subset of Y.16. Thus, combining the above bound +with the fact that subgaussian distributions at small total variation distance have means +that are close, we can conclude the algorithm recovers the centers of the mixture. +While this program suggests a path to recover the first property, it also possesses a +fatal flaw: the projection 푊′ of 푊 ∈ 풦(푌) onto 풦(푌) ∩ 풦(푌′) may be far in the sense that +13Notice for typical inputs Y from Model 1.2 one expect ∥푊(Y)∥F ≈ 푛2/푘 . +14While this is far from being true, it turns out that having access to a pseudo-distribution satisfying 풫(푌) +is enough for our subsequent argument to work, albeit with some additional technical work required. +15We remark that a priori it is also not clear how to encode the subgaussian constraint in a way that +we could recover a degree-푡 pseudo-distribution satisfying 풫(푌) in polynomial time. By now this is well +understood, we discuss this in Section 3. +16More generally, we may think of a vector 푣 ∈ ℝ푛 as the vector inducing the distribution given by 푣/∥푣∥1 +onto the set 푌 of 푛 elements. +10 + +|∥푊 ∥2 +F − ∥푊′∥2 +F| ⩾ Ω(∥푊 ∥2 +F + ∥푊′∥2 +F) ⩾ Ω(푛2/푘) . The reason behind this phenomenon can +be found in the constraint � +푖 푧푖ℓ = 푛/푘 . The set indicated by the vector (푧1ℓ . . . , 푧푛ℓ) may +be subgaussian in the sense of 풫(푌) for input 푌 but, upon changing a single sample, this +may no longer be true. We work around this obstacle in two steps: +1. We replace the above constraint with � +푖 푧푖ℓ ⩽ 푛/푘 . +2. We compute ˆ푊 := argmin푊 solving 풫(푌)∥퐽 − 푊 ∥2 +F , where 퐽 ∈ ℝ푛×푛 is the all-ones +matrix.17 +The catch now is that the program is satisfiable for any input푌. Moreover, we can guarantee +property (ii) (required by our sensitivity argument) for 훼 ⩽ 푂(푛/푘), since we can obtain +푊′ ∈ 풦(푌)∩풦(푌′) simply zeroing out the row/column in 푊 corresponding to the sample +differing in 푌 and 푌′. Then for typical inputs Y, the correlation with the true solution is +guaranteed by the new strongly convex objective function. +From low sensitivity of the indicators to low sensitivity of the estimates. +For adjacent +inputs 푌, 푌′ let ˆ푊, ˆ푊′ be respectively the matrices computed by the above strongly convex +programs. Our discussion implies that, applying our sensitivity bound, we can show ∥ ˆ푊 − +ˆ푊′∥2 +F ⩽ 푂(푛/푘) . The problem is that simply applying a randomized rounding approach +here cannot work. The reason is that even tough the vector ˆ푊푖 induces a subgaussian +distribution, the vector ˆ푊푖 + 푣 for 푣 ∈ ℝ푛, might not. Without the subgaussian constraint +we cannot provide any meaningful utility bound. In other words, the root of our problem +is that there exists heavy-tailed distributions that are arbitrarily close in total variation +distance to any given subgaussian distribution. +On the other hand, our sensitivity bound implies ∥ ˆ푊 − ˆ푊′∥2 +1 ⩽ 표(∥ ˆ푊 ∥1) and thus, all +but a vanishing fraction of rows 푖 ∈ [푛] must satisfy ∥ ˆ푊푖 − ˆ푊′ +푖 ∥1 ⩽ 표(∥ ˆ푊푖∥1). For each row +푖 , let 휇푖 , 휇′ +푖 be the means of the distributions induced respectively by ˆ푊푖 , ˆ푊′ +푖 . We thus +find ourselves in the following settings: +1. For a set of (1 − 표(1)) · 푛 good rows +��휇푖 − 휇′ +푖 +�� +2 ⩽ 표(1) , +2. For the set ℬ of remaining bad rows, the distance +��휇푖 − 휇′ +푖 +�� +2 may be unbounded. +We hide differences of the first type as follows: pick a random subsample 퓢 of [푛] +of size 푛푐, for some small 푐 > 0, and for each picked row use the Gaussian mechanism. +The subsampling step is useful as it allows us to decrease the standard deviation of the +entry-wise random noise by a factor 푛1−푐 . +We hide differences of the second type as follows: use the classic high dimensional +(휀, 훿)-private histogram learner on 퓢 and for the 푘 largest bins of highest count privately +return their average. The crux of the argument here is that the cardinality of ℬ ∩ 퓢 is +17We remark that for technical reasons our function in Section 6.1 will be slightly different. We do not +discuss it here to avoid obfuscating our main message. +11 + +sufficiently small that the privacy guarantees of the histogram learner can be extended +even for inputs that differ in |ℬ ∩ 퓢| many samples. +Finally, standard composition arguments will guarantee privacy of the whole algo- +rithm. +Comparison with previous works on empirical risk minimization. +Results along the +lines of the sensitivity bound described at the beginning of the section (see Lemma 4.1 for a +formal statement) have been extensively used in the context of empirical risk minimization +[CMS11, KST12, SCS13, BST14, WYX17, MSVV21]. Most results focus on the special case +of unconstrained optimization of strongly convex functions. In contrast, our sensitivity +bound applies to the significantly more general settings where both the objective functions +and the constrained set may depend on the input.18 +Most notably for our settings of interest, [CMS11] studied unconstrained optimization +of (smooth) strongly convex functions depending on the input, with bounded gradient.We +recover such a result for 푋′ = 푋 in (ii) +In [MSVV21], the authors considered constraint optimization of objective functions +where the domain (but not the function) may depend on the input data. They showed +how one can achieve differential privacy while optimize the desired objective function +by randomly perturbing the constraints. It is important to remark that, in [MSVV21], the +notion of utility is based on the optimization problem (and their guarantees are tight only +up to logarithmic factors). In the settings we consider, even in the special case where 푓 +does not depend on the input, this notion of utility may not correspond to the notion of +utility required by the estimation problem, and thus, the corresponding guarantees may +turn out to be too loose to ensure the desired error bounds. +Exponential time pure-DP algorithm for SBM. +Our exponential time algorithm is based +on the exponential mechanism [MT07]. In particular, for a given graph 퐺, recall that +푌 = +1 +훾푑 +� +퐴(퐺) − 푑 +푛 퐽� +, where 퐴(퐺) is the adjacency matrix of 퐺 and 퐽 the all-ones matrix. +Define the function 푠 : {±1}푛 → ℝ as 푠(푥) = ⟨푥, 푌푥⟩ and Δ = +2 +훾푑. In privacy terms, these +are called the score function and the sensitivity - the maximum amount 푠 can change on +adjacent graphs - which can be readily seen to be +2 +훾푑. The exponential mechanism then +amounts to outputting a sample from the distribution with density +푝(푥) ∝ exp� 휀 +2Δ푠(푥)� +. +(2.1) +Standard arguments show that this procedure is 휀-private. Note, that it is well-known +that if G ∼ SBM푛(푑, 훾, 푥∗) and 훾2푑 is larger than some universal constant, 푌 is close to +1 +푛 푥∗(푥∗)⊤ in cut-norm (or (∞ → 1)-norm). That is, the quadratic form 푌 − 1 +푛 푥∗(푥∗)⊤ is +18The attentive reader may argue that one could cast convex optimization over a constrained domain +as unconstrained optimization of a new convex function with the appropriate penalty terms. In practice +however, this turns out to be hard to do for constraints such as Definition 3.19. +12 + +close to zero over the hypercube [GV16]. It follows, that the maximizer of score function +푠 over the hypercube is close to +1 +√푛 푥∗. It is not too hard to show that with exponentially +high probability (in 푛), this remains true also for samples from the above distribution. +On an intuitive level this follows since we assign exponentially larger mass to points +achieving comparable scores as the maximizer than to points achieving smaller scores +(see Section 5.3). +While this algorithm matches known non-private thresholds and rates up to constants +and has close to optimal privacy guaratnees (see the discusion in Section 1.1), there are +several obstacles to making it efficient. We discuss several approaches: First, one could try +to sample from the distribution in Eq. (2.1) directly. Note that this corresponds to an Ising +model over the hypercube with interaction matrix 퐽 ≔ 휀 +훿푌. However, known samplers, e.g. +[EKZ22, KLR22], require strong assumptions on the spectrum of 퐽 which are not satisfied +in our setting - in particular, 퐽 could have arbitrarly many eigenvalues of magnitude +larger than 1. A second approach would be to relax the support of the distribution to all +positive semi-definite matrices with diagonal entries equal to 1/푛 - similar to the set 풦 +considered for our approximate DP algorithm - and the score function to the inner product +of 푌 with such matrices. Although such "convexification" techniques of the exponential +mechanism have recently seen success in the design of pure-DP algorithms [HKM22] and +this particular relaxation is known to work in the non-private setting [GV16, FC20], it fails +in this case: The volume of the set of matrices achieving large enough score is smaller by +a factor of exp� +−푛2� +than the set of all feasible matrices. Hence, the exponential boost by +the reweighing of Eq. (2.1) is not enough to ensure outputting such a candidate. A third +strategy would be the following: In the non-private setting, for the restricted regime of +훾2푑 ⩾ 퐶 log 푛, for a large enough constant 퐶 > 0, standard matrix concentration bounds +show that PCA can recover the label of a large constant fraction of the vertices. However, +known lower bounds for pure-DP PCA algorithms [KT13], prevent us from recovering +this result in the private setting: In particular, define two matrices to be adjacent if there +difference has spectral norm at most 1. In this setting, any 휀-DP algorithm, which outputs +a vector achieving constant correlation with the top eigenvector of an 푛 × 푛 input matrix +needs to have spectral norm at least Ω� 푛 +휀 +� +. Translated to our scaling used above, this would +mean ∥푌∥ ⩾ Ω(푛훾푑 +휀 ), whereas we have ∥푌∥ ⩽ 푂(1). +Information theoretic privacy lower bounds for stochastic block models. +Our informa- +tion theoretic lower bound for stochastic block models is based on the following idea. Sup- +pose we have an 휀-differentially private exact recovery algorithm of SBM such that, over +the randomness of the algorithm and the input G ∼ SBM푛(푑, 훾, 푥), the algorithm outputs +ˆ푥(G) ∈ {±푥} with probability at least 2/3. Note for any 푥 ∈ {±1}푛, SBM푛(푑, 훾, 푥) is just a +product distribution of �푛 +2 +� +Bernoulli distributions. Fixing an arbitrary balanced 푦 ∈ {±1}푛, +there exist balanced 푦1, . . . , 푦푛 ∈ {±1}푛 such that Ham(푦, 푦푖) = 2 for 푖 ∈ [푛]. For each +푖 ∈ [푛], one may find a coupling 휔푖 of distributions SBM푛(푑, 훾, 푦) and SBM푛(푑, 훾, 푦푖) such +13 + +that, if(G, G′) ∼ 휔푖 then G and G′ typicallydifferbyonly 푂(훾푑)edges. Then bythe assump- +tion our algorithm is 휀-differentially private, it follows that, on input G ∼ SBM푛(푑, 훾, 푦), +our private algorithm outputs ±푦푖 with probability at least 푒−휀·푂(훾푑)· 2 +3 for each 푖 ∈ [푛]. Since +the sum of probabilities of disjoint events does not exceed one, we get 푛 · 푒−푂(휀훾푑) · 2 +3 ⩽ 1, +which implies 휀 ⩾ Ω(log 푛 +훾푑 ). +3 +Preliminaries +We use boldface characters for random variables. We hide multiplicative factors logarithmic +in 푛 using the notation ˜푂(·) , ˜Ω(·). Similarly, we hide absolute constant multiplicative +factors using the standard notation 푂(·) , Ω(·) , Θ(·). Often times we use the letter 퐶 do +denote universal constants independent of the parameters at play. We write 표(1), 휔(1) for +functions tending to zero (resp. infinity) as 푛 grows. We say that an event happens with +high probability if this probability is at least 1 − 표(1). Throughout the paper, when we say +"an algorithm runs in time 푂(푞)" we mean that the number of basic arithmetic operations +involved is 푂(푞). That is, we ignore bit complexity issues. +Vectors, matrices, tensors. +We use Id푛 to denote the 푛-by-푛 dimensional identity matrix, +퐽푛 ∈ ℝ푛×푛 the all-ones matrix and 0푛 , 1푛 ∈ ℝ푛 to denote respectively the zero and the +all-ones vectors. When the context is clear we drop the subscript. For matrices 퐴, 퐵 ∈ ℝ푛×푛 +we write 퐴 ⪰ 퐵 if 퐴 − 퐵 is positive semidefinite. For a matrix 푀, we denote its eigenvalues +by 휆1(푀) , . . . , 휆푛(푀), we simply write 휆푖 when the context is clear. We denote by ∥푀∥ +the spectral norm of 푀. We denote by ℝ푑⊗푡 the set of real-valued order-푡 tensors. for a +푑 × 푑 matrix 푀, we denote by 푀⊗푡 the 푡-fold Kronecker product 푀 ⊗ 푀 ⊗ · · · ⊗ 푀 +�������������������������������������� +푡 times +. We define +the flattening, or vectorization, of 푀 to be the 푑푡-dimensional vector, whose entries are the +entries of 푀 appearing in lexicographic order. With a slight abuse of notation we refer to +this flattening with 푀, ambiguities will be clarified form context. We denote by 푁 � +0, 휎2�푑⊗푡 +the distribution over Gaussian tensors with 푑푡 entries with standard deviation 휎. Given +푢, 푣 ∈ {±1}푛, we use Ham(푢, 푣) := �푛 +푖=1 +1[푢푖 ≠ 푣푖] to denote their Hamming distance. +Given a vector 푢 ∈ ℝ푛, we let sign(푢) ∈ {±1}푛 denote its sign vector. A vector 푢 ∈ {±1}푛 +is said to be balanced if �푛 +푖=1 푢푖 = 0. +Graphs. +We consider graphs on 푛 vertices and let 풢푛 be the set of all graphs on 푛 vertices. +For a graph 퐺 on 푛 vertices we denote by 퐴(퐺) ∈ ℝ푛×푛 its adjacency matrix. When the +context is clear we simply write 퐴 . Let 푉(퐺) (resp. 퐸(퐺)) denote the vertex (resp. edge) set +of graph 퐺. Given two graphs 퐺, 퐻 on the same vertex set 푉, let 퐺 \ 퐻 := (푉, 퐸(퐺) \ 퐻(퐺)). +Given a graph 퐻, 퐻′ ⊆ 퐻 means 퐻′ is a subgraph of 퐻 such that 푉(퐻′) = 푉(퐻) and +퐸(퐻) ⊆ 퐸(퐻). The Hamming distance between two graphs 퐺, 퐻 is defined to be the size +of the symmetric difference between their edge sets, i.e. Ham(퐺, 퐻) := |퐸(퐺)△퐸(퐻)|. +14 + +3.1 +Differential privacy +In this section we introduce standard notions of differential privacy [DMNS06]. +Definition 3.1 (Differential privacy). An algorithm ℳ : 풴 → 풪 is said to be (휀, 훿)- +differentially private for 휀, 훿 > 0 if and only if, for every 푆 ⊆ 풪 and every neighboring +datasets 푌, 푌′ ∈ 풴 we have +ℙ[ℳ(푌) ∈ 푆] ⩽ 푒휀 · ℙ[ℳ(푌′) ∈ 푆] + 훿 . +To avoid confusion, for each problem we will exactly state the relevant notion of neigh- +boring datasets. Differential privacy is closed under post-processing and composition. +Lemma 3.2 (Post-processing). If ℳ : 풴 → 풪 is an (휀, 훿)-differentially private algorithm and +ℳ′ : 풴 → 풵 is any randomized function. Then the algorithm ℳ′(ℳ(푌)) is (휀, 훿)-differentially +private. +In order to talk about composition it is convenient to also consider DP algorithms +whose privacy guarantee holds only against subsets of inputs. +Definition 3.3 (Differential Privacy Under Condition). An algorithm ℳ : 풴 → 풪 is said +to be (휀, 훿)-differentially private under condition Ψ (or (휀, 훿)-DP under condition Ψ) for +휀, 훿 > 0 if and only if, for every 푆 ⊆ 풪 and every neighboring datasets 푌, 푌′ ∈ 풴 both +satisfying Ψ we have +ℙ[ℳ(푌) ∈ 푆] ⩽ 푒휀 · ℙ[ℳ(푌′) ∈ 푆] + 훿 . +It is not hard to see that the following composition theorem holds for privacy under +condition. +Lemma 3.4 (Composition for Algorithm with Halting, [KMV22]). Let ℳ1 : 풴 → 풪1 ∪ +{⊥} , ℳ2 : 풪1 × 풴 → 풪2 ∪ {⊥} , . . . , ℳ푡 : 풪푡−1 × 풴 → 풪푡 ∪ {⊥} be algorithms. Furthermore, +let ℳ denote the algorithm that proceeds as follows (with 풪0 being empty): For 푖 = 1 . . . , 푡 compute +표푖 = ℳ푖(표푖−1, 푌) and, if 표푖 = ⊥, halt and output ⊥. Finally, if the algorithm has not halted, then +output 표푡. Suppose that: +• For any 1 ⩽ 푖 ⩽ 푡, we say that 푌 satisfies the condition Ψ푖 if running the algorithm on 푌 +does not result in halting after applying ℳ1, . . . , ℳ푖. +• ℳ1 is (휀1, 훿1)-DP. +• ℳ푖 is (휀푖, 훿푖)-DP (with respect to neighboring datasets in the second argument) under +condition Ψ푖−1 for all 푖 = {2, . . . , 푡} . +Then ℳ is (� +푖 휀푖, � +푖 훿푖)-DP. +15 + +3.1.1 +Basic differential privacy mechanisms +The Gaussian and the Laplace mechanism are among the most widely used mechanisms +in differential privacy. They work by adding a noise drawn from the Gaussian (respectively +Laplace) distribution to the output of the function one wants to privatize. The magnitude +of the noise depends on the sensitivity of the function. +Definition 3.5 (Sensitivity of function). Let 푓 : 풴 → ℝ푑 be a function, its ℓ1-sensitivity +and ℓ2-sensitivity are respectively +Δ 푓 ,1 := +max +푌 ,푌′∈풴 +푌 ,푌′ are adjacent +��푓 (푌) − 푓 (푌′) +�� +1 +Δ 푓 ,2 := +max +푌 ,푌′∈풴 +푌 ,푌′ are adjacent +�� 푓 (푌) − 푓 (푌′) +�� +2 . +For function with bounded ℓ1-sensitivity the Laplace mechanism is often the tool of +choice to achieve privacy. +Definition 3.6 (Laplace distribution). The Laplace distribution with mean 휇 and parameter +푏 > 0, denoted by Lap(휇, 푏), has PDF 1 +2푏 푒−|푥−휇|/푏 . Let Lap(푏) denote Lap(0, 푏). +A standard tail bound concerning the Laplace distribution will be useful throughout +the paper. +Fact 3.7 (Laplace tail bound). Let 풙 ∼ Lap(휇, 푏). Then, +ℙ +� +|풙 − 휇| > 푡 +� +⩽ 푒−푡/푏 . +The Laplace distribution is useful for the following mechanism +Lemma 3.8 (Laplace mechanism). Let 푓 : 풴 → ℝ푑 be any function with ℓ1-sensitivity at most +Δ 푓 ,1. Then the algorithm that adds Lap +� Δ푓 ,1 +휀 +�⊗푑 +to 푓 is (휀, 0)-DP. +It is also useful to consider the "truncated" version of the Laplace distribution where +the noise distribution is shifted and truncated to be non-positive. +Definition 3.9 (Truncated Laplace distribution). The (negatively) truncated Laplace distri- +bution w with mean 휇 and parameter 푏 on ℝ, denoted by tLap(휇, 푏), is defined as Lap(휇, 푏) +conditioned on the value being non-positive. +Lemma 3.10 (Truncated Laplace mechanism). Let 푓 : 풴 → ℝ be any function with ℓ1- +sensitivity at most Δ 푓 ,1. Then the algorithm that adds tLap +� +−Δ 푓 ,1 +� +1 + log(1/훿) +휀 +� +, Δ 푓 ,1/휀 +� +to 푓 is +(휀, 훿)-DP. +The following tail bound is useful when reasoning about truncated Laplace random +variables. +16 + +Lemma 3.11 (Tail bound truncated Laplace). Suppose 휇 < 0 and 푏 > 0. Let 풙 ∼ tLap(휇, 푏). +Then,for 푦 < 휇 we have that +ℙ +� +풙 < 푦 +� +⩽ 푒(푦−휇/푏) +2 − 푒휇/푏 . +In constrast, when the function has bounded ℓ2-sensitivity, the Gaussian mechanism +provides privacy. +Lemma 3.12 (Gaussian mechanism). Let 푓 : 풴 → ℝ푑 be any function with ℓ2-sensitivity +at most Δ 푓 ,2. Let 0 < 휀 , 훿 ⩽ 1. Then the algorithm that adds 푁 +� +0, +Δ2 +푓 ,2·2 log(2/훿) +휀2 +· Id +� +to 푓 is +(휀, 훿)-DP. +3.1.2 +Private histograms +Here we present a classical private mechanism to learn a high dimensional histogram. +Lemma 3.13 (High-dimensional private histogram learner, see [KV18]). Let 푞, 푏 , 휀 > 0 +and 0 < 훿 < 1/푛. Let {퐼푖}∞ +푖=−∞ be a partition of ℝ into intervals of length 푏, where 퐼푖 := +� +푥 ∈ ℝ +�� 푞 + (푖 − 1) · 푏 ⩽ 푥 < 푞 + 푖 · 푏 +� +. Consider the partition of ℝ푑 into sets +� +퐵푖1,...,푖푑 +�∞ +푖1,...,푖푑=1 +where +퐵푖1,...,푖푑 := +� +푥 ∈ ℝ푑 �� ∀푗 ∈ [푑] , 푥푗 ∈ 퐼푖푗 +� +Let 푌 = +� +푦1, . . . , 푦푛 +� +⊆ ℝ푑 be a database of 푛 points. For each 퐵푖1,...,푖푑, let 푝푖1,...,푖푑 = +1 +푛 +��� +푗 ∈ [푛] +�� 푦푗 ∈ 퐵푖1,...,푖푑 +���. For 푛 ⩾ +8 +휀훼 ·log 2 +훿훽 , there exists an efficient (휀, 훿)-differentially private +algorithm that returns ˆ풑1,...,1, . . . , ˆ풑푖1,...,푖푑, . . . satisfying +ℙ +� +max +푖1,...,푖푑∈ℕ|푝푖1,...,푖푑 − ˆ풑푖1,...,푖푑| ⩾ 훼 +� +⩽ 훽 . +Proof. We consider the following algorithm, applied to each 푖1, . . . , 푖푑 ∈ ℕ on input 푌: +1. If 푝푖1,...,푖푑 = 0 set ˆ푝푖1,...,푖푑 = 0 , otherwise let ˆ풑푖1,...,푖푑 = 푝푖1,...,푖푑 + 흉 where 흉 ∼ Lap� +0, 2 +푛휀 +� +. +2. If ˆ풑푖1,...,푖푑 ⩽ 3 log(2/훿) +휀푛 +set ˆ풑푖1,...,푖푑 = 0. +First we argue utility. By construction we get ˆ풑푖1,...,푖푑 = 0 whenever 푝푖1,...,푖푑 = 0, thus +we may focus on non-zero 푝푖1,...,푖푑. There are at most 푛 non zero 푝푖1,...,푖푑. By choice of 푛, 훿 +and by Fact 3.7 the maximum over 푛 independent trials 흉 ∼ Lap� +0, 2 +푛휀 +� +is bounded by 훼 +in absolute value with probability at least 훽. +It remains to argue privacy. Let 푌 = +� +푦1, . . . , 푦푛 +� +, 푌′ = +� +푦′ +1, . . . , 푦′ +푛 +� +be adjacent +databases. For 푖1, . . . , 푖푑 ∈ ℕ, let +푝푖1,...,푖푑 = +��� +푗 ∈ [푛] +�� 푦푗 ∈ 퐵푖1,...,푖푑 +��� +17 + +푝′ +푖1,...,푖푑 = +��� +� +푗 ∈ [푛] +��� 푦′ +푗 ∈ 퐵푖1,...,푖푑 +���� . +Since 푌, 푌′ are adjacent there exists only two set of indices ℐ := {푖1, . . . , 푖푑} and 풥 := +� +푗1, . . . , 푗푑 +� +such that 푝ℐ ≠ 푝′ +ℐ and 푝풥 ≠ 푝′ +풥 . Assume without loss of generality 푝ℐ > 푝′ +ℐ. +Then it must be 푝ℐ = 푝′ +ℐ +1/푛 and 푝풥 = 푝′ +풥 −1/푛 . Thus by the standard tail bound on the +Laplace distribution in Fact 3.7 and by Lemma 3.8, we immediately get that the algorithm +is (휀, 훿)-differentially private. +□ +3.2 +Sum-of-squares and pseudo-distributions +We introduce here the sum-of-squares notion necessary for our private algorithm learning +mixtures of Gaussians. We remark that these notions are not needed for Section 5. +Let 푤 = (푤1, 푤2, . . . , 푤푛) be a tuple of 푛 indeterminates and let ℝ[푤] be the set of +polynomials with real coefficients and indeterminates 푤, . . . , 푤푛. We say that a polynomial +푝 ∈ ℝ[푤] is a sum-of-squares (sos) if there are polynomials 푞1, . . . , 푞푟 such that 푝 = 푞2 +1 + · · ·+ +푞2 +푟. +3.2.1 +Pseudo-distributions +Pseudo-distributions are generalizations of probability distributions. We can represent a +discrete (i.e., finitely supported) probability distribution over ℝ푛 by its probability mass +function 퐷 : ℝ푛 → ℝ such that 퐷 ⩾ 0 and � +푤∈supp(퐷) 퐷(푤) = 1. Similarly, we can describe +a pseudo-distribution by its mass function. Here, we relax the constraint 퐷 ⩾ 0 and only +require that 퐷 passes certain low-degree non-negativity tests. +Concretely, a level-ℓ pseudo-distribution is a finitely-supported function 퐷 : ℝ푛 → ℝ +such that � +푤 퐷(푤) = 1 and � +푤 퐷(푤)푓 (푤)2 ⩾ 0 for every polynomial 푓 of degree at most +ℓ/2. (Here, the summations are over the support of 퐷.) A straightforward polynomial- +interpolation argument shows that every level-∞-pseudo distribution satisfies 퐷 ⩾ 0 and +is thus an actual probability distribution. We define the pseudo-expectation of a function 푓 +on ℝ푑 with respect to a pseudo-distribution 퐷, denoted ˜피퐷(푤) 푓 (푤), as +˜피퐷(푤) 푓 (푤) = +� +푤 +퐷(푤)푓 (푤) . +(3.1) +The +degree-ℓ +moment +tensor +of +a +pseudo-distribution +퐷 +is +the +tensor +피퐷(푤)(1, 푤1, 푤2, . . . , 푤푛)⊗ℓ. In particular, the moment tensor has an entry corresponding +to the pseudo-expectation of all monomials of degree at most ℓ in 푤. The set of all +degree-ℓ moment tensors of probability distribution is a convex set. Similarly, the set +of all degree-ℓ moment tensors of degree 푑 pseudo-distributions is also convex. Key +to the algorithmic utility of pseudo-distributions is the fact that while there can be no +efficient separation oracle for the convex set of all degree-ℓ moment tensors of an actual +probability distribution, there’s a separation oracle running in time 푛푂(ℓ) for the convex +set of the degree-ℓ moment tensors of all level-ℓ pseudodistributions. +18 + +Fact 3.14 ([Sho87, Par00, Nes00, Las01]). For any 푛, ℓ ∈ ℕ, the following set has a 푛푂(ℓ)-time +weak separation oracle (in the sense of [GLS81]): +� ˜피퐷(푤)(1, 푤1, 푤2, . . . , 푤푛)⊗푑 | degree-d pseudo-distribution 퐷 over ℝ푛� +. +(3.2) +This fact, together with the equivalence of weak separation and optimization [GLS81] +allows us to efficiently optimize over pseudo-distributions (approximately)—this algo- +rithm is referred to as the sum-of-squares algorithm. +The level-ℓ sum-of-squares algorithm optimizes over the space of all level-ℓ pseudo- +distributions that satisfy a given set of polynomial constraints—we formally define this +next. +Definition 3.15 (Constrained pseudo-distributions). Let 퐷 be a level-ℓ pseudo-distribution +over ℝ푛. Let 풜 = { 푓1 ⩾ 0, 푓2 ⩾ 0, . . . , 푓푚 ⩾ 0} be a system of 푚 polynomial inequality con- +straints. We say that 퐷 satisfies the system of constraints 풜 at degree 푟, denoted 퐷 푟 풜, if for ev- +ery 푆 ⊆ [푚] and every sum-of-squares polynomial ℎ with deg ℎ + � +푖∈푆 max{deg 푓푖, 푟} ⩽ ℓ, +˜피퐷ℎ · +� +푖∈푆 +푓푖 ⩾ 0 . +We write 퐷 +풜 (without specifying the degree) if 퐷 +0 풜 holds. Furthermore, we +say that 퐷 +푟 풜 holds approximately if the above inequalities are satisfied up to an error +of 2−푛ℓ · ∥ℎ∥ · � +푖∈푆∥ 푓푖∥, where ∥·∥ denotes the Euclidean norm19 of the coefficients of a +polynomial in the monomial basis. +We remark that if 퐷 is an actual (discrete) probability distribution, then we have 퐷 +풜 +if and only if 퐷 is supported on solutions to the constraints 풜. +We say that a system 풜 of polynomial constraints is explicitly bounded if it contains a +constraint of the form {∥푤∥2 ⩽ 푀}. The following fact is a consequence of Fact 3.14 and +[GLS81], +Fact 3.16 (Efficient Optimization over Pseudo-distributions). There exists an (푛 + 푚)푂(ℓ)- +time algorithm that, given any explicitly bounded and satisfiable system20 풜 of 푚 polynomial +constraints in 푛 variables, outputs a level-ℓ pseudo-distribution that satisfies 풜 approximately. +3.2.2 +Sum-of-squares proof +Let 푓1, 푓2, . . . , 푓푟 and 푔 be multivariate polynomials in 푤. A sum-of-squares proof that the +constraints { 푓1 ⩾ 0, . . . , 푓푚 ⩾ 0} imply the constraint {푔 ⩾ 0} consists of sum-of-squares +polynomials (푝푆)푆⊆[푚] such that +푔 = +� +푆⊆[푚] +푝푆 · Π푖∈푆 푓푖 . +(3.3) +19The choice of norm is not important here because the factor 2−푛ℓ swamps the effects of choosing another +norm. +20Here, we assume that the bit complexity of the constraints in 풜 is (푛 + 푚)푂(1). +19 + +We say that this proof has degree ℓ if for every set 푆 ⊆ [푚], the polynomial 푝푆Π푖∈푆 푓푖 has +degree at most ℓ. If there is a degree ℓ SoS proof that { 푓푖 ⩾ 0 | 푖 ⩽ 푟} implies {푔 ⩾ 0}, we +write: +{ 푓푖 ⩾ 0 | 푖 ⩽ 푟} ℓ {푔 ⩾ 0} . +(3.4) +Sum-of-squares proofs satisfy the following inference rules. For all polynomials +푓 , 푔 : ℝ푛 → ℝ and for all functions 퐹: ℝ푛 → ℝ푚, 퐺: ℝ푛 → ℝ푘, 퐻 : ℝ푝 → ℝ푛 such +that each of the coordinates of the outputs are polynomials of the inputs, we have: +풜 ℓ { 푓 ⩾ 0, 푔 ⩾ 0} +풜 ℓ { 푓 + 푔 ⩾ 0} +, +풜 ℓ { 푓 ⩾ 0}, 풜 ℓ′ {푔 ⩾ 0} +풜 ℓ+ℓ′ { 푓 · 푔 ⩾ 0} +(addition and multiplication) +풜 ℓ ℬ, ℬ ℓ′ 퐶 +풜 ℓ·ℓ′ 퐶 +(transitivity) +{퐹 ⩾ 0} ℓ {퐺 ⩾ 0} +{퐹(퐻) ⩾ 0} ℓ·deg(퐻) {퐺(퐻) ⩾ 0} +. +(substitution) +Low-degree sum-of-squaresproofsare sound and complete ifwe take low-level pseudo- +distributions as models. +Concretely, sum-of-squares proofs allow us to deduce properties of pseudo- +distributions that satisfy some constraints. +Fact 3.17 (Soundness). If 퐷 +푟 풜 for a level-ℓ pseudo-distribution 퐷 and there exists a sum-of- +squares proof 풜 +푟′ ℬ, then 퐷 +푟·푟′+푟′ ℬ. +If the pseudo-distribution 퐷 satisfies 풜 only approximately, soundness continues to +hold if we require an upper bound on the bit-complexity of the sum-of-squares 풜 +푟′ 퐵 +(number of bits required to write down the proof). +In our applications, the bit complexity of all sum of squares proofs will be 푛푂(ℓ) (assum- +ing that all numbers in the input have bit complexity 푛푂(1)). This bound suffices in order +to argue about pseudo-distributions that satisfy polynomial constraints approximately. +The following fact shows that every property of low-level pseudo-distributions can be +derived by low-degree sum-of-squares proofs. +Fact 3.18 (Completeness). Suppose 푑 ⩾ 푟′ ⩾ 푟 and 풜 is a collection of polynomial constraints +with degree at most 푟, and 풜 ⊢ {�푛 +푖=1 푤2 +푖 ⩽ 퐵} for some finite 퐵. +Let {푔 ⩾ 0} be a polynomial constraint. If every degree-푑 pseudo-distribution that satisfies +퐷 +푟 풜 also satisfies 퐷 +푟′ {푔 ⩾ 0}, then for every 휀 > 0, there is a sum-of-squares proof +풜 +푑 {푔 ⩾ −휀}. +20 + +3.2.3 +Explictly bounded distributions +We will consider a subset of subgaussian distributions denoted as certifiably subgaussians. +Many subgaussians distributions are known to be certifiably subgaussian (see [KSS18]). +Definition 3.19 (Explicitly bounded distribution). Let 푡 ∈ ℕ. A distribution 퐷 over ℝ푑 +with mean 휇 is called 2푡-explicitly 휎-bounded if for each even integer 푠 such that 1 ⩽ 푠 ⩽ 푡 +the following equation has a degree 푠 sum-of-squares proof in the vector variable 푢 +푢 +2푠 +� +피 +x∼퐷⟨x − 휇, 푢⟩2푠 ⩽ (휎푠)푠 · ∥푢∥2푠 +2 +� +Furthermore, we say that 퐷 is explicitly bounded if it is 2푡-explicitly 휎-bounded for every +푡 ∈ ℕ. A finite set 푋 ⊆ ℝ푑 is said to be 2푡-explicitly 휎-bounded if the uniform distribution +on 푋 is 2푡-explicitly 휎-bounded. +Sets that are 2푡-explicitly 휎-bounded with large intersection satisfy certain key proper- +ties. Before introducing them we conveniently present the following definition. +Definition 3.20 (Weight vector inducing distribution). Let 푌 be a set of size 푛 and let +푝 ∈ [0, 1]푛 be a vector satisfying +��푝 +�� +1 = 1 . We say that 푝 induces the distribution 퐷 with +support 푌 if +ℙy∼퐷 +� +y = 푦푖 +� += 푝푖 . +Theorem 3.21 ([KSS18, HL18]). Let 푌 ⊆ ℝ푑 be a set of cardinality 푛. Let 푝, 푝′ ∈ [0, 1]푛 be +weight vectors satisfying +��푝 +�� +1 = +��푝′�� +1 = 1 and +��푝 − 푝′�� +1 ⩽ 훽 . Suppose that 푝 (respectively 푝′) +induces a 2푡-explicitly 휎1-bounded (resp. 휎2) distribution over 푌 with mean 휇(푝) (resp. 휇(푝′)). There +exists an absolute constant 훽∗ such that, if 훽 ⩽ 훽∗, then for 휎 = 휎1 + 휎2 : +��휇(푝) − 휇(푝′) +�� ⩽ 훽1−1/2푡 · 푂 +�√ +휎푡 +� +. +In the context of learning Gaussian mixtures, we will make heavy use of the statement +below. +Theorem 3.22 ([KSS18, HL18]). Let 푌 be a 2푡-explicitly 휎-bounded set of size 푛. Let 푝 ∈ ℝ푛 be +the weight vector inducing the uniform distribution over 푌. Let 푝′ ∈ ℝ푛 be a unit vector satisfying +��푝 − 푝′�� +1 ⩽ 훽 for some 훽 ⩽ 훽∗ where 훽∗ is a small constant. Then 푝′ induces a 2푡-explicitly +(휎 + 푂(훽1−1/2푡))-bounded distribution over 푌. +4 +Stability of strongly-convex optimization +In this section, we prove ℓ2 sensitivity bounds for the minimizers of a general class of +(strongly) convex optimization problems. In particular, we show how to translate a uniform +point-wise sensitivity bound for the objective functions into a ℓ2 sensitivity bound for the +minimizers. +21 + +Lemma 4.1 (Stability of strongly-convex optimization). Let 풴 be a set of databases. Let 풦(풴) +be a family closed convex subsets of ℝ푚 parametrized by 푌 ∈ 풴 and let ℱ (풴) be a family of +functions 푓푌 : 풦(푌) → ℝ , parametrized by 푌 ∈ 풴 , such that: +(i) for adjacent databases 푌, 푌′ ∈ 풴 , and 푋 ∈ 풦(푌) there exist 푋′ ∈ 풦(푌′) ∩ 풦(푌) satisfying +�� 푓푌(푋) − 푓푌′(푋′) +�� ⩽ 훼 and +��푓푌′(푋′) − 푓푌(푋′) +�� ⩽ 훼 . +(ii) 푓푌 is 휅-strongly convex in 푋 ∈ 풦(푌). +Then for 푌, 푌′ ∈ 풴, ˆ푋 := arg min푋∈풦(푌) 푓푌(푋) and ˆ푋′ := arg min푋′∈풦(푌′) 푓푌′(푋′) , it holds +��� ˆ푋 − ˆ푋′��� +2 +2 ⩽ 12훼 +휅 +. +Proof. Let 푋′ ∈ 풦(푌)∩풦(푌′) be a point such that +��� 푓푌(푋′) − 푓푌′( ˆ푋′) +��� ⩽ 훼 . By Proposition C.2 +it holds +��� ˆ푋 − ˆ푋′��� +2 +2 ⩽ 2 +��� ˆ푋 − 푋′��� +2 +2 + 2 +���푋′ − ˆ푋′��� +2 +2 +⩽ 4 +휅 +� +푓푌(푋′) − 푓푌( ˆ푋) + 푓푌′(푋′) − 푓푌′( ˆ푋′) +� +. +Suppose now w.l.o.g. 푓푌( ˆ푋) ⩾ 푓푌′( ˆ푋′), a symmetric argument works in the other case. +Then +푓푌′( ˆ푋′) + 훼 ⩾ 푓푌(푋′) ⩾ 푓푌( ˆ푋) ⩾ 푓푌′( ˆ푋′) +and +푓푌′( ˆ푋′) + 2훼 ⩾ 푓푌(푋′) + 훼 ⩾ 푓푌′(푋′) ⩾ 푓푌′( ˆ푋′) . +It follows as desired +푓푌(푋′) − 푓푌( ˆ푋) + 푓푌′(푋′) − 푓푌′( ˆ푋′) ⩽ 3훼 . +□ +5 +Private recovery for stochastic block models +In this section, we present how to achieve exact recovery in stochastic block models pri- +vately and thus prove Theorem 1.3. To this end, we first use the stability of strongly convex +optimization (Lemma 4.1) to obtain a private weak recovery algorithm in Section 5.1. Then +we show how to privately boost the weak recovery algorithm to achieve exact recovery in +Section 5.2. In Section 5.4, we complement our algorithmic results by providing an almost +tight lower bound on the privacy parameters. We start by defining the relevant notion of +adjacent databases. +22 + +Definition 5.1 (Adjacent graphs). Let 퐺 , 퐺′ be graphs with vertex set [푛]. We say that +퐺 , 퐺′ are adjacent if |퐸(퐺)△퐸(퐺′)| = 1 . +Remark 5.2 (Parameters as public information). We remark that we assume the parameters +푛, 훾, 푑 to be public information given in input to the algorithm. +5.1 +Private weak recovery for stochastic block models +In this section, we show how to achieve weak recovery privately via stability of strongly +convex optimization (Lemma 4.1). We first introduce one convenient notation. The error +rate of an estimate ˆ푥 ∈ {±1}푛 of the true partition 푥 ∈ {±1}푛 is defined as err( ˆ푥, 푥) := +1 +푛 · min{Ham( ˆ푥, 푥), Ham( ˆ푥, −푥)}.21 Our main result is the following theorem. +Theorem 5.3. Suppose 훾 +√ +푑 ⩾ 12800 , 휀, 훿 ⩾ 0. There exists an (Algorithm 5.4) such that, for +any 푥 ∈ {±1}푛, on input G ∼ SBM푛(훾, 푑, 푥), outputs ˆ푥(G) ∈ {±1}푛 satisfying +err( ˆ푥(G), 푥) ⩽ 푂 +� +1 +훾 +√ +푑 ++ 1 +훾푑 · log(2/훿) +휀2 +� +with probability 1 − exp(−Ω(푛)). Moreover, the algorithm is (휀, 훿)-differentially private for any +input graph and runs in polynomial time. +Before presenting the algorithm we introduce some notation. Given a graph 퐺, let +푌(퐺) := +1 +훾푑(퐴(퐺) − 푑 +푛 퐽) where 퐴(퐺) is the adjacency matrix of 퐺 and 퐽 denotes all-one ma- +trices. Define 풦 := +� +푋 ∈ ℝ푛×푛 �� 푋 ⪰ 0 , 푋푖푖 = 1 +푛 ∀푖 +� +. The algorithm starts with projecting +matrix 푌(퐺) to set 풦. To ensure privacy, then it adds Gaussian noise to the projection 푋1 +and obtains a private matrix 푋2. The last step applies a standard rounding method. +Algorithm 5.4 (Private weak recovery for SBM). +Input: Graph 퐺. +Operations: +1. Projection: 푋1 ← argmin푋∈풦 ∥푌(퐺) − 푋∥2 +퐹. +2. Noise addition: X2 ← 푋1 + W where W ∼ 풩 +� +0, 24 +푛훾푑 +log(2/훿) +휀2 +�푛×푛 +. +3. Rounding: Compute the leading eigenvector v of X2 and return sign(v). +In the rest of this section, we will show Algorithm 5.4 is private in Lemma 5.6 and its +utility guarantee in Lemma 5.7. Then Theorem 5.3 follows directly from Lemma 5.6 and +Lemma 5.7. +21Note |⟨ ˆ푥, 푥⟩| = (1 − 2 err( ˆ푥, 푥)) · 푛 for any ˆ푥, 푥 ∈ {±1}푛. +23 + +Privacy analysis. +Let 풴 be the set of all matrices 푌(퐺) = +1 +훾푑(퐴(퐺) − 푑 +푛 퐽) where 퐺 is a +graph on 푛 vertices. We further define 푓 : 풴 → ℝ to be the function +푓 (푌) := min +푋∈풦 ∥푌 − 푋∥2 +F. +(5.1) +We first use Lemma 4.1 to prove that function 푓 is stable. +Lemma 5.5 (Stability). The function 푓 as defined in Eq. (5.1) has ℓ2-sensitivity Δ 푓 ,2 ⩽ +� +24 +푛훾푑. +Proof. Let 푔 : 풴 × 풦 → ℝ be the function 푔(푌, 푋) := ∥푋∥2 +F − 2⟨푌, 푋⟩. By Lemma 4.1 it +suffices to prove that 푔 has ℓ1-sensitivity +4 +푛훾푑 with respect to 푌 and that it is 2-strongly con- +vex with respect to 푋. The sensitivity bound follows by observing that adjacent databases +푌, 푌′ satisfy ∥푌 − 푌′∥1 ⩽ +2 +훾푑 and that any 푋 ∈ 풦 satisfies ∥푋∥∞ ⩽ 1 +푛 . Thus it remains to +prove strong convexity with respect to 푋 ∈ 풦. Let 푋, 푋′ ∈ 풦 then +∥푋′∥2 +F = ∥푋∥2 +F + 2⟨푋′ − 푋, 푋⟩ + ∥푋 − 푋′∥2 +F += ∥푋∥2 +F + 2⟨푋′ − 푋, 푋 + 푌 − 푌⟩ + ∥푋 − 푋′∥2 +F += 푔(푌, 푋) + ⟨푋′ − 푋, ∇푔(푋, 푌)⟩ + 2⟨푋′, 푌⟩ + ∥푋 − 푋′∥2 +F . +That is 푔(푌, 푋) is 2-strongly convex with respect to 푋. Note any 푋 ∈ 풦 is symmetric. Then +the result follows by Lemma 4.1. +□ +Then it is easy to show the algorithm is private. +Lemma 5.6 (Privacy). The weak recovery algorithm (Algorithm 5.4) is (휀, 훿)-DP. +Proof. Since any 푋 ∈ 풦 is symmetric, we only need to add a symmetric noise matrix +to obtain privacy. Combining Lemma 5.5 with Lemma 3.12, we immediately get that the +algorithm is (휀, 훿)-private. +□ +Utility analysis. +Now we show the utility guarantee of our priavte weak recovery algo- +rithm. +Lemma 5.7 (Utility). For any 푥 ∈ {±1}푛, on input G ∼ SBM푛(훾, 푑, 푥), Algorithm 5.4 efficiently +outputs ˆ푥(G) ∈ {±1}푛 satisfying +err( ˆ푥(G), 푥) ⩽ 6400 +훾 +√ +푑 ++ 7000 +훾푑 · log(2/훿) +휀2 +, +with probability 1 − exp(−Ω(푛)). +To prove Lemma 5.7, we need the following lemma which is an adaption of a well- +known result in SBM [GV16, Theorem 1.1]. Its proof is deferred to Appendix D. +24 + +Lemma 5.8. Consider the settings of Lemma 5.7. With probability 1 − exp(−Ω(푛)), +����푋1(G) − 1 +푛 푥푥⊤ +���� +2 +퐹 +⩽ 800 +훾 +√ +푑 +. +Proof of Lemma 5.7. By Lemma 5.8, we have +����푋1(G) − 1 +푛 푥푥⊤ +���� ⩽ +����푋1(G) − 1 +푛 푥푥⊤ +���� +퐹 +⩽ +� +800 +훾 +√ +푑 +=: 푟(훾, 푑) +with probability 1 − exp(−Ω(푛)). We condition our following analysis on this event hap- +pening. +Let u be the leading eigenvector of 푋1(G). Let 흀1 and 흀2 be the largest and second largest +eigenvalues of 푋1(G). By Weyl’s inequality (Lemma B.1) and the assumption 훾 +√ +푑 ⩾ 12800, +we have +흀1 − 흀2 ⩾ 1 − 2푟(훾, 푑) ⩾ 1 +2. +Let v be the leading eigenvector of 푋1(G) + W. By Davis-Kahan’s theorem (Lemma B.2), +we have +∥u − v∥ ⩽ 2∥W∥ +흀1 − 흀2 ⩽ 4∥W∥, +��u − 푥/ +√ +푛 +�� ⩽ 2 +����푋1(G) − 1 +푛 푥푥⊤ +���� ⩽ 2푟(훾, 푑). +Putting things together and using Fact A.1, we have +��v − 푥/ +√ +푛 +�� ⩽ ∥u − v∥ + +��u − 푥/ +√ +푛 +�� ⩽ 24 +√ +6 +� +훾푑 +� +log(2/훿) +휀 ++ 2푟(훾, 푑) +with probability 1 − exp(−Ω(푛)). +Observe Ham(sign(푦), 푥) ⩽ ∥푦 − 푥∥2 for any 푦 ∈ ℝ푛 and any 푥 ∈ {±1}푛. Then with +probability 1 − exp(−Ω(푛)), +Ham(sign(v), 푥) ⩽ +��√ +푛 · v − 푥 +��2 ⩽ 6400 +훾 +√ +푑 ++ 7000 +훾푑 · log(2/훿) +휀2 +. +□ +Proof of Theorem 5.3. By Lemma 5.6 and Lemma 5.7. +□ +25 + +5.2 +Private exact recovery for stochastic block models +In this section, we prove Theorem 1.3. We show how to achieve exact recovery in stochastic +block models privately by combining the private weak recovery algorithm we obtained in +the previous section and a private majority voting scheme. +Since exact recovery is only possible with logarithmic average degree (just to avoid iso- +lated vertices), it is more convenient to work with the following standard parameterization +of stochastic block models. Let 훼 > 훽 > 0 be fixed constants. The intra-community edge +probability is 훼 · log 푛 +푛 , and the inter-community edge probability is 훽 · log 푛 +푛 . In the language +of Model 1.1, it is SBM푛( 훼+훽 +2 +· log 푛, 훼−훽 +훼+훽 , 푥). Our main result is the following theorem. +Theorem 5.9 (Private exact recovery of SBM, restatement of Theorem 1.3). Let 휀, 훿 ⩾ 0. +Suppose 훼, 훽 are fixed constants satisfying22 +√ +훼 − +� +훽 ⩾ 4 +and +훼 − 훽 ⩾ Ω +� +1 +휀2 · log(2/훿) +log 푛 ++ 1 +휀 +� +, +(5.2) +Then there exists an algorithm (Algorithm 5.11) such that, for any balanced23 푥 ∈ {±1}푛, on input +G ∼ SBM푛( 훼+훽 +2 · log 푛, 훼−훽 +훼+훽 , 푥), outputs ˆ푥(G) ∈ {푥, −푥} with probability 1 − 표(1). Moreover, the +algorithm is (휀, 훿)-differentially private for any input graph and runs in polynomial time. +Remark 5.10. In a standard regime ofprivacyparameterswhere 휀 ⩽ 푂(1)and 훿 = 1/poly(푛), +the private exact recovery threshold Eq. (5.2) reads +√ +훼 − +� +훽 ⩾ 4 +and +훼 − 훽 ⩾ Ω� +휀−2 + 휀−1� +, +Recall the non-private exact recovery threshold is √훼 − +� +훽 > +√ +2. Thus the non-private +part in Eq. (5.2), i.e. 4, is close to optimal. +Algorithm 5.11 starts with randomly splitting the input graph 퐺 into two subgraphs +G1 and G2. Setting the graph-splitting probability to 1/2, each subgraph will contain about +half of the edges of 퐺. Then we run an (휀, 훿)-DP weak recovery algorithm (Algorithm 5.4) +on G1 to get a rough estimate ˜푥(G1) of accuracy around 90%. Finally, we boost the accuracy +to 100% by doing majority voting (Algorithm 5.12) on G2 based on the rough estimate +˜푥(G1). That is, if a vertex has more neighbors from the opposite community (according to +˜푥(G1)) in G2, then we assign this vertex to the opposite community. To make the majority +voting step private, we add some noise to the vote. +22In the language of Model 1.1, for any 푡 we have √훼 − +� +훽 ⩾ 푡 if and only if +푑 +log 푛 (1 − +� +1 − 훾2) ⩾ 푡2 +2 . +23Recall a vector 푥 ∈ {±1}푛 is said to be balanced if �푛 +푖=1 푥푖 = 0. +26 + +Algorithm 5.11 (Private exact recovery for SBM). +Input: Graph 퐺 +Operations: +1. Graph-splitting: Initialize G1 to be an empty graph on vertex set 푉(퐺). Indepen- +dently put each edge of 퐺 in G1 with probability 1/2. Let G2 = 퐺 \ G1. +2. Rough estimation on G1: Run the (휀, 훿)-DP partial recovery algorithm +(Algorithm 5.4) on G1 to get a rough estimate ˜푥(G1). +3. Majority +voting +on +G2: +Run +the +(휀, 0)-DP +majority +voting +algorithm +(Algorithm 5.12) with input (G2, ˜푥(G1)) and get output ˆx. +4. Return ˆx. +Algorithm 5.12 (Private majority voting). +Input: Graph 퐺, rough estimate ˜푥 ∈ {±1}푛 +Operations: +1. For each vertex 푣 ∈ 푉(퐺), let Z푣 = S푣 − D푣 where +• D푣 = � +{푢,푣}∈퐸(퐺) +1[ ˜푥푢 ≠ ˜푥푣] , +• S푣 = � +{푢,푣}∈퐸(퐺) +1[ ˜푥푢 = ˜푥푣] . +Set ˆx푣 = sign(Z푣 + W푣) · ˜푥(G1)푣 where W푣 ∼ Lap(2/휀). +2. Return ˆx. +In the rest of this section, we will show Algorithm 5.11 is private in Lemma 5.14 and +it recovers the hidden communities exactly with high probability in Lemma 5.16. Then +Theorem 5.9 follows directly from Lemma 5.14 and Lemma 5.16. +Privacy analysis. +We first show the differential privcay of the majority voting algorithm +(Algorithm 5.12) with respect to input graph 퐺 (i.e. assuming fixed the input rough esti- +mate). +Lemma 5.13. Algorithm 5.12 is (휀, 0)-DP with respect to input 퐺. +Proof. Observing the ℓ1-sensitivity of the degree count function 푍 in step is 2, the (휀, 0)-DP +follows directly from Laplace mechanism (Lemma 3.12) and post-processing (Lemma 3.2). +□ +Then the privacy of the private exact recovery algorithm (Algorithm 5.11) is a conse- +quence of composition. +27 + +Lemma 5.14 (Privacy). Algorithm 5.11 is (휀, 훿)-DP. +Proof. Let 풜1 : 풢푛 → {±1}푛 denote the (휀, 훿)-DP recovery algorithm in step 2. Let 풜2 : +풢푛 × {±1}푛 → {±1}푛 denote the (휀, 훿)-DP majority voting algorithm in step 3. Let 풜 be +the composition of 풜1 and 풜2. +We first make several notations. Given a graph 퐻 and an edge 푒, 퐻푒 is a graph obtained +b adding 푒 to 퐻. Given a graph 퐻, G1(퐻) is a random subgraph of 퐻 by keeping each +edge of 퐻 with probability 휆 independently. +Now, fix two adjacent graphs 퐺 and 퐺푒 where edge 푒 appears in 퐺푒 but not in 퐺. Also, +fix two arbitrary possible outputs 푥1, 푥2 ∈ {±1}푛 of algorithm 풜.24 It is direct to see, +ℙ(풜(퐺) = (푥1, 푥2)) = +� +퐻⊆퐺 +ℙ(풜1(퐻) = 푥1) ℙ(풜2(퐺 \ 퐻, 푥1) = 푥2) ℙ(G1(퐺) = 퐻). +(5.3) +Since ℙ(G1(퐺) = 퐻) = ℙ(G1(퐺푒) = 퐻) + ℙ(G1(퐺푒) = 퐻푒) for any 퐻 ⊆ 퐺, we have +ℙ(풜(퐺푒) = (푥1, 푥2)) = +� +퐻⊆퐺 +ℙ(풜1(퐻) = 푥1) ℙ(풜2(퐺푒 \ 퐻, 푥1) = 푥2) ℙ(G1(퐺푒) = 퐻) ++ ℙ(풜1(퐻푒) = 푥1) ℙ(풜2(퐺푒 \ 퐻푒, 푥1) = 푥2) ℙ(G1(퐺푒) = 퐻푒) (5.4) +Since both 풜1 and 풜2 are (휀, 훿)-DP, we have for each 퐻 ⊆ 퐺, +ℙ(풜1(퐻푒) = 푥1) ⩽ 푒휀 ℙ(풜1(퐻) = 푥1) + 훿, +(5.5) +ℙ(풜2(퐺푒 \ 퐻, 푥1) = 푥2) ⩽ 푒휀 ℙ(풜2(퐺 \ 퐻, 푥1) = 푥2) + 훿. +(5.6) +Plugging Eq. (5.5) and Eq. (5.6) into Eq. (5.4), we obtain +ℙ(풜(퐺푒) = (푥1, 푥2)) ⩽ +� +퐻⊆퐺 +[푒휀 ℙ(풜1(퐻) = 푥1) ℙ(풜2(퐺 \ 퐻, 푥1) = 푥2) + 훿] ℙ(G1(퐺) = 퐻) += 푒휀 ℙ(풜(퐺) = (푥1, 푥2)) + 훿. +Similarly, we can show +ℙ(풜(퐺) = (푥1, 푥2)) ⩽ 푒휀 ℙ(풜(퐺푒) = (푥1, 푥2)) + 훿. +(5.7) +□ +Utility analysis. +We first show the utility guarantee of the priavte majority voting algo- +rithm. +Lemma 5.15. Suppose G is generated by first sampling G ∼ SBM푛( 훼+훽 +2 +· log 푛, 훼−훽 +훼+훽 , 푥) for some +balanced 푥 and then for each vertex removing at most Δ ⩽ 푂(log2 푛) adjacent edges arbitrarily. +Then on input G and a balanced rough estimate ˜푥 satisfying Ham( ˜푥, 푥) ⩽ 푛/16, Algorithm 5.12 +efficiently outputs ˆ푥(G) such that for each vertex 푣, +ℙ( ˆ푥(G)푣 ≠ 푥푣) ⩽ exp +� +− 1 +64 · 휀(훼 − 훽) · log 푛 +� ++ 2 · exp +� +− 1 +162 · (훼 − 훽)2 +훼 + 훽 +· log 푛 +� +. +24We can imagine that algorithm 풜 first outputs (푥1, 푥2) and then outputs 푥2 as a post-processing step. +28 + +Proof. Let us fix an arbitrary vertex 푣 and analyze the probability ℙ( ˆ푥(G)푣 ≠ 푥푣). Let +푟 := Ham( ˜푥, 푥)/푛. Then it is not hard to see +ℙ( ˆ푥(G)푣 ≠ 푥푣) ⩽ ℙ(B + A′ − A − B′ + W > 0) +(5.8) +where +• A ∼ Binomial((1/2 − 푟)푛 − Δ, 훼 log 푛 +푛 ), corresponding to the number of neighbors that +are from the same community and correctly labeled by ˜푥, +• B′ ∼ Binomial(푟푛 − Δ, 훽 log 푛 +푛 ), corresponding to the number of neighbors that are +from the different community but incorrectly labeled by ˜푥, +• B ∼ Binomial((1/2 − 푟)푛, 훽 log 푛 +푛 ), corresponding to the number of neighbors that are +from the different community and correctly labeled by ˜푥, +• A′ ∼ Binomial(푟푛, 훼 log 푛 +푛 ), corresponding to the number of neighbors that are from +the same community but incorrectly labeled by ˜푥, +• W ∼ Lap(0, 2/휀), independently. +The Δ term appearing in both A and B′ corresponds to the worst case where Δ “favorable” +edges are removed. If 푟 ⩾ Ω(1), then Δ = 푂(log2 푛) is negligible to 푟푛 = Θ(푛) and we can +safely ignore the effect of removing Δ edges. If 푟 = 표(1), then we can safely assume ˜푥 is +correct on all vertices and ignore the effect of removing Δ edges as well. Thus, we will +assume Δ = 0 in the following analysis. +For any 푡, 푡′, we have +ℙ(A′ + B − A − B′ + W > 0) ⩽ ℙ(A′ + B + W > 푡) + ℙ(A + B′ ⩽ 푡) +⩽ ℙ(A′ + B ⩾ 푡 − 푡′) + ℙ(W ⩾ 푡′) + ℙ(A + B′ ⩽ 푡). +We choose 푡, 푡′ by first picking two constants 푎, 푏 > 0 satisfying 푎 + 푏 < 1 and then solving +• 피[A′ + B] − 푡 = 푎 · (피[A + B′] − 피[A′ + B]) and +• 푡′ = (1 − 푎 − 푏) · (피[A + B′] − 피[A′ + B]). +By Fact 3.7, +ℙ(W > 푡′) ⩽ exp +� +−푡′휀 +2 +� +⩽ exp +� +−(1/4 − 푟)(1 − 푎 − 푏) +2 +· 휀(훼 − 훽) · log 푛 +� +. +By Fact A.4 and the assumption 푟 ⩽ 1/16, we have +ℙ(A + B′ ⩽ 푡) ⩽ exp +� +−(피[A + B′] − 푡)2 +2 피[A + B′] +� +⩽ exp +� +−(1/4 − 푟)2푎2 · (훼 − 훽)2 +훼 + 훽 +· log 푛 +� +. +29 + +Setting 푏 = 1/2, by Fact A.4 and the assumption 푟 ⩽ 1/16, we have +ℙ(A′ + B ⩾ 푡 − 푡′) ⩽ exp +� +−(푡 − 푡′ − 피[A′ + B])2 +푡 − 푡′ + 피[A′ + B] +� +⩽ exp +� +−2(1/4 − 푟)2 +7 +· (훼 − 훽)2 +훼 + 훽 +· log 푛 +� +. +Further setting 푎 = 1/3, we have +ℙ( ˆ푥(G)푣 ≠ 푥푣) ⩽ exp +� +−1/4 − 푟 +12 +· 휀(훼 − 훽) · log 푛 +� ++ 2 · exp +� +−(1/4 − 푟)2 +9 +· (훼 − 훽)2 +훼 + 훽 +· log 푛 +� +. +Finally, plugging the assumption 푟 ⩽ 1/16 to conclude. +□ +Then it is not difficult to show the utility guarantee of our priavte exact recovery +algorithm. +Lemma 5.16 (Utility). Suppose 훼, 훽 are fixed constants satisfying +√ +훼 − +� +훽 ⩾ 4 +and +훼 − 훽 ⩾ Ω +� +1 +휀2 · log(2/훿) +log 푛 ++ 1 +휀 +� +. +Then for any balanced 푥 ∈ {±1}푛, on input G ∼ SBM푛( 훼+훽 +2 +· log 푛, 훼−훽 +훼+훽 , 푥), Algorithm 5.11 +efficiently outputs ˆ푥(G) satisfying ˆ푥(G) ∈ {푥, −푥} with probability 1 − 표(1). +Proof. We will show the probability of a fixed vertex being misclassified is at most 표(1/푛). +Then by union bound, exact recovery can be achieved with probability 1 − 표(1). +As the graph-splitting probability is 1/2, G1 follows SBM푛( 훼 +2 · log 푛 +푛 , 훽 +2 · log 푛 +푛 , 푥). By +Theorem 5.3, the rough estimate ˜푥(G1) satisfies25 +err( ˜푥(G1), 푥) ⩽ 푟 := 표(1) + +14000 +(훼 − 훽)휀2 · log(2/훿) +log 푛 +. +(5.9) +with probability at least 1 − exp(−Ω(푛)). Without loss of generality, we can assume +Ham( ˜푥(G1), 푥) ⩽ 푟푛, since we consider −푥 otherwise. By Fact A.2, the maximum de- +gree of G1 is at most Δ := 2 log2 푛 with probability at least 1 − 푛 exp(−(log 푛)2/3). In the +following, we condition our analysis on the above two events regarding ˜푥(G1) and G1. +Now, let us fix a vertex and analyze the probability 푝푒 that it is misclassified after +majority voting. With 퐺1 being fixed, G2 can be thought of as being generated by first +sampling G and then removing 퐺1 from G. To make 푟 ⩽ 1/16, it suffices to ensure +훼 − 훽 > 5002 +휀2 · log(2/훿) +log 푛 +by Eq. (5.9).Then by Lemma 5.15, we have +푝푒 ⩽ exp +� +− 1 +64 · 휀(훼 − 훽) · log 푛 +� ++ 2 · exp +� +− 1 +162 · (훼 − 훽)2 +훼 + 훽 +· log 푛 +� +. +25It is easy to make the output of Algorithm 5.4 balanced at the cost of increasing the error rate by a factor +of at most 2. +30 + +To make 푝푒 at most 표(1/푛), it suffices to ensure +1 +64 · 휀(훼 − 훽) > 1 +and +1 +162 · (훼 − 훽)2 +훼 + 훽 +> 1. +Note (훼 − 훽)2/(훼 + 훽) > (√훼 − +� +훽)2 for 훼 > 훽. Therefore, as long as +√ +훼 − +� +훽 ⩾ 4 +and +훼 − 훽 ⩾ 5002 +휀2 +· log(2/훿) +log 푛 ++ 64 +휀 , +Algorithm 5.11 recovers the hidden communities exactly with probability 1 − 표(1). +□ +Proof of Theorem 5.9. By Lemma 5.14 and Lemma 5.16. +□ +5.3 +Inefficient recovery using the exponential mechanism +In this section, we will present an inefficient algorithm satisfying pure privacy which +succeeds for all ranges of parameters - ranging from weak to exact recovery. The algorithm +is based on the exponential mechanism [MT07] combined with the majority voting scheme +introduced in section Section 5.2. In particular, we will show +Theorem 5.17 (Full version of Theorem 1.4). Let 훾 +√ +푑 ⩾ 12800 and 푥 ∈ {±1}푛 be balanced. +Let 휁 ⩾ 2 exp +� +− 훾2푑 +512 +� +. For any 휀 ⩾ 64 log(2/휁) +훾푑 +, there exists an algorithm, Algorithm 5.18, which on +input G ∼ SBM푛(훾, 푑, 푥∗) outputs an estimate ˆ푥(G) ∈ {±1}푛 satisfying +err( ˆ푥(G), 푥∗) ⩽ 휁 +with probability at least 1−휁. In addition, the algorithm is 휀-private. Further, by slightly modifying +the algorithm, we can achieve error 20/ +� +log(1/휁) with probability 1 − 푒−푛.26 +A couple of remarks are in order. First, our algorithm works across all degree-regimes +in the literature and matches known non-private thresholds and rates up to constants. We +remark that for ease of exposition we did not try to optimize these constants. In particular, +for 훾2푑 a constant we achieve weak recovery. We reiterate, that 훾2푑 > 1 is the optimal +non-private threshold. For the regime, where 훾2푑 = 휔(1), it is known that the optimal +error rate is exp� +−(1 − 표(1))훾2푑� +even non-privately [ZZ16], where 표(1) goes to zero as +훾2푑 tends to infinity. We match this up to constants. Moreover, our algorithm achieves +exact recovery as soon as 훾2푑 ⩾ 512 log 푛 since then 휁 < +1 +푛. This also matches known +non-private threshholds up to constants [ABH15, MNS15a]. Also, our dependence on the +privacy parameter 휀 is also optimal as shown by the information-theoretic lower bounds +in Section 5.4. +26The first, smaller, error guarantee additionally needs the requirement that 휁 ⩽ exp(−640). The second +one does not. +31 + +We also emphasize, that if we only aim to achieve error on the order of +1 +훾 +√ +푑 += Θ +� +1 +� +log(1/휁) +� +, +we can achieve exponentially small failure probability in 푛, while keeping the privacy pa- +rameter 휀 the same. This can be achieved, by ommitting the boosting step in our algorithm +and will be clear from the proof of Theorem 5.17. We remark that in this case, we can also +handle non-balanced communities. +Again, for an input graph 퐺, consider the matrix 푌(퐺) = +1 +훾푑 +� +퐴(퐺) − 푑 +푛 퐽� +. For 푥 ∈ {±1}푛 +we define the score function +푠퐺(푥) = ⟨푥, 푌(퐺)푥⟩ . +Since the entries of 퐴(퐺) are in [0, 1] and adjacent graphs differ in at most one edge, it +follows immediately, that this score function has sensitivity at most +Δ = max +퐺∼퐺′ , +푥∈{±1}푛 +|푠퐺(푥) − 푠퐺′(푥)| = 2 +훾푑 · max +퐺∼퐺′ , +푥∈{±1}푛 +|⟨푥, (퐴(퐺) − 퐴(퐺′))푥⟩| ⩽ 2 +훾푑 . +Algorithm 5.18 (Inefficient algorithm for SBM). +Input: Graph 퐺, privacy parameter 휀 > 0 +Operations: +1. Graph-splitting: Initialize G1 to be an empty graph on vertex set 푉(퐺). Indepen- +dently assign each edge of 퐺 to G1 with probability 1/2. Let G2 = 퐺 \ G1. +2. Rough estimation on G1: Sample ˜푥 from the distribution with density +푝(푥) ∝ exp +� 휀 +2Δ⟨푥, 푌(G1)푥⟩ +� +, +where Δ = +2 +훾푑. +3. Majority voting on G2: Run the 휀-DP majority voting algorithm (Algorithm 5.12) +with input (G2, ˜푥(G1)). Denote its output by ˆx. +4. Return ˆx. +We first analyze the privacy guarantees of the above algorithm. +Lemma 5.19. Algorithm 5.18 is 휀-DP. +Proof. For simplicity and clarity of notation, we will show that the algorithm satisfies 2휀-DP. +Clearly, the graph splitting step is 0-DP. Step 2 corresponds to the exponential mechanism. +32 + +Since the sensitivity of the score function is at most Δ = +2 +훾푑 it follows by the standard +analysis of the mechanism that this step is 휀-DP [MT07]. By Lemma 5.13, the majority +voting step is also 휀-DP. Hence, the result follows by composition (cf. Lemma 3.4). +□ +Next, we will analyze its utility. +Lemma 5.20. Let 훾 +√ +푑 ⩾ 12800 and 푥 ∈ {±1}푛 be balanced. Let exp(−640) ⩾ 휁 ⩾ +2 exp +� +− 훾2푑 +512 +� +, 휀 ⩾ +64 log(2/휁) +훾푑 +, and G +∼ +SBM푛(훾, 푑, 푥∗), the output +ˆ푥(G) +∈ +{±1}푛 of +Algorithm 5.18 satisfies +err( ˆ푥(G), 푥∗) ⩽ 휁 +with probability at least 1 − 휁. +Proof. We will first show that the rough estimate ˜푥 obtained in step 2 achieves +err( ˜푥, 푥∗) ⩽ +20 +� +log(1/휁) +with probability 푒−푛. This will prove the second part of the theorem - for this we don’t +need that 휁 ⩽ exp(−640). In fact, arbitrary 휁 works. The final error guarantee will then +follow by Lemma 5.15. First, notice that similar to the proof of [GV16, Lemma 4.1], using +Bernstein’s inequality and a union bound, we can show that (cf. Fact D.2 for a full proof) +max +푥∈{±1}푛 +����⟨푥, +� +푌(G) − 1 +푛 푥∗(푥∗)⊤ +� +푥⟩ +���� ⩽ 100푛 +훾 +√ +푑 +⩽ +5 +� +log(1/휁) +with probability at least 1 − exp−10푛. Recall that 푠G(푥) = ⟨푥, 푌(G)푥⟩. Let 훼 = +5 +√ +log(1/휁). We +call 푥 ∈ {±1}푛 good if 푠G(푥) ⩾ (1 − 3훼)푛. It follows that for good 푥 it holds that +1 +푛 · ⟨푥, 푥∗⟩2 ⩾ ⟨푥, 푌(G)푥⟩ − +���� +� +푥, +� +푌(G) − 1 +푛 푥∗(푥∗)⊤ +� +푥 +����� ⩾ (1 − 4훼)푛 . +Which implies that +2 err(푥, 푥∗) ⩽ 1 − +√ +1 − 4훼 = 1 − 1 − 4훼 +√ +1 − 4훼 +⩽ 1 − 1 − 4훼 +1 − 2훼 = +2훼 +1 − 2훼 ⩽ 4훼 , +where we used that 훼 ⩽ 1/4 and that +√ +1 − 4푥 ⩽ 1 − 2푥 for 푥 ⩾ 0. Hence, we have for good +푥 that +err(푥, 푥∗) ⩽ +20 +� +log(1/휁) +. +Since 푠G(푥∗) ⩾ (1 − 훼)푛,there is at least one good candidate. Hence, we can bound the +probability that we do not output a good 푥 as +exp� 휀 +2Δ(1 − 3훼)푛� +· 푒푛 +exp� 휀 +2Δ(1 − 훼)푛� +· 1 += exp +�� +1 − 2휀훼 +Δ +� +푛 +� +⩽ 푒−푛 , +33 + +where we used that +2휀훼 +Δ +⩾ 64 log(2/휁) +훾푑 +· +5훾푑 +� +log(1/휁) +⩾ 320 +� +log(1/휁) ⩾ 2 . +We will use Lemma 5.15 to proof the final conclusion of the theorem. In what follows, +assume without loss of generality that Ham(푥, 푥∗) < Ham(푥, −푥∗). The above discussion +implies that +Ham(푥, 푥∗) ⩽ 8훼푛 ⩽ +40푛 +� +log(1/휁) +⩽ 푛 +16 , +where the last inequality uses 휁 ⩽ 푒−640. Further, by Fact A.2 it also follows that the maxi- +mum degree of G2 is at most 푂 +� +log2 푛 +� +(by some margin). Recall that G2 ∼ SBM(푑, 훾, 푥∗). +In the parametrization of Lemma 5.15 this means that +훼 = (1 + 훾)푑 +log 푛 +, +훽 = (1 − 훾)푑 +log 푛 +, +훼 − 훽 = 2훾푑 +log 푛 , +훼 + 훽 = +2푑 +log 푛 . +Thus, it follows that the output ˆ푥 of the majority voting step satisfies for every vertex 푣 +ℙ( ˆ푥(G)푣 ≠ 푥푣) ⩽ exp +� +− 1 +64 · 휀(훼 − 훽) · log 푛 +� ++ 2 · exp +� +− 1 +162 · (훼 − 훽)2 +훼 + 훽 +· log 푛 +� +⩽ exp +� +− 1 +32 · 휀훾푑 +� ++ exp +� +− 1 +162 · 훾2푑 +� +⩽ 휁2/4 + 휁2/4 ⩽ 휁2 . +By Markov’s Inequality it now follows that +ℙ(err( ˆ푥(G), 푥∗) ⩾ 휁) ⩽ 휁 . +□ +5.4 +Lower bound on the parameters for private recovery +In this section, we prove a tight lower bound for private recovery for stochastic block +models. Recall the definition of error rate, err(푢, 푣) := 1 +푛 · min{Ham(푢, 푣), Ham(푢, −푣)} for +푢, 푣 ∈ {±1}푛. Our main result is the following theorem. +Theorem 5.21 (Full version ofTheorem 1.5). Suppose there exists an 휀-differentially private algo- +rithm such that for any balanced 푥 ∈ {±1}푛, on input G ∼ SBM푛(푑, 훾, 푥), outputs ˆ푥(G) ∈ {±1}푛 +satisfying +ℙ(err( ˆ푥(G), 푥) < 휁) ⩾ 1 − 휂, +34 + +where27 1/푛 ⩽ 휁 ⩽ 0.04 and the randomness is over both the algorithm and stochastic block models. +Then, +푒2휀 − 1 ⩾ Ω +�log(1/휁) +훾푑 ++ log(1/휂) +휁푛훾푑 +� +. +(5.10) +Remark 5.22. Both terms in lower bound Eq. (5.10) are tight up to constants by the fol- +lowing argument. Considering typical privacy parameters 휀 ⩽ 1, then 푒2휀 − 1 ≈ 2휀. For +exponentially small failure probability, i.e. 휂 = 2−Ω(푛), the lower bound reads 휀 ⩾ Ω( 1 +훾푑 · 1 +휁), +which is achieved by Algorithm 5.18 without the boosting step - see the discussion af- +ter Theorem 5.17. For polynomially small failure probability, i.e.휂 = 1/poly(푛), the lower +bound Eq. (5.10) reads 휀 ⩾ Ω( 1 +훾푑 · log 1 +휁), which is achieved by Theorem 5.17. +By setting 휁 = 1/푛 in Theorem 5.21, we directly obtain a tight lower bound for private +exact recovery as a corollary. +Corollary 5.23. Suppose there exists an 휀-differentially private algorithm such that for any bal- +anced 푥 ∈ {±1}푛, on input G ∼ SBM푛(푑, 훾, 푥), outputs ˆ푥(G) ∈ {±1}푛 satisfying +ℙ( ˆ푥(G) ∈ {푥, −푥}) ⩾ 1 − 휂, +where the randomness is over both the algorithm and stochastic block models. Then, +푒2휀 − 1 ⩾ Ω +�log(푛) + log 1 +휂 +훾푑 +� +. +(5.11) +Remark 5.24. The lower bound Eq. (5.11) for priavte exact recovery is tight up to constants, +since there exists an (inefficient) 휀-differentially priavte exact recovery algorithm with +휀 ⩽ 푂(log 푛 +훾푑 ) and 휂 = 1/poly(푛) by Theorem 5.17 and [SNVT22, Theorem 3.7]. +In rest of this section, we will prove Theorem 5.21. The proof applies the packing lower +bound argument similar to [HKM22, Theorem 7.1]. To this end, we first show err(·, ·) is a +semimetric over {±1}푛. +Lemma 5.25. err(·, ·) is a semimetric over {±1}푛. +Proof. Symmetry and non-negativity are obvious from the definition. We will show err(·, ·) +satisfies triangle inequality via case analysis. Let 푢, 푣, 푤 ∈ {±1}푛 be three arbitrary sign +vectors. By symmetry, we only need to consider the following four cases. +Case 1: Ham(푢, 푣), Ham(푢, 푤), Ham(푣, 푤) ⩽ 푛/2. This case is reduced to showing Ham- +ming distance satisfies triangle inequality, which is obvious. +Case 2: Ham(푢, 푣), Ham(푢, 푤) ⩽ 푛/2 and Ham(푣, 푤) ⩾ 푛/2. We need to check two +subcases. First, +err(푢, 푣) ⩽ err(푢, 푤) + err(푣, 푤) ⇔ Ham(푢, 푣) + Ham(푣, 푤) ⩽ Ham(푢, 푤) + 푛 +27Error rate less than 1/푛 already means exact recovery. Thus it does not make sense to set 휁 to any +value strictly smaller than 1/푛. The upper bound 휁 ⩽ 0.04 is just a technical condition our proof needs for +Eq. (5.12). +35 + +⇐ Ham(푢, 푣) + 퐻(푢, 푣) + 퐻(푢, 푤) ⩽ Ham(푢, 푤) + 푛 +⇔ Ham(푢, 푣) ⩽ 푛/2. +Second, +err(푣, 푤) ⩽ err(푢, 푣) + err(푢, 푤) ⇔ 푛 ⩽ Ham(푣, 푤) + Ham(푢, 푣) + Ham(푢, 푤) +⇐ 푛 ⩽ 2 Ham(푣, 푤). +Case 3: Ham(푢, 푣) ⩽ 푛/2 and Ham(푢, 푤), Ham(푣, 푤) ⩾ 푛/2. This case can be reduced +to case 1 by considering 푢, 푣, −푤. +Case 4: Ham(푢, 푣), Ham(푢, 푤), Ham(푣, 푤) ⩾ 푛/2. This case can be reduced to case 2 by +considering −푢, 푣, 푤. +□ +Proof of Theorem 5.21. Suppose there exists an 휀-differentially private algorithm satisfying +the theorem’s assumption. +We first make the following notation. Given a semimetric 휌 over {±1}푛, a center 푣 ∈ +{±1}푛, and a radius 푟 ⩾ 0, define 퐵휌(푣, 푟) := {푤 ∈ {±1}푛 : 1⊤푤 = 0, 휌(푤, 푣) ⩽ 푟}. +Pick an arbitrary balanced 푥 ∈ {±1}푛. Let 푀 = {푥1, 푥2, . . . , 푥푚} be a maximal 2휁- +packing of 퐵err(푥, 4휁) in semimetric err(·, ·). By maximality of 푀, we have 퐵err(푥, 4휁) ⊆ +∪푚 +푖=1퐵err(푥푖, 2휁), which implies +|퐵err(푥, 4휁)| ⩽ +푚 +� +푖=1 +��퐵err(푥푖, 2휁) +�� +=⇒ |퐵Ham(푥, 4휁)| ⩽ +푚 +� +푖=1 +2 · +��퐵Ham(푥푖, 2휁) +�� = 2푚 · |퐵Ham(푥, 2휁)| +=⇒ 2푚 ⩾ |퐵Ham(푥, 4휁푛)| +|퐵Ham(푥, 2휁푛)| = +�푛/2 +2휁푛 +�2 +�푛/2 +휁푛 +�2 ⩾ +� +1 +4휁 +�4휁푛 +� +푒 +2휁 +�2휁푛 = +� +1 +8푒휁 +�2휁푛 +(5.12) +For each 푖 ∈ [푚], define 푌푖 := {푤 ∈ {±1}푛 : err(푤, 푥푖) ⩽ 휁}. Then 푌푖’s are pairwise +disjoint. For each 푖 ∈ [푚], let 푃푖 be the distribution over 푛-vertex graphs generated by +SBM푛(푑, 훾, 푥푖). By our assumption on the algorithm, we have for any 푖 ∈ [푚] that +ℙ +G∼푃푖 +( ˆ푥(G) ∈ 푌푖) ⩾ 1 − 휂. +Combining the fact that 푌푖’s are pairwise disjoint, we have +푚 +� +푖=1 +ℙ +G∼푃1 +( ˆ푥(G) ∈ 푌푖) = +ℙ +G∼푃1 +� ˆ푥(G) ∈ ∪푚 +푖=1푌푖 +� ⩽ 1 =⇒ +푚 +� +푖=2 +ℙ +G∼푃1 +( ˆ푥(G) ∈ 푌푖) ⩽ 휂. +(5.13) +In the following, we will lower bound ℙG∼푃1( ˆ푥(G) ∈ 푌푖) for each 푖 ∈ [푚] \ {1} using group +privacy. +36 + +Note each 푃푖 is a product of �푛 +2 +� +independent Bernoulli distributions. Thus for any +푖, 푗 ∈ [푚], there exists a coupling 휔푖푗 of 푃푖 and 푃푗 such that, if (G, H) ∼ 휔, then +Ham(G, H) ∼ Binomial(푁푖푗, 푝), +where 푝 = 2훾푑/푛 and 푁푖푗 = Ham(푥푖, 푥푗) · (푛 − Ham(푥푖, 푥푗)). Applying group privacy, we +have for any two graphs 퐺, 퐻 and for any 푆 ⊆ {±1}푛 that28 +ℙ( ˆ푥(퐺) ∈ 푆) ⩽ exp(휀 · Ham(퐺, 퐻)) · ℙ( ˆ푥(퐻) ∈ 푆). +(5.14) +For each 푖 ∈ [푚], taking expectations on both sides of Eq. (5.14) with respect to coupling +휔푖1 and setting 푆 = 푌푖, we have +피 +(G,H)∼휔푖1 +ℙ( ˆ푥(G) ∈ 푌푖) ⩽ +피 +(G,H)∼휔푖1 +exp(휀 · Ham(G, H)) · ℙ( ˆ푥(H) ∈ 푌푖). +(5.15) +The left side of Eq. (5.15) is equal to +피 +(G,H)∼휔푖1 +ℙ( ˆ푥(G) ∈ 푌푖) = +ℙ +G∼푃푖( ˆ푥(G) ∈ 푌푖) ⩾ 1 − 휂. +Upper bounding the right side of Eq. (5.15) by Cauchy-Schwartz inequality, we have +피 +(G,H)∼휔푖1 +exp(휀 · Ham(G, H)) · ℙ( ˆ푥(H) ∈ 푌푖) +⩽ +� +피 +(G,H)∼휔푖1 +exp(2휀 · Ham(G, H)) +�1/2 +· +� +피 +(G,H)∼휔푖1 +ℙ( ˆ푥(H) ∈ 푌푖)2 +�1/2 += +� +피 +X∼Binomial(푁푖1,푝) exp(2휀 · X) +�1/2 +· +� +피 +H∼푃1 ℙ( ˆ푥(H) ∈ 푌푖)2 +�1/2 +. +Using the formula for the moment generating function of binomial distributions, we have +피 +X∼Binomial(푁푖1,푝) exp(2휀 · X) = (1 − 푝 + 푝 · 푒2휀)푁푖1, +and it is easy to see +피 +H∼푃1 +ℙ( ˆ푥(H) ∈ 푌푖)2 = +피 +H∼푃1 +(피 +1[ ˆ푥(H) ∈ 푌푖])2 ⩽ +ℙ +H∼푃1 +( ˆ푥(H) ∈ 푌푖). +Putting things together, Eq. (5.15) implies for each 푖 ∈ [푚] that +ℙ +H∼푃1 +( ˆ푥(H) ∈ 푌푖) ⩾ +(1 − 휂)2 +(1 − 푝 + 푝 · 푒2휀)푁푖1 . +(5.16) +Since 푥푖 ∈ 퐵err(푥, 4휁) for 푖 ∈ [푚], by assuming 휁 ⩽ 1/16, we have +푁푖1 = Ham(푥푖, 푥1) · (푛 − Ham(푥1, 푥푖)) ⩽ 8휁푛(푛 − 8휁푛). +(5.17) +28In Eq. (5.14), the randomness only comes from the algorithm. +37 + +Recalling 푝 = 2훾푑/푛 and combining Eq. (5.12), Eq. (5.13), Eq. (5.16) and Eq. (5.17), we have +(푚 − 1) · +(1 − 휂)2 +(1 − 푝 + 푝 · 푒2휀)8휁푛(푛−8휁푛) ⩽ 휂. +By taking logarithm on both sides, using 푡 ⩾ log(1 + 푡) for any 푡 > −1, and assuming +휁 ⩽ 1/(8푒), we have +푒2휀 − 1 ≳ +log +1 +8푒휁 +훾푑 ++ +log 1 +휂 +휁푛훾푑 . +□ +6 +Private algorithms for learning mixtures of spherical +Gaussians +In this section we present a private algorithm for recovering the centers of a mixtures of 푘 +Gaussians (cf. Model 1.2). Let 풴 ⊆ � +ℝ푑�⊗푛 be the collection of sets of 푛 points in ℝ푑. We +consider the following notion of adjacency. +Definition 6.1 (Adjacent databases). We say that 푌, 푌′ ∈ 풴 are adjacent if |푌 ∩ 푌′| ⩾ 푛 −1 . +Remark 6.2 (Problem parametersaspublicinformation). We considerthe parameters 푛, 푘, Δ +to be public information given as input to the algorithm. +Next we present the main theorem of the section. +Theorem 6.3 (Privately learning spherical mixtures of Gaussians). Consider an instance of +Model 1.2. Let 푡 ∈ ℕ be such that Δ ⩾ 푂 +�√ +푡푘1/푡� +. For 푛 ⩾ Ω� +푘푂(1) · 푑푂(푡)� +, 푘 ⩾ (log 푛)1/5 , +there exists an algorithm, running in time (푛푑)푂(푡), that outputs vectors ˆ흁1, . . . , ˆ흁ℓ satisfying +max +ℓ∈[푘] +�� ˆ흁ℓ − 휇휋(ℓ) +�� +2 ⩽ 푂(푘−12) , +with high probability, for some permutation 휋 : [푘] → [푘] .29 Moreover, for 휀 ⩾ 푘−10 , 훿 ⩾ 푛−10 , +the algorithm is (휀, 훿)-differentially private for any input 푌. +We remark that our algorithm not only works for mixtures of Gaussians but for all +mixtures of 2푡-explicitly bounded distributions (cf. Definition 3.19). +Our algorithm is based on the sum-of-squares hierarchy and at the heart lies the +following sum-of-squares program. The indeterminates 푧11, . . . , 푧1푘, . . . , 푧푛푘 and vector- +valued indeterminates 휇′ +1, . . . , 휇′ +푘, will be central to the proof of Theorem 6.3. Let 푛, 푘, 푡 be +fixed parameters. +29We remark that we chose constants to optimize readibility and not the smallest possible ones. +38 + + + +푧2 +푖ℓ = 푧푖ℓ +∀푖 ∈ [푛] , ℓ ∈ [푘] +(indicators) +� +ℓ∈[푘] +푧푖ℓ ⩽ 1 +∀푖 ∈ [푛] +(cluster mem.) +푧푖ℓ · 푧푖ℓ′ = 0 +∀푖 ∈ [푛] , ℓ ∈ [푘] +(uniq. mem.) +� +푖 +푧푖ℓ ⩽ 푛/푘 +∀ℓ ∈ [푘] +(size of clusters) +휇′ +ℓ = 푘 +푛 +� +푖 +푧푖ℓ · 푦푖 +∀ℓ ∈ [푘] +(means of clusters) +∀푣 ∈ ℝ푑 : 푘 +푛 +푛 +� +푖=1 +푧푖ℓ ⟨푦푖 − 휇′ +ℓ , 푣⟩2푠 + +��푄푣⊗푠��2 = (2푠)푠 · ∥푣∥2푠 +2 +∀푠 ⩽ 푡, ℓ ∈ [푘] +(푡 moment) + + +(풫푛,푘,푡(푌)) +We remark that the moment constraint encodes the 2푡-explicit 2-boundedness con- +straint introduced in Definition 3.19. Note that in the form stated above there are infinitely +many constraints, one for each vector 푣. This is just for notational convenience. This con- +straint postulates equality of two polynomials in 푣. Formally, this can also be encoded by +requiring there coefficients to agree and hence eliminating the variable 푣. It is not hard to +see that this can be done adding only polynomially many constraints. Further, the matrix +variable 푄 represents the SOS proof of the 2푡-explicit 2-boundedness constraint and we +can hence deduce that for all 0 ⩽ 푠 ⩽ 푡 +풫 +2푠 +푣 +� +푘 +푛 +푛 +� +푖=1 +푧푖ℓ ⟨푦푖 − 휇′ +ℓ , 푣⟩2푠 ⩽ (2푠)푠∥푠∥2푠 +2 +� +. +Before presenting the algorithm we will introduce some additional notation which +will be convenient. We assume 푡, 푛, 푘 to be fixed throughout the section and drop the cor- +responding subscripts. For 푌 ∈ 풴, let 풵(푌) be the set of degree-10푡 pseudo-distributions +satisfying 풫(푌). For each 휁 ∈ 풵(푌) define 푊(휁) as the 푛-by-푛 matrix satisfying +푊(휁)푖푗 = ˜피휁 + +� +ℓ∈[푘] +푧푖ℓ · 푧푗ℓ + +. +We let 풲(푌) := {푊(휁) | 휁 ∈ 풵(푌)} . +Recall that 퐽 denotes the all-ones matrix. We define the function 푔 : ℝ푛×푛 → ℝ as +푔(푊) = ∥푊 ∥2 +F − (10)10푘300⟨퐽, 푊⟩ +and let +푊(ˆ휁(푌)) ≔ argmin푊∈풲(푌) 푔(푊) . +We also consider the following function +39 + +Definition 6.4 (Soft thresholding function). We denote by 휙 : [0, 1] → [0, 1] the function +휙(푥) = + + +0 +if 푥 ⩽ 0.8 , +1 +if 푥 ⩾ 0.9 , +푥−0.8 +0.9−0.8 +otherwise . +Notice that 휙(·) is +1 +0.9−0.8 = 10 Lipschitz. Next we introduce our algorithm. Notice the +algorithm relies on certain private subroutines. We describe them later in the section to +improve the presentation. +40 + +Algorithm 6.5 (Private algorithm for learning mixtures of Gaussians). +Input: Set of 푛 points 푌 ⊆ ℝ푑 , 휀 , 훿 > 0 , 푘, 푡 ∈ ℕ , 푑∗ = 100 log 푛 , 푏 = 푘−15 . +1. Compute 푊 = 푊(ˆ휁(푌)). +2. Pick 흉 ∼ tLap +� +−푛1.6� +1 + log(1/훿) +휀 +� +, 푛1.6 +휀 +� +. +3. If |흉| ⩾ 푛1.7 or +��휙(푊) +�� +1 ⩽ 푛2 +푘 · +� +1 − +1 +푛0.1 − +1 +푘100 +� ++ 흉 reject. +4. For all 푖 ∈ [푛] , compute the 푛-dimensional vector +휈(푖) = +� +0 +if +��휙(푊푖) +�� +1 = 0 +��휙(푊푖) +��−1 +1 +� +푗 휙(푊푖푗) · 푦푗 +otherwise. +5. Pick a set 퓢 of 푛0.01 indices 푖 ∈ [푛] uniformly at random. +6. For each 푖 ∈ 퓢 let ¯흂(푖) = 휈(푖) + w where w ∼ 푁 +� +0, 푛−0.18 · log(2/훿) +휀2 +· Id +� +. +7. Pick 횽 ∼ 푁 � +0, 1 +푑∗ +�푑∗×푑 , q 푢.푎.푟. +∼ +[0, 푏] and run the histogram learner of Lemma 3.13 +with input 횽 ¯흂(1), . . . , 횽 ¯흂(푛0.01) and parameters +q, 푏, 훼 = 푘−10, 훽 = 푛−10, 훿∗ = 훿 +푛 , 휀∗ = 휀 · 10푘50 +푛0.01 . +Let B1, . . . , B푘 be the resulting 푑∗-dimensional bins with highest counts. Break +ties randomly. +8. Reject if min푖∈[푘] +��� +푗 +�� 횽 ¯흂(푗) ∈ B푖 +��� < 푛0.01 +2푘 . +9. For each 푙 ∈ [푘] output +ˆ흁푙 ≔ +1 +��� +푗 +�� 횽 ¯흂(푗) ∈ B푖 +��� · �� +� +� +횽 ¯흂(푗)∈B푙 +¯흂(푗)�� +� ++ w′ , +where w′ ∼ 푁 +� +0, 푁 +� +0, 32 · 푘−120 · log(2푘푛/훿) +휀2 +· Id +�� +. +For convenience, we introduce some preliminary facts. +Definition 6.6 (Good 푌). Let Y be sampled according to Model 1.2. We say that Y is good +if: +1. for each ℓ ∈ [푘], there are at least 푛 +푘 − 푛0.6 and most 푛 +푘 + 푛0.6 points sampled from 퐷ℓ +in Y. Let Yℓ ⊆ Y be such set of points. +41 + +2. Each Yℓ is 2푡-explicitly 2-bounded. +It turns out that typical instances Y are indeed good. +Lemma 6.7. [HL18, KSS18] Consider the settings of Theorem 6.3. Then Y is good with high +probability. Further, in this case the sets 풵(푌) and 풲(푌) are non-empty. +6.1 +Privacy analysis +In this section we show that our clustering algorithm is private. +Lemma 6.8 (Differential privacy of the algorithm). Consider the settings of Theorem 6.3. Then +Algorithm 6.5 is (휀, 훿)-differentially private. +We split our analysis in multiple steps and combine them at the end. On a high level, +we will argue that on adjacent inputs 푌, 푌′ many of the vectors 휈(푖) by the algorithm are +close to each other and a small part can be very far. We can then show that we can mask +this small difference using the Gaussian mechanism and afterwards treat this subset of the +vectors as privatized (cf. Lemma E.4). Then we can combine this with known histogram +learners to deal with the small set of 휈(푖)’s that is far from each other on adjacent inputs. +6.1.1 +Sensitivity of the matrix W +Here we use Lemma 4.1 to reason about the sensitivity of 휙(푊(ˆ휁(푌))). For adjacent datasets +푌, 푌′ ∈ 풴 we let ˆ휁 , ˆ휁′ be the pseudo-distribution corresponding to 푊(ˆ휁(푌)) and 푊(ˆ휁(푌′)) +computed in step 1 of the algorithm, respectively. We prove the following result. +Lemma 6.9 (ℓ1-sensitivity of 휙(푊)). Consider the settings of Theorem 6.3. Let 푊, 푊′ be re- +spectively be the matrices computed in step 1 by Algorithm 6.5 on adjacent inputs 푌, 푌′ ∈ 풴. +Then +��휙(푊) − 휙(푊′) +�� +1 ⩽ 푛1.6 . +For all but 푛0.8 rows 푖 of 휙(푊), 휙(푊′), it holds +��휙(푊)푖 − 휙(푊′)푖 +�� +1 ⩽ 푛0.8 . +Proof. The second inequality is an immediate consequence of the first via Markov’s in- +equality. Thus it suffices to prove the first. Since 휙(·) is 10-Lipschitz, we immediately +obtain the result if +���푊(ˆ휁(푌)) − 푊(ˆ휁(푌′)) +��� +1 ⩽ 푛1.55 . +42 + +Thus we focus on this inequality. To prove it, we verify the two conditions of Lemma 4.1. +First notice that 푔 is 2-strongly convex with respect to its input 푊. Indeed for 푊, 푊′ ∈ +풲(푌), since ∀푖, 푗 ∈ [푛] , 푊푖푗 ⩾ 0 it holds that +∥푊′∥2 +F = ∥푊 ∥2 +F + ∥푊 − 푊′∥2 +F + 2⟨푊′ − 푊, 푊⟩ += ∥푊 ∥2 +F + ∥푊 − 푊′∥2 +F + 2⟨푊′ − 푊, 푊⟩ + ⟨푊′ − 푊, (10)10푘300(퐽 − 퐽)⟩ += 푔(푊) + ∥푊 − 푊′∥2 +F + ⟨푊′ − 푊, ∇푔(푊)⟩ + ⟨푊′, (10)10푘300퐽⟩ , +where we used that ∇푔(푊) = 2푊 − (10)10푘300퐽. Thus it remain to prove (i) of Lemma 4.1. +Let ˆ휁 ∈ 풵(푌) , ˆ휁′ ∈ 풵(푌′) be the pseudo-distributions such that 푊푌(ˆ휁) = 푊 and +푊푌(ˆ휁′) = 푊′. We claim that there always exists 휁adj ∈ 풵(푌) ∩ 풵(푌′) such that +1. |푔(푊(휁)) − 푔(푊(휁adj)| ⩽ 2푛 +푘 · � +(10)10푘300 + 1� ⩽ 3 · (10)10푘300푛 , +2. |푔푌′(푊(휁adj)) − 푔(푊(휁adj)| = 0 . +Note that in this case the second point is always true since 푔 doesn’t depend on 푌. Together +with Lemma 4.1 these two inequalities will imply that +���푊(ˆ휁(푌)) − 푊(ˆ휁(푌′)) +��� +2 +F ⩽ 18 · (10)10푘300푛 . +By assumption on 푛, an application of Cauchy-Schwarz will give us the desired result. +So, let 푖 be the index at which 푌, 푌′ differ. We construct 휁adj as follows: for all polyno- +mials 푝 of degree at most 10푡 we let +˜피휁adj +� +푝 +� += +� +˜피휁 +� +푝 +� +if 푝 does not contain variables 푧푖ℓ for any ℓ ∈ [푘] +0 +otherwise. +By construction 휁adj ∈ 풵(푌)∩풵(푌′). Moreover, 푊(휁), 푊(휁adj) differ in at most 2푛/푘 entries. +Since all entries of the two matrices are in [0, 1], the first inequality follows by definition +of the objective function. +□ +6.1.2 +Sensitivity of the resulting vectors +In this section we argue that if the algorithm does not reject in step 3 then the vectors 휈(푖) +are stable on adjacent inputs. Concretely our statement goes as follows: +Lemma 6.10 (Stability of the 휈(푖)’s). Consider the settings of Theorem 6.3. Suppose Algorithm 6.5 +does not reject in step 3, on adjacent inputs 푌 , 푌′ ∈ 풴. Then for all but 6푛 +푘50 indices 푖 ∈ [푛], it +holds: +���휈(푖) +푌 − 휈(푖) +푌′ +��� +2 ⩽ 푂� +푛−0.1� +. +The proof of Lemma 6.10 crucially relies on the next statement. +43 + +Lemma 6.11 (Covariance bound). Consider the settings of Theorem 6.3. Let 푊 be the matrix +computed by Algorithm 6.5 on input 푌 ∈ 풴. For 푖 ∈ [푛], if +��휙(푊푖) +�� +1 ⩾ 푛 +푘 · +� +1 − 10 +푘50 +� +then 휈(푖) +induces a 2-explicitly 40-bounded distribution over 푌. +Proof. First, by assumption notice that there must be at least 푛 +푘 · +� +1 − 10 +푘50 +� +entries of 휙(푊푖) +larger than 0.8. We denote the set of 푗 ∈ [푛] such that 푊푖푗 ⩾ 0.8 by 풢 . Let 휁 ∈ 풵(푌) be the +degree 10푡 pseudo-distribution so that 푊 = 푊(휁(푌)). Since 휁 satisfies 풫(푌), for ℓ ∈ [푘] it +follows from the moment bound constraint for 푠 = 1 that for all unit vectors 푢 it holds that +풫 +4 + + +0 ⩽ 푘 +푛 +푛 +� +푗=1 +푧푗ℓ ⟨y푗 − 휇′ +푙, 푢⟩2 ⩽ 2 + + +, +Using the SOS triangle inequality (cf. Fact E.2) +2 +푎,푏 (푎 + 푏)2 ⩽ 2(푎2 + 푏2) it now follows that +0 ⪯ ˜피휁 + +푘2 +푛2 +� +푗 ,푗′∈[푛] +푧푗ℓ 푧푗′ℓ · � +푦푗 − 푦푗′�⊗2 + +⪯ 8Id +and thus +0 ⪯ ˜피휁 + +푘2 +푛2 +� +ℓ∈[푘] +� +푗 ,푗′∈[푛] +푧푖ℓ 푧푗ℓ 푧푗′ℓ · � +푦푗 − 푦푗′�⊗2 + +⪯ 8Id . +Furthermore using 풫(푌) 2 {푧푖ℓ 푧푖ℓ′ = 0} for ℓ ≠ ℓ′ we have +˜피휁 + +� +ℓ∈[푘] +� +푗 ,푗′∈[푛] +푧푖ℓ 푧푗ℓ 푧푗′ℓ + += ˜피휁 + +�� +� +� +ℓ∈[푘] ,푗∈[푛] +푧푖ℓ 푧푗ℓ�� +� +· �� +� +� +ℓ′∈[푘] ,푗′∈[푛] +푧푖ℓ′푧푗′ℓ′�� +� + +. +Now, for fixed 푗 , 푗′ ∈ [푛], using +� +푎2 = 푎 , 푏2 = 푏 +� +푂(1) +� +1 + 푎푏 − 푎 − 푏 = 1 − 푎푏 − (푎 − 푏)2 ⩾ 0 +� +with 푎 = � +ℓ∈[푘] 푧푖ℓ 푧푗ℓ and 푏 = � +ℓ′∈[푘] 푧푖ℓ′푧푗′ℓ′ we get +˜피휁 + +�� +� +� +ℓ∈[푘] +푧푖ℓ 푧푗ℓ�� +� +�� +� +� +ℓ′∈[푘] +푧푖ℓ′푧푗′ℓ′�� +� + +⩾ ˜피휁 + +� +ℓ∈[푘] +푧푖ℓ 푧푗ℓ + +� +ℓ′∈[푘] +푧푖ℓ′푧푗′ℓ′ + +− 1 += 푊푖푗 + 푊푖푗′ − 1 . +Now if 푗, 푗′ ∈ 풢 we must have +� +ℓ∈[푘] +˜피휁 +� +푧푖ℓ 푧푗ℓ 푧푗′ℓ +� += ˜피휁 + +�� +� +� +ℓ∈[푘] +푧푖ℓ 푧푗ℓ�� +� +�� +� +� +ℓ′∈[푘] +푧푖ℓ′푧푗′ℓ′�� +� + +⩾ 0.6 . +44 + +Since 휙(푊푖푗) ⩽ 1 by definition and +��휙(푊푖) +�� +1 ⩾ 푛 +푘 · +� +1 − 10 +푘50 +� +, we conclude +��휙(푊푖) +�� +1 +−2 + +� +푗 ,푗′∈[푛] +휙(푊푖푗)휙(푊푖푗′)� +푦푗 − 푦푗′�⊗2 + +⪯ 5 · 푘2 +푛2 +� +푗 ,푗′∈[푛] ,ℓ∈[푘] +˜피휁 +� +푧푖ℓ 푧푗ℓ 푧푗′ℓ +� +· � +푦푗 − 푦푗′�⊗2 +⪯ 40Id . +as desired. +□ +We can now prove Lemma 6.10. +Proof of Lemma 6.10. Let 푊, 푊′ be the matrices computed by Algorithm 6.5 in step 1 on +input 푌, 푌′, respectively. Let 풢 ⊆ [푛] be the set of indices 푖 such that +��휙(푊)푖 − 휙(푊′)푖 +�� +1 ⩽ 푛0.8 . +Notice that |풢| ⩾ 푛 − 푛0.8 by Lemma 6.9. Since on input 푌 the algorithm did not reject in +step 3 we must have +��휙(푊) +�� +1 ⩾ 푛2 +푘 · +� +1 − +1 +푛0.1 − +1 +푘100 +� +− 푛1.7 ⩾ 푛2 +푘 · +� +1 − +2 +푘100 +� +. +Let 푔푊 be the number of indices 푖 ∈ 풢 such that +��휙(푊)푖 +�� +1 ⩾ 푛 +푘 · +� +1 − +1 +푘50 +� +. It holds that +푛2 +푘 · +� +1 − +2 +푘100 +� +⩽ 푔푊 · 푛 +푘 + (푛 − |풢|) · 푛 +푘 + � +|퐺| − 푔푤 +� 푛 +푘 · +� +1 − 1 +푘50 +� +⩽ 푔푊 · 푛 +푘 · 1 +푘50 + 푛1.8 +푘 ++ 푛2 +푘 · +� +1 − 1 +푘50 +� +⩽ 푔푊 · 푛 +푘 · 1 +푘50 + 푛2 +푘 · +� +1 + +1 +푘100 − 1 +푘50 +� +. +Rearring now yields +푔푊 ⩾ 푛 · +� +1 − 3 +푘50 +� +. +Similarly, let 푔푊′ be the number of indices 푖 ∈ 풢 such that +��휙(푊′)푖 +�� +1 ⩾ 푛 +푘 · +� +1 − +1 +푘50 +� +. By an +analogous argument it follows that 푔푊′ ⩾ 푛 · +� +1 − +3 +푘50 +� +. Thus, by the pigeonhole principle +there are at least 푔푊 ⩾ 푛 · +� +1 − +6 +푘50 +� +indices 푖 such that +1. +��휙(푊)푖 +�� +1 ⩾ 푛 +푘 +� +1 − +1 +푘50 +� +, +45 + +2. +��휙(푊′)푖 +�� +1 ⩾ 푛 +푘 +� +1 − +1 +푘50 +� +, +3. +��휙(푊)푖 − 휙(푊′)푖 +�� +1 ⩽ 푛0.8 . +Combining these with Lemma 6.11 we may also add +4. the distribution induced by +��휙(푊푖) +��−1 +1 휙(푊푖) is 2-explicitly 40-bounded, +5. the distribution induced by +��휙(푊′ +푖 ) +��−1 +1 휙(푊′ +푖 ) is 2-explicitly 40-bounded. +Using that for non-zero vectors 푥, 푦 it holds that +��� 푥 +∥푥∥ − +푦 +∥푦∥ +��� ⩽ +2 +∥푥∥ +��푥 − 푦 +�� points 1 to 3 +above imply that +��� +��휙(푊푖) +��−1 +1 휙(푊푖) − +��휙(푊′ +푖 ) +��−1 +1 휙(푊′ +푖 ) +��� +1 ⩽ +2푛0.8 +푛 +푘 · +� +1 − +1 +푘50 +� = 푂� +푛−0.2� +. +Hence, applying Theorem 3.21 with 푡 = 1 it follows that +���휈(푖) +푌 − 휈(푖) +푌′ +��� +2 ⩽ 푂� +푛−0.1� +. +□ +6.1.3 +From low sensitivity to privacy +In this section we argue privacy of the whole algorithm, proving Lemma 6.8. Before doing +that we observe that low-sensitivity is preserved with high probability under subsampling. +Fact 6.12 (Stability of 퓢). Consider the settings of Theorem 6.3. Suppose Algorithm 6.5 does not +reject in step 3, on adjacent inputs 푌 , 푌′ ∈ 풴. With probability at least 1 − 푒−푛Ω(1) over the random +choices of 퓢, for all but 10푛0.01 +푘50 +indices 푖 ∈ 퓢, it holds: +���휈(푖) +푌 − 휈(푖) +푌′ +��� +2 ⩽ 푂� +푛−0.1� +. +Proof. There are at most 6푛 +푘50 such indices in [푛] by Lemma 6.10. By Chernoff’s bound, cf. +Fact A.4, the claim follows. +□ +Finally, we prove our main privacy lemma. +Proof of Lemma 6.8. For simplicity, we will prove that the algorithm is (5휀, 5훿)-private. +Let 푌, 푌′ ∈ 풴 be adjacent inputs. By Lemma 3.10 and Lemma 6.9 the test in step 3 of +Algorithm 6.5 is (휀, 훿)-private. +Thus suppose now the algorithm did not reject in step 3 on inputs푌, 푌′. By composition +(cf. Lemma 3.4) it is enough to show that the rest of the algorithm is (휀, 훿)-private with +respect to 푌, 푌′ under this condition. Next, let 휈(1) +푌 , . . . , 휈(푛) +푌 and 휈(1) +푌′ , . . . , 휈(푛) +푌′ be the vectors +46 + +computed in step 4 of the algorithm and 풮 be the random set of indices computed in step +5.30 By Lemma 6.10 and Fact 6.12 with probability 1 − 푒−푛Ω(1) over the random choices of +퓢 we get that for all but 10푛0.01 +푘50 +indices 푖 ∈ 퓢, it holds that +���휈(푖) +푌 − 휈(푖) +푌′ +��� +2 ⩽ 푂� +푛−0.1� +. +Denote this set of indices by 풢. Note, that we may incorporate the failure probability +푒−푛Ω(1) ⩽ min{휀/2, 훿/2} into the final privacy parameters using Fact E.3. +Denote by V, V′ the |퓢|-by-푑 matrices respectively with rows 휈(푖1) +푌 , . . . , 휈 +(푖|퓢|) +푌 +and +휈(푖1) +푌′ , . . . , 휈 +(푖|퓢|) +푌′ +, where 푖1, . . . , 푖|퓢| are the indices in 퓢 . Recall, that |풢| rows of V and +V′ differ by at most 푂� +푛−0.1� +in ℓ2-norm. Thus, by the Gaussian mechanism used in step 6 +(cf. Lemma 3.12) and Lemma E.4 it is enough to show that step 7 to step 9 of the algo- +rithm are private with respect to pairs of inputs 푉 and 푉′ differing in at most 1 row.31 In +particular, suppose these steps are (휀1, 훿1)-private. Then, for 푚 = 푛0.01 − |풢| ⩽ 10푛0.01 +푘50 , by +Lemma E.4 it follows that step 6 to step 9 are (휀′, 훿′)-differentially private with +휀′ ≔ 휀 + 푚휀1 , +훿′ ≔ 푒휀푚푒(푚−1)휀1훿1 + 훿 . +Consider steps 7 and 8. Recall, that in step 7 we invoke the histogram learner with +parameters +푏 = 푘−15, q 푢.푎.푟. +∼ +[0, 푏], 훼 = 푘−10, 훽 = 푛−10, 훿∗ = 훿 +푛 , 휀∗ = 휀 · 10푘50 +푛0.01 . +Hence, by Lemma 3.13 this step is (휀∗, 훿∗)-private since +8 +휀∗훼 · log +� +2 +훿∗훽 +� +⩽ 200 · 푘10 · 푛0.01 +10 · 푘50 · 휀 +· log 푛 = 20 · 푛0.01 +푘40 · 휀 +· log 푛 ⩽ 푛 , +for 휀 ⩾ 푘−10. Step 8 is private by post-processing. +Next, we argue that step 9 is private by showing that the average over the bins has +small ℓ2-sensitivity. By Lemma 3.4 we can consider the bins B1, . . . , B푘 computed in the +previous step as fixed. Further, we can assume that the algorithm did not reject in step 8, +i.e., that each bin contains at least 푛0.01 +2푘 points of 푉 and 푉′ respectively. As a consequence, +every bin contains at least two (projections of) points of the input 푉 or 푉′ respectively. In +particular, it contains at least one (projection of a) point which is present in both 푉 and 푉′. +Fix a bin B푙 and let ¯휈∗ be such that it is both in 푉 and 푉′ and 횽¯휈∗ ∈ B푙. Also, define +푆푙 ≔ +��� +� +푗 +��� 횽¯휈(푗) +푌 ∈ B푖 +���� , +30Note that since this does not depend on 푌 or 푌′, respectively, we can assume this to be the same in both +cases. Formally, this can be shown, e.g., via a direct calculation or using Lemma 3.4. +31Note that for the remainder of the analysis, these do not correspond to V and V′, since those differ in 푚 +rows. Lemma E.4 handles this difference. +47 + +푆′ +푙 ≔ +��� +� +푗 +��� 횽¯휈(푗) +푌′ ∈ B푖 +���� . +Assume 푉 and 푉′ differ on index 푗. We consider two cases. First, assume that 횽¯휈(푗) +푌 +and 횽¯휈(푗) +푌′ both lie in B푙. In this case, 푆푙 = 푆′ +푙 and using Lemma A.5 it follows that with +probability 푛−100 ⩽ min{휀/2, 훿/2} it holds that +���¯휈(푗) +푌 − ¯휈(푗) +푌′ +��� +2 ⩽ +���¯휈(푗) +푌 − ¯휈∗��� +2 + +���¯휈∗ − ¯휈(푗) +푌′ +��� ⩽ 10 · +����횽¯휈(푗) +푌 − 횽¯휈∗��� +2 + +���횽¯휈(푗) +푌′ − 횽¯휈∗��� +2 +� +⩽ 20 · +√ +푑∗ · 푏 ⩽ 200 · 푘−12 . +And hence we can bound +������� +1 +푆푙 +· +��� +� +� +횽¯휈(푗) +푌 ∈B푙 +¯휈(푗) +푌 +��� +� +− 1 +푆′ +푙 +· +��� +� +� +횽¯휈(푗) +푌′∈B푙 +¯휈(푗) +푌′ +��� +� +������� +2 +⩽ +���¯휈(푗) +푌 − ¯휈(푗) +푌′ +��� +2 +푆푙 +⩽ 400 · 푘−11 +푛0.01 +. +Next, assume that 횽¯휈(푗) +푌 ∉ B푙 and 횽¯휈(푗) +푌′ ∈ B푙 (the other case works symetrically). It follows +that 푆푙 = 푆′ +푙 − 1 and we can bound +������� +1 +푆푙 +· +��� +� +� +횽¯휈(푗) +푌 ∈B푙 +¯휈(푗) +푌 +��� +� +− 1 +푆′ +푙 +· +��� +� +� +횽¯휈(푗) +푌′∈B푙 +¯휈(푗) +푌′ +��� +� +������� +2 += +1 +푆푙 · 푆′ +푙 +· +������� +푆′ +푙 +��� +� +� +횽¯휈(푗) +푌 ∈B푙 +¯휈(푗) +푌 +��� +� +− � +푆′ +푙 − 1���� +� +� +횽¯휈(푗) +푌′∈B푙 +¯휈(푗) +푌′ +��� +� +������� +2 += +1 +푆푙 · 푆′ +푙 +· +������� +푆′ +푙 · ¯휈(푗) +푌′ + +��� +� +� +횽 ¯휈(푗) +푌′∈B푙 +¯휈(푗) +푌′ +��� +� +������� +2 += 1 +푆푙 +· +������� +¯휈(푗) +푌′ − 1 +푆′ +푙 +��� +� +� +횽¯휈(푗) +푌′∈B푙 +¯휈(푗) +푌′ +��� +� +������� +2 +⩽ +√ +푑∗ · 푏 +푆푙 +⩽ 20 · 푘−11 +푛0.01 +. +Hence, the ℓ2-sensitivity is at most Δ ≔ 400·푘−11 +푛0.01 . Since +2Δ2 · log(2/(훿∗/푘)) +(휀∗/푘)2 += 32 · 푘−120 · log(2푘푛/훿) +휀2 +and w′ ∼ 푁 +� +0, 32 · 푘−120 · log(2푘푛/훿) +휀2 +· Id +� +it follows that outputing ˆ흁푙 is (휀∗/푘, 훿∗/푘)-DP by +the Gaussian Mechanism that. By Lemma 3.4 it follows step 9 is (휀∗, 훿∗)-private. +Hence, by Lemma 3.4 it follows that step 7 to step 9 are (2휀∗, 2훿∗)-differentially private. +Using 푚 ⩽ 10푛0.01 +푘10 +it now follows by Lemma E.4 that step 6 to step 9 are (휀′, 훿′)-private for +휀′ = 휀 + 2푚휀∗ ⩽ 3휀 , +48 + +훿′ = 2푒휀푚푒(푚−1)2휀∗훿∗ + 훿 ⩽ 2푚푒3휀 · 훿 +푛 + 훿 ⩽ 3훿 . +Thus, combined with the private check and Fact E.3 in step 3 the whole algorithm is +(5휀, 5훿)-private. +□ +6.2 +Utility analysis +In this section we reason about the utility of Algorithm 6.5 and prove Theorem 6.3. We +first introduce some notation. +Definition 6.13 (True solution). Let Y be an input sampled from Model 1.2. Denote by +푊∗(Y) ∈ 풲(Y) the matrix induced by the true solution (or ground truth). I.e., let +푊∗(Y)푖푗 = +� +1 +if 푖 , 푗 were both sampled from the same component of the mixture, +0 +otherwise. +Whenever the context is clear, we simply write W∗ to ease the notation. +First, we show that in the utility case step 3 of Algorithm 6.5 rejects only with low +probability. +Lemma 6.14 (Algorithm does not reject on good inputs). Consider the settings of Theorem 6.3. +Suppose Y is a good set as per Definition 6.6. Then +���푊(ˆ휁(Y)) +��� +1 ⩾ 푛2 +푘 · +� +1 − 푛−0.4 − +1 +(10)10푘300 +� +and +Algorithm 6.5 rejects with probability at most exp� +−Ω� +푛1.7�� +. +Proof. Since Y is good, there exists W∗ ∈ 풲(Y), corresponding to the indicator matrix of +the true solution, such that +푔(W∗) = ∥W∗∥2 +F − 1010푘300⟨퐽, W∗⟩ ⩽ 푛2 +푘 + 푛1.6 − (10)10푘300 +� +푛2 +푘 − 푛1.6 +� += 푛2 +푘 +� +1 + +푘 +푛0.4 − (10)10푘300 +� +1 − +푘 +푛0.4 +�� +. +Since 푔(푊(ˆ휁(Y))) ⩽ 푔(W∗) it follows that +(10)10푘300⟨퐽, 푊(ˆ휁(Y))⟩ ⩾ |푔(푊(ˆ휁(Y)))| ⩾ 푛2 +푘 +� +(10)10푘300 +� +1 − +푘 +푛0.4 +� +− 1 − +푘 +푛0.4 +� +. +Since, +���푊(ˆ휁(Y)) +��� +1 ⩾ ⟨퐽, 푊(ˆ휁(Y))⟩ the first claim follows rearranging the terms. This means +that the algorithm rejects only if |흉| ⩾ 푛1.7. Recall that 흉 ∼ tLap +� +−푛1.6� +1 + log(1/훿) +휀 +� +, 푛1.6 +휀 +� +. +Hence,by Lemma 3.11 it follows that +ℙ� +|흉| ⩾ 푛1.7� ⩽ exp� +−푛1.7 + 휀 + log(1/훿)� +2 − exp� +−휀 − log(1/훿)� += exp� +−Ω� +푛1.7�� +. +□ +49 + +The next step shows that on a good input Y the matrix 휙(푊(ˆ휁(Y))) is close to the true +solution. +Lemma 6.15 (Closeness to true solution on good inputs). Consider the settings of Theorem 6.3. +Suppose Y is a good set as per Definition 6.6. Let 푊(Y) ∈ 풲(Y) be the matrix computed by +Algorithm 6.5. Suppose the algorithm does not reject. Then +��휙(푊(Y)) − W∗�� +1 ⩽ 푛2 +푘 · 3 +푘98 . +The proof is similar to the classical utility analysis of the sum-of-squares program +found, e.g., in [HL18, FKP+19]. We defer it to Appendix E. +Together, the above results imply that the vectors 휈(푖) computed by the algorithm are +close to the true centers of the mixture. +Lemma 6.16 (Closeness to true centers). Consider the settings of Theorem 6.3. Suppose Y is a +good set as per Definition 6.6. Let W ∈ 풲(Y) be the matrix computed by Algorithm 6.5. Suppose +the algorithm does not reject in step 3. Then for each ℓ ∈ [푘], there exists 푛 +푘 · +� +1 − +2 +푘47 +� +indices +푖 ∈ [푛], such that +��휈(푖)(W) − 휇ℓ +�� +2 ⩽ 푂� +푘−25� +. +Proof. We aim to show that for most indices 푖 ∈ [푛] the vectors +��휙(W푖) +��−1 +1 휙(W푖) and +��W∗ +푖 +��−1 +1 W∗ +푖 induce a 2-explicitly 40-bounded distribution over Y. If additionally the two +vectors are close in ℓ1-norm, the result will follow by Theorem 3.21. +Note that +��W∗ +푖 +��−1 +1 W∗ +푖 induces a 2-explicitly 40-bounded distribution by Lemma 6.7. By +Markov’s inequality and Lemma 6.15 there can be at most 푛/푘48 indices 푗 ∈ [푛] such that +���휙(W)푗 − W∗ +푗 +��� +1 ⩾ 푛 +푘 · 3 +푘50 . +Consider all remaining indices 푖. It follows that +��휙(W푖) +�� +1 ⩾ +��W∗ +푖 +�� +1 − +��휙(W)푖 − W∗ +푖 +�� +1 ⩾ 푛 +푘 · +� +1 − +푘 +푛0.4 − 3 +푘50 +� +⩾ 푛 +푘 · +� +1 − 10 +푘50 +� +. +Hence, by Lemma 6.11 the distribution induced by +��휙(W푖) +��−1 +1 휙(W푖) is 2-explicitly 40- +bounded distribution. Further, using +��W∗ +푖 +�� +1 ⩾ 푛 +푘 +� +1 − +푘 +푛0.4 +� +we can bound +��� +��휙(W푖) +��−1 +1 휙(W푖) − +��W∗ +푖 +��−1 +1 W∗ +푖 +��� +1 = +��휙(W푖) +��−1 +1 +��W∗ +푖 +��−1 +1 · +����W∗ +푖 +�� +1휙(W푖) − +��휙(W푖) +�� +1W∗ +푖 +�� +1 +⩽ +��휙(W푖) +��−1 +1 +��W∗ +푖 +��−1 +1 · +�����휙(W푖) +�� +1 − +��W∗ +푖 +�� +1 +�� · +��휙(W푖) +�� +1 + +��휙(W푖) +�� +1 · +��휙(W푖) − W∗ +푖 +�� +1 +� +⩽ +��W∗ +푖 +��−1 +1 · 2 +��휙(W푖) − W∗ +푖 +�� +1 ⩽ +6 +푘50 · +� +1 − +푘 +푛0.4 +� ⩽ +7 +푘50 . +50 + +Hence, by Theorem 3.21 for each 푙 ∈ [푘] there are at least 푛 +푘 − 푛0.6 − +푛 +푘48 ⩾ 푛 +푘 · +� +1 − +2 +푘47 +� +indices 푖 such that +������ +휈(푖)(W) − +��W∗ +푖 +��−1 +1 +푛 +� +푗=1 +W∗ +푖,푗y푗 +������ +2 +⩽ 푂� +푘−25� +. +The result now follows by standard concentration bounds applied to the distribution +induced by +��W∗ +푖 +��−1 +1 W∗ +푖. +□ +An immediate consequence of Lemma 6.16 is that the vectors ¯흂(푖) inherits the good +properties of the vectors 휈(푖) with high probability. +Corollary 6.17 (Closeness to true centers after sub-sampling). Consider the settings of +Theorem 6.3. Suppose Y is a good set as per Definition 6.6. Let W ∈ 풲(Y) be the matrix computed +by Algorithm 6.5. Suppose the algorithm does not reject. Then with high probability for each ℓ ∈ [푘], +there exists 푛0.01 +푘 +· +� +1 − 150 +푘47 +� +indices 푖 ∈ 퓢, such that +�� ¯흂(푖) − 휇ℓ +�� +2 ⩽ 푂� +푘−25� +. +Proof. For each ℓ ∈ [푘], denote by 풯ℓ the set of indices in [푛] satisfying +��휈(푖)(W) − 휇ℓ +�� +2 ⩽ 푂� +푘−25� +. +By Lemma 6.16 we know that 풯ℓ has size at least 푛 +푘 · +� +1 − +2 +푘47 +� +. Further, let 풮 be the set of +indices selected by the algorithm. By Chernoff’s bound Fact A.4 with probability 1−푒−푛Ω(1) , +we have |퓢 ∩ 풯ℓ | ⩾ 푛0.01 +푘 +· +� +1 − 150 +푘47 +� +. Taking a union bound over all ℓ ∈ [푘] we get that with +probability 1 − 푒−푛Ω(1) , for each ℓ ∈ [푘], there exists 푛0.01 +푘 +· +� +1 − 150 +푘47 +� +indices 푖 ∈ 퓢 such that +��휈(푖)(W) − 휇ℓ +�� +2 ⩽ 푂� +푘−25� +. +Now, we obtain the corollary observing (cf. Fact A.1 with 푚 = 1) that with probability at +least 1 − 푒−푛Ω(1), for all 푖 ∈ 퓢 +�� ¯흂(푖) − 휈(푖)(W) +�� +2 = ∥w∥2 ⩽ 푛−0.05 · +� +log(2/훿) +휀 +· +√ +푑 ⩽ 푛−0.04 ⩽ 푂� +푘−25� +. +□ +For each ℓ, denote by 퓖ℓ ⊆ 퓢 the set of indices 푖 ∈ 퓢 satisfying +�� ¯흂(푖) − 휇ℓ +�� +2 ⩽ 푂� +푘−25� +. +Let 퓖 := � +ℓ∈[푘] +퓖ℓ . We now have all the tools to prove utility of Algorithm 6.5. We achieve +this by showing thst with high probability, each bin returned by the algorithm at step +7 satisfies 퓖ℓ′ ⊆ Bℓ for some ℓ , ℓ′ ∈ [푘] . Choosing the bins small enough will yield the +desired result. +51 + +Lemma 6.18 (Closeness of estimates). Consider the settings of Theorem 6.3. Suppose Y is a +good set as per Definition 6.6. Let W ∈ 풲(Y) be the matrix computed by Algorithm 6.5. Suppose +the algorithm does not reject. Then with high probability, there exists a permutation 휋 : [푘] → [푘] +such that +max +ℓ∈[푘] +��휇ℓ − ˆ흁휋(ℓ) +�� +2 ⩽ 푂� +푘−20� +Proof. Consider distinct ℓ, ℓ′ ∈ [푘]. By Corollary 6.17 for each ¯흂(푖) , ¯흂(푗) ∈ 퓖ℓ it holds that +�� ¯흂(푖) − ¯흂(푗)�� +2 ⩽ 퐶 · 푘−25 , +for some universal constant 퐶 > 0. Moreover, by assumption on 휇ℓ, 휇ℓ′ for each ¯흂(푖) ∈ 퓖ℓ +and ¯흂(푗) ∈ 퓖ℓ′ +�� ¯흂(푖) − ¯흂(푗)�� +2 ⩾ Δ − 푂� +푘−25� +. +Thus, by Lemma A.5 with probability at least 1 − 푒Ω(푑∗) ⩾ 1 − 푛−100 it holds that or each +¯흂(푖) , ¯흂(푗) ∈ 퓖ℓ and ¯흂푟 ∈ 풢ℓ′ with ℓ′ ≠ ℓ , +��횽 ¯흂(푖) − 횽 ¯흂(푗)�� +2 ⩽ 퐶∗ · 푘−25 +and +��횽 ¯흂(푖) − 횽 ¯흂(푟)�� +2 ⩾ Δ − 퐶∗ · 푘−25 +for some other universal constant 퐶∗ > 퐶. Let 푄횽(퓖ℓ) ⊆ ℝ푑∗ be a ball of radius 퐶∗ · � +푘−25� +such that ∀푖 ∈ 퓖ℓ it holds 횽 ¯흂(푖) ∈ 푄횽(퓖ℓ). That is, 푄횽(퓖ℓ) contains the projection of all +points in 퓖ℓ . +Recall that 푑∗ = 100 log(푛) ⩽ 100푘5 and 푏 = 푘−15. Let 퓑 = {B푖}∞ +푖=1 be the sequence of +bins computed by the histogram learner of Lemma 3.13 for ℝ푑∗ at step 7 of the algorithm. +By choice of 푏, and since q is chosen uniformly at random in [0, 푏], the probability that +there exists a bin B ∈ 퓑 containing 푄횽(퓖ℓ) is at least +1 − 푑∗ · 퐶∗ +푏 · � +푘−25� ⩾ 1 − 100퐶∗ +푏 +· 푘−20 ⩾ 1 − 푂� +푘−5� +, +where we used that 푑∗ = 100 log 푛 ⩽ 100푘5. A simple union bound over ℓ ∈ [푘] yields +that with high probability for all ℓ ∈ [푘] , there exists B ∈ 퓑 such that 푄횽(퓖ℓ) ⊆ B . For +simplicity, denote such bin by Bℓ. +We continue our analysis conditioning on the above events, happening with high +probability. First, notice that for all 푙 ∈ [푘] +max +푢,푢′∈Bℓ ∥푢 − 푢′∥2 +2 ⩽ 푑∗ · 푏2 ⩽ 100푘−25 ⩽ Δ − 퐶∗푘−25 +푘10 +, +and thus there cannot be ℓ, ℓ′ ∈ [푘] such that 푄횽(퓖ℓ) ⊆ Bℓ and 푄횽(퓖′ +ℓ) ⊆ Bℓ . Moreover, +by Corollary 6.17 and +min +ℓ∈[푘]|퓖ℓ | ⩾ 푛0.01 +푘 +· +� +1 − 150 +푘47 +� +, +52 + +and hence +|퓢 \ 퓖| ⩽ 푛0.01 · 150 +푘47 = 푛0.01 +푘 +· 150 +푘46 +it must be that step 7 returned bins B1, . . . , B푘. This also implies that the algorithm does +not reject. Further, by Lemma A.5 for all ¯흂(푖), ¯흂(푗) such that 횽 ¯흂(푖), 횽 ¯흂(푗) ∈ B푙 it holds that +�� ¯흂(푖) − ¯흂(푗)�� +2 ⩽ 퐶∗ · +��횽 ¯흂(푖) − 횽 ¯흂(푗)�� +2 ⩽ 퐶∗ · +√ +푑∗ · 푏 ⩽ 푂� +푘−12� +. +And hence, by triangle inequality, we get +�� ¯흂(푖) − 휇푙 +�� +2 ⩽ 푂� +푘−12� +. +Finally, recall that for each ℓ ∈ [푘], +ˆ흁푙 ≔ +1 +��� +푗 +�� 횽 ¯흂(푗) ∈ B푖 +��� · �� +� +� +횽 ¯흂(푗)∈B푙 +¯흂(푗)�� +� ++ w′ , +where w′ ∼ 푁 +� +0, 푁 +� +0, 32 · 푘−120 · log(2푘푛/훿) +휀2 +· Id +�� +. Since by choice of 푛, 푘, 휀 it holds that +32 · 푘−120 · log(2푘푛/훿) +휀2 +⩽ 푂� +푘−90� +, +we get with probability at least 1 − 푒−푘Ω(1) for each ℓ ∈ [푘], by Fact A.1, with 푚 = 1, and a +union bound that +∥w′∥ ⩽ 푂� +푘−20� +. +Since all ¯흂(푖) such that 횽 ¯흂(푖) ∈ B푙 are at most 푂� +푘−12� +-far from 휇푙, also their average is. +We conclude that +�� ˆ흁ℓ − 휇푙 +�� +2 ⩽ 푂(푘−12) + ∥w∥2 ⩽ 푂(푘−12) . +This completes the proof. +□ +Now Theorem 6.3 is a trivial consequence. +Proof of Theorem 6.3. The error guarantees and privacy guarantees immediately follows +combining Lemma 6.8, Lemma 6.15, Lemma 6.14 and Lemma 6.18. 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Wainwright, High-dimensional statistics: A non-asymptotic viewpoint, +Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge Uni- +versity Press, 2019. 63 +58 + +[WYX17] +Di Wang, Minwei Ye, and Jinhui Xu, Differentially private empirical risk minimiza- +tion revisited: Faster and more general, Advances in Neural Information Processing +Systems 30 (2017). 12 +[ZZ16] +Anderson Y Zhang and Harrison H Zhou, Minimax rates of community detection +in stochastic block models, The Annals of Statistics 44 (2016), no. 5, 2252–2280. 5, +31 +59 + +A +Concentration inequalities +We introduce here several useful and standard concentration inequalities. +Fact A.1 (Concentration of spectral norm of Gaussian matrices). Let W ∼ 풩(0, 1)푚×푛. Then +for any 푡, we have +ℙ +�√ +푚 − +√ +푛 − 푡 ⩽ 휎min(W) ⩽ 휎max(W) ⩽ +√ +푚 + +√ +푛 + 푡 +� +⩾ 1 − 2 exp +� +−푡2 +2 +� +, +where 휎min(·) and 휎max(·) denote the minimum and the maximum singular values of a matrix, +respectively. +Let W′ be an 푛-by-푛 symmetric matrix with independent entries sampled from 푁(0, 휎2). Then +∥W′∥ ⩽ 3휎√푛 with probability at least 1 − exp(−Ω(푛)). +Fact A.2 (Maximum degree of Erdős-Rényi graphs). Let 퐺 be an Erdős-Rényi graph on 푛 +vertices with edge probability 푝. Then with probability at least 1 − 푛 exp(−푛푝/3), any vertex in 퐺 +has degree at most 2푛푝. +Fact A.3 (Gaussian concentration bounds). Let X ∼ 풩(0, 휎2). Then for any 푡 ⩾ 0, +max{ℙ(X ⩾ 푡), ℙ(X ⩽ −푡)} ⩽ exp +� +− 푡2 +2휎2 +� +. +Fact A.4 (Chernoff bound). Let X1, . . . , Xn be independent random variables taking values in +{0, 1}. Let X := �푛 +푖=1 Xi and let 휇 := 피 X. Then for any 훿 > 0, +ℙ� +X ⩽ (1 − 훿)휇� ⩽ exp +� +−훿2휇 +2 +� +, +ℙ� +X ⩾ (1 + 훿)휇� ⩽ exp +� +− 훿2휇 +2 + 훿 +� +. +Lemma A.5 ([Joh84]). Let Φ be a 푑-by-푛 Gaussian matrix, with each entry independently chosen +from 푁(0, 1/푑). Then, for every vector 푢 ∈ ℝ푛 and every 훼 ∈ (0, 1) +ℙ(∥Φ푢∥ = (1 ± 훼)∥푢∥) ⩾ 1 − 푒−Ω(훼2푑) . +B +Linear algebra +Lemma B.1 (Weyl’s inequality). Let 퐴 and 퐵 be symmetric matrices. Let 푅 = 퐴 − 퐵. Let +훼1 ⩾ · · · ⩾ 훼푛 be the eigenvalues of 퐴. Let 훽1 ⩾ · · · ⩾ 훽푛 be the eigenvalues of 퐵. Then for each +푖 ∈ [푛], +��훼푖 − 훽푖 +�� ⩽ ∥푅∥. +60 + +Lemma B.2 (Davis-Kahan’s theorem). Let 퐴 and 퐵 be symmetric matrices. Let 푅 = 퐴 − 퐵. +Let 훼1 ⩾ · · · ⩾ 훼푛 be the eigenvalues of 퐴 with corresponding eigenvectors 푣1, . . . , 푣푛. Let +훽1 ⩾ · · · ⩾ 훽푛 be the eigenvalues of 퐵 with corresponding eigenvectors 푢1, . . . , 푢푛. Let 휃푖 be the +angle between ±푣푖 and ±푢푖 . Then for each 푖 ∈ [푛], +sin(2휃푖) ⩽ +2∥푅∥ +min푗≠푖 +��훼푖 − 훼푗 +��. +C +Convex optimization +Proposition C.1. Let 푓 : ℝ푚 → ℝ be a convex function. Let 풦 ⊆ ℝ푚 be a convex set. Then +푦∗ ∈ 풦 is a minimizer of 푓 over 풦 if and only if there exists a subgradient 푔 ∈ 휕 푓 (푦∗) such that +� +푦 − 푦∗, 푔 +� +⩾ 0 +∀푦 ∈ 풦. +Proof. Define indicator function +퐼풦(푦) = +� +0, +푦 ∈ 풦, +∞, +푦 ∉ 풦. +Then for 푦 ∈ 풦, one has +휕퐼풦(푦) = +� +푔 ∈ ℝ푚 : +� +푔, 푦 − 푦′� +⩾ 0 ∀푦′ ∈ 풦 +� +. +Note 푦∗ is a minimizer of 푓 over 풦, if and only if 푦∗ is a minimizer of 푓 + 퐼풦 over ℝ푚, if +and only if 0푚 ∈ 휕(푓 + 퐼풦)(푦∗) = 휕 푓 (푦∗) + 휕퐼풦(푦∗), if and only if there exists 푔 ∈ 휕 푓 (푦∗) +such that ⟨푔, 푦 − 푦∗⟩ ⩾ 0 for any 푦 ∈ 풦. +□ +Proposition C.2 (Pythagorean theorem from strong convexity). Let 푓 : ℝ푚 → ℝ be a convex +function. Let 풦 ⊆ ℝ푚 be a convex set. Suppose 푓 is 휅-strongly convex over 풦. Let 푥∗ ∈ 풦 be a +minimizer of 푓 over 풦. Then for any 푥 ∈ 풦, one has +∥푥 − 푥∗∥2 ⩽ 2 +휅(푓 (푥) − 푓 (푥∗)). +Proof. By strong convexity, for any subgradient 푔 ∈ 휕 푓 (푥∗) one has +푓 (푥) ⩾ 푓 (푥∗) + +� +푥 − 푥∗, 푔 +� ++ 휅 +2 ∥푥 − 푥∗∥2. +By Proposition C.1, ⟨푥 − 푥∗, 푔⟩ ⩾ 0 for some 푔 ∈ 휕 푓 (푥∗). Then the result follows. +□ +61 + +D +Deferred proofs SBM +We prove Lemma 5.8 restated below. +Lemma D.1 (Restatement of Lemma 5.8). Consider the settings of Lemma 5.7. With probability +1 − exp(−Ω(푛)) over G ∼ SBM푛(훾, 푑, 푥), +���� ˆ푋(푌(G)) − 1 +푛 푥푥⊤ +���� +2 +퐹 +⩽ 800 +훾 +√ +푑 +. +Proof. Recall 풦 = {푋 ∈ ℝ푛×푛 : 푋 ⪰ 0, 푋푖푖 = 1/푛 ∀푖}. Let 푋∗ := 1 +푛 푥푥⊤. Since ˆ푋 = ˆ푋(푌(G)) +is a minimizer of min푋∈풦 ∥푌(G) − 푋∥2 +퐹 and 푋∗ ∈ 풦, we have +��� ˆ푋 − 푌(G) +��� +2 +퐹 ⩽ ∥푋∗ − 푌(G)∥2 +퐹 ⇐⇒ +��� ˆ푋 − 푋∗��� +2 +퐹 ⩽ 2 +� +ˆ푋 − 푋∗, 푌(G) − 푋∗� +. +The infinity-to-one norm of a matrix 푀 ∈ ℝ푚×푛 is defined as +∥푀∥∞→1 := max{⟨푢, 푀푣⟩ : 푢 ∈ {±1}푚, 푣 ∈ {±1}푛}. +By [GV16, Fact 3.2], every 푍 ∈ 풦 satisfies +|⟨푍, 푌(G) − 푋∗⟩| ⩽ 퐾퐺 +푛 · ∥푌(G) − 푋∗∥∞→1, +where 퐾퐺 ⩽ 1.783 is Grothendieck’s constant. Similar to the proof of [GV16, Lemma 4.1], +using Bernstein’s inequality and union bound, we can show (cf. Fact D.2) +∥푌(G) − 푋∗∥∞→1 ⩽ 100푛 +훾 +√ +푑 +with probability 1 − exp(−Ω(푛)). Putting things together, we have +���� ˆ푋(푌(G)) − 1 +푛 푥푥⊤ +���� +2 +퐹 +⩽ 400 · 퐾퐺 +훾 +√ +푑 +, +with probability 1 − exp(−Ω(푛)). +□ +Fact D.2. Let 훾 > 0, 푑 ∈ ℕ, 푥∗ ∈ {±1}푛, and G ∼ SBM(훾, 푑, 푥∗). Let 푌(G) = +1 +훾푑 +� +퐴(G) − 푑 +푛 퐽� +, +where 퐴((퐺)) is the adjacency matrix of (퐺) with entries 푑/푛 on the diagonal. Then +max +푥∈{±1}푛 +��푥⊤� +푌(G) − 1 +푛 푥∗(푥∗)⊤� +푥 +�� ⩽ 100푛 +훾 +√ +푑 +with probability at least 1 − 푒−10푛. +62 + +Proof. The result will follow using Bernstein’s Inequality and a union bound. Define +푬 ≔ 푌(G) − 1 +푛 푥∗(푥∗)⊤. Fix 푥 ∈ {±1}푛 and for 1 ⩽ 푖 < 푗 ⩽ 푛, let 풁푖,푗 ≔ 푬푖,푗푥푖푥푗. Then +푥⊤푬푥 = 2 � +1⩽푖<푗⩽푛 풁푖,푗. Note that +피 풁푖,푗 = 0 , +��풁푖,푗 +�� ⩽ 1 +훾푛 · +�푛 +푑 − 1 +� ++ +1 +훾푑푛 ⩽ 1 +훾푑 , +피 풁2 +푖,푗 = Var +� +풀(G)푖,푗 +� +⩽ 피풀(G)2 +푖,푗 ⩽ (1 + 훾) 푑 +푛 · +1 +훾2푛2 +��푛 +푑 − 1 +�2 +− +1 +훾2푛2 +� ++ +1 +훾2푛2 +⩽ (1 + 훾) +1 +푑훾2푛 + +1 +훾2푛2 ⩽ +3 +훾2푑푛 . +By Bernstein’s Inequality (cf. [Wai19, Proposition 2.14]) it follows that +ℙ�� +� +� +푖<푗 +풁푖,푗 ⩾ 50푛 +훾 +√ +푑 +�� +� +⩽ ℙ�� +� +� +푖<푗 +풁푖,푗 ⩾ 푛2 +2 · 100푛 +훾 +√ +푑 +�� +� +⩽ 2 exp�� +� +− +104 +훾2푑 +3 +훾2푑푛 + +100 +3훾2푑3/2푛 +�� +� += 2 exp +� +− 104푛 +3 + 100 +√ +푑 +� +⩽ exp(−50푛) . +Hence, by a union bound over all 푥 ∈ {±1}푛 it follows that +max +푥∈{±1}푛 +��푥⊤� +푌(G) − 1 +푛 푥∗(푥∗)⊤� +푥 +�� ⩽ 100푛 +훾 +√ +푑 +with probability at least 1 − 푒−10푛. +□ +E +Deferred proofs for clustering +In this section, we will prove Lemma 6.15 restated below. +Lemma (Restatement of Lemma 6.15). Consider the settings of Theorem 6.3. Suppose Y is a +good set as per Definition 6.6. Let 푊(Y) ∈ 풲(Y) be the matrix computed by Algorithm 6.5. +Suppose the algorithm does not reject. Then +��휙(푊(Y)) − W∗�� +1 ⩽ 푛2 +푘 · 3 +푘98 . +We will need the following fact about our clustering program. Similar facts where +used, e.g., in [HL18, FKP+19]. One difference for us is that we don’t have a constraint on +the lower bound on the cluster size indicated by our SOS variables. However, since we +maximize a variant of the ℓ1 norm of the second moment matrix of the pseudo-distribution +this will make up for this. +63 + +Fact E.1. Consider the same setting as in Lemma 6.15. Let 0 < 훿 ⩽ +1 +1.5·1010 · +1 +푘201 and denote by +C1, . . . , C푘 ⊆ [푛] the indices belonging to each true cluster. Then 푊(Y) satisfies the following +three properties: +1. For all 푖, 푗 ∈ [푛] it holds that 0 ⩽ W푖,푗 ⩽ 1, +2. for all 푖 ∈ [푛] it holds that �푛 +푗=1 W푖,푗 ⩽ 푛 +푘 and for at least (1 − +1 +1000푘100)푛 indices 푖 ∈ [푛] it +holds that �푛 +푗=1 W푖,푗 ⩾ (1 − +1 +(10)6푘200) · 푛 +푘 , +3. for all 푟 ∈ [푘] it holds that � +푖∈C푟 ,푗∉C푟 W푖,푗 ⩽ 훿 · 푛2 +푘 . +We will prove Fact E.1 at the end of this section. With this in hand, we can proof +Lemma 6.15. +Proof of Lemma 6.15. For brevity, we write W = 푊(Y). Since 휙(W∗) = W∗ and 휙 is 10- +Lipschitz we can also bound +��휙(W) − W∗�� +1 ⩽ 10 · ∥W − W∗∥1 . +Let 훿 ⩽ +1 +1.5·1010 · +1 +푘201 and again let C1, . . . , C푘 ⊆ [푛] denote the indices belonging to each +true cluster. Note that by assumption that Y is a good sample it holds for each 푟 ∈ [푘] that +푛 +푘 − 푛0.6 ⩽ |C푟| ⩽ 푛 +푘 + 푛0.6. +Let 푟, 푟′ ∈ [푘]. We can write +∥W − W∗∥1 = +푘 +� +푟=1 +� +푖,푗∈C푟 +��W푖,푗 − 1 +�� + +푘 +� +푟=1 +� +푖∈C푟,푗∉C푟 +��W푖,푗 − 0 +�� +(E.1) +Note that we can bound the second sum by 푘 · 훿 푛2 +푘 using Item 3. Further, in what follows +consider only indices 푖 such that �푛 +푗=1 W푖,푗 ⩾ (1 − +1 +(10)6푘200 ) · 푛 +푘 . By Item 2 we can bound the +contribution of the other indices by +1 +1000푘100 푛 · +�푛 +푘 + 푛0.6� +⩽ +2 +1000푘100 · 푛2 +푘 . +Focusing only on such indices, for the first sum in Eq. (E.1), fix 푟 ∈ [푘]. We will aim to +show that most entries of W are large if and only if the corresponding entry of W∗ is 1. +By Item 3 and Markov’s Inequality, it follows that for at least a (1 − +1 +1000푘100)-fraction of the +indices 푖 ∈ C푟 it holds that +� +푗∉C푟 +W푖,푗 ⩽ 1000푘100 · 훿 +푛2 +푘·|C푟| ⩽ 1000푘100훿 · +푛 +1−푘·푛−0.4 ⩽ 2000푘101훿 · 푛 +푘 , +where we used that |C푟| ⩾ 푛 +푘 − 푛0.6. Call such indices good. Notice that for good indices it +follows using Item 2 that +� +푗∈C푟 +W푖,푗 ⩾ 푛 +푘 · (1 − +1 +(10)6푘200 − 2000푘101훿) . +64 + +Denote by 퐺 the number of 푗 ∈ C푟 such that W푖,푗 ⩾ 1 − +1 +1000푘100. Using the previous display +and that W푖,푗 ⩽ 1 we obtain +푛 +푘 · +� +1 − +1 +(10)6푘200 − 2000푘101훿 +� +⩽ +� +푗∈C푟 +W푖,푗 ⩽ 퐺 · 1 + (|C푟| − 퐺) · (1 − +1 +1000푘100) +⩽ 퐺 · +1 +1000푘100 + 푛 +푘 · (1 + +1 +푘푛0.4) · (1 − +1 +1000푘100) +⩽ 퐺 · +1 +1000푘100 + 푛 +푘 · (1 + +1 +푘푛0.4) , +where we also used |C푟| ⩽ 푛 +푘 + 푛0.6. Rearranging now yields +퐺 ⩾ 푛 +푘 · +� +1 − +1 +1000푘100 − 103푘99 +푛0.4 +− 2 · 106푘101훿 +� +⩾ 푛 +푘 · +� +1 − +2 +1000푘100 − 2 · 106푘101훿 +� +. +We can now bound +� +푖,푗∈C푟 +��W푖,푗 − 1 +�� = +� +푖,푗∈C푟 ,푖 is good +��W푖,푗 − 1 +�� + +� +푖,푗∈C푟 ,푖 is not good +��W푖,푗 − 1 +�� +⩽ |C푟| · +� +(|C푟| − 퐺) · 1 + |C푟| · +1 +1000푘100 +� ++ +1 +1000푘100 · |C푟|2 +⩽ |C푟|2(1 + +1 +500푘100) − 퐺 · |C푟| +⩽ 푛2 +푘2 (1 + +푘 +푛0.4)2(1 + +1 +500푘100) − 푛2 +푘2 (1 − +2 +1000푘100 − 2 · 106푘101훿)(1 − +푘 +푛0.4 ) +⩽ 푛2 +푘2 · (30 · 106푘101훿 + +11 +500푘100) ⩽ 푛2 +푘 · (30 · 106푘100훿 + +11 +500푘101) +⩽ 푛2 +푘 · +3 +125푘101 . +Putting everything together, it follows that +��휙(W) − W∗��2 +F ⩽ +��휙(W) − W∗�� +1 ⩽ 10 · 푛2 +푘 +� +훿푘 + +2 +1000푘100 + +3 +125푘100 +� +⩽ 푛2 +푘 · +4 +푘100 ⩽ 푛2 +푘 · +3 +푘98 . +□ +It remains to verify Fact E.1. +Proof of Fact E.1. Let 풫 = 풫푛,푘,푡(Y) be the system of Eq. (풫푛,푘,푡(푌)). Recall that W푖,푗 = +˜피 � +푙∈[푘] 푧푖,푙푧푗,푙. Since +풫 +4 + + +0 ⩽ +� +푙∈[푘] +푧푖,푙푧푗,푙 ⩽ +� +푙∈[푘] +푧푖,푙 ⩽ 1 + + +, +it follows that 0 ⩽ W푖,푗 ⩽ 1. Further, for each 푖 ∈ [푛] it holds that +풫 +4 + + +� +푗∈[푛],푙∈[푘] +푧푗,푙푧푖,푙 ⩽ 푛 +푘 +� +푙∈[푘] +푧푖,푙 ⩽ 푛 +푘 + + +65 + +implying that � +푗∈[푛] W푖,푗 ⩽ 푛 +푘 . Further, by Lemma 6.14 +∥W∥1 ⩾ 푛2 +푘 · +� +1 − 푛−0.4 − +1 +(10)10푘300 +� +⩾ 푛2 +푘 · +� +1 − +1 +(10)9푘300 +� +. +Denote by W푖 the 푖-th row of W and by 퐿 the number of rows which have ℓ1 norm at least +(1 − +1 +(10)6푘200 ) · 푛 +푘 . Since for all 푖 it holds that ∥W푖∥1 ⩽ 푛 +푘 it follows that +푛2 +푘 · +� +1 − +1 +(10)9푘300 +� +⩽ +� +푖∈[푛] +∥W푖∥1 ⩽ 퐿 · 푛 +푘 + (푛 − 퐿) · +� +1 − +1 +(10)6푘200 +� +· 푛 +푘 += 퐿 · +1 +(10)6푘200 · 푛 +푘 + 푛2 +푘 · +� +1 − +1 +(10)6푘200 +� +Rearranging then yields 퐿 ⩾ (1 − +1 +1000푘100) · 푛 which proofs Item 2. +It remains to verify Item 3. Fix 푟, 푙 ∈ [푘] and define 푧푙(C푟) = 푘 +푛 +� +푖∈C푟 푧푖,푙. Let 푡 > 0 be an +integer. We aim to show that for all unit vectors 푣 it holds that +풫 +10푡 +� +푧푙(C푟) · 1 +Δ2푡 +� +푟′≠푟 +푧푙(C푟′)⟨휇푟 − 휇푟′, 푣⟩2푡 ⩽ 훿 +푘 +� +, +(E.2) +where Δ is the minimal separation between the true means. Before proving this, let us +examine how we can use this fact to prove Item 3. Note, that for all 푟 ≠ 푟′ it holds that +� +푠,푢∈[푘] +� +휇푟 − 휇푟′, +휇푠−휇푢 +∥휇푠−휇푢∥ +�2푡 +⩾ Δ2푡 . +Hence, if the above SOS proof indeed exists, we obtain +� +푖∈C푟 ,푗∉C푟 +W푖,푗 = +푘 +� +푙=1 +˜피 +� +푖∈C푟 ,푗∉C푟 +푧푖,푙푧푗,푙 = 푛2 +푘2 ˜피푧푙(C푟) · +� +푟′≠푟 +푧푙(C푟′) +⩽ +푛2 +Δ2푡푘2 +� +푠,푢∈[푘] +˜피푧푙(C푟) · +� +푟′≠푟 +푧푙(C푟) +� +휇푟 − 휇푟′, +휇푠−휇푢 +∥휇푠−휇푢∥ +�2푡 +⩽ 훿 +푘 푘2 · 푛2 +푘2 = 훿 · 푛2 +푘 . +In the remainder of this proof we will prove Eq. (E.2). We will use the following SOS +version of the triangle Inequality (cf. Fact E.2) +2푡 +푥,푦 (푥 + 푦)2푡 ⩽ 22푡−1(푥2푡 + 푦2푡) . +Recall that 휇′ +푙 = 푘 +푛 +�푛 +푖=1 푧푖,푙 푦푖 and denote by 휇휋(푖) the true mean corresponding to the 푖-th +sample. Let 푣 be an arbitrary unit vector, it follows that +풫 +10푡 {푧푙(C푟) · 1 +Δ2푡 +� +푟′≠푟 +푧푙(C푟′)⟨휇푟 − 휇푟′, 푣⟩2푡 +66 + +⩽ 푧푙(C푟) · 22푡−1 +Δ2푡 +� +푟′≠푟 +푧푙(C푟′)� +⟨휇푟 − 휇′ +푙, 푣⟩2푡 + ⟨휇푟′ − 휇′ +푙, 푣⟩2푡� +⩽ 22푡−1 +Δ2푡 +푘 +� +푟=1 +푧푙(C푟)⟨휇푟 − 휇′ +푙, 푣⟩2푡 = 22푡−1 +Δ2푡 · 푘 +푛 +푛 +� +푖=1 +푧푖,푙⟨휇휋(푖) − 휇′ +푙, 푣⟩2푡} , +where we used that 풫 +1 +�푘 +푟=1 푧푙(C푟) ⩽ 1. Using the SOS triangle inequality again and that +풫 +2 푧푖,푙 ⩽ 1 we obtain +풫 +10푡 {푧푙(C푟) · 1 +Δ2푡 +� +푟′≠푟 +푧푙(C푟′)⟨휇푟 − 휇푟′, 푣⟩2푡 +⩽ 24푡−1 +Δ2푡 · +� +푘 · 1 +푛 +푛 +� +푖=1 +⟨y푖 − 휇휋(푖), 푣⟩2푡 + 푘 +푛 +푛 +� +푖=1 +푧푖,푙⟨y푖 − 휇′ +푙, 푣⟩2푡 +� +} . +We start by bounding the first sum. Recall that by assumption the uniform distribution +over each true cluster is 2푡-explicitly 2-bounded. It follows that +2푡 { 1 +푛 +푛 +� +푖=1 +⟨y푖 − 휇휋(푖), 푣⟩2푡 = 1 +푘 +푘 +� +푟=1 +푘 +푛 +� +푖∈C푟 +⟨y푖 − 휇푟, 푣⟩2푡 ⩽ 1 +푘 +푘 +� +푟=1 +푘 +푛 · |C푟| · (2푡)푡 · ∥푣∥2푡 +2 +(E.3) +⩽ +� +1 + +푘 +푛0.4 +� +· (2푡)푡 ⩽ 2(2푡)푡} , +(E.4) +where we used that |C푟| ⩽ 푛 +푘 + 푛0.6. To bound the second sum, we will use the moment +bound constraints. In particular, we know that +풫 +10푡 +� +푘 +푛 +푛 +� +푖=1 +푧푖,푙⟨y푖 − 휇′ +푙, 푣⟩2푡 ⩽ (2푡)푡 +� +. +(E.5) +Combining Eq. (E.4) and Eq. (E.5) now yields +풫 +10푡 +� +푧푙(C푟) · 1 +Δ2푡 +� +푟′≠푟 +푧푙(C푟′)⟨휇푟 − 휇푟′, 푣⟩2푡 ⩽ 푘22푡+1(2푡)푡 +Δ2푡 +⩽ 푘 +� +8푡 +Δ2 +�푡� +. +Note that by assumption Δ ⩾ 푂( +√ +푡푘1/푡). Overloading notation, we can choose the 푡 +parameter in the SOS proof to be 202 times the 푡 parameter in the lower bound in the +separation to obtain32 +� +푖∈C푟 ,푗∉C푟 +W푖,푗 ⩽ 훿 · 푛2 +푘 . +□ +32Note that this influences the exponent in the running time and sample complexity only by a constant +factor and hence doesn’t violate the assumptions of Theorem 6.3. +67 + +E.1 +Small Lemmas +Fact E.2 (Lemma A.2 in [KSS18]). For all integers 푡 > 0 it holds that +2푡 +푥,푦 +(푥 + 푦)2푡 ⩽ 22푡−1(푥2푡 + 푦2푡) . +Fact E.3. Let 휀, 훿 > 0. Let ℳ : 풴 → 풪 be a randomized algorithm that, for every pair of +adjacent inputs, with probability at least 1 − 훾 ⩾ 1/2 over the internal randomness of 풴33 satisfies +(휀, 훿)-privacy. Then ℳ is (휀 + 2훾, 훿 + 훾)-private. +Proof. Let 푋, 푋′ be adjacent input and let 퐵 be the event under which ℳ is (휀, 훿)-private. +By assumption, we know that ℙ(퐵) ⩾ 1 − 훾. Let 푆 ∈ 풪, it follows that +ℙ(ℳ(푋) ∈ 푆) = ℙ(퐵) · ℙ(ℳ(푋) ∈ 푆 | 퐵) + ℙ(퐵푐) · ℙ(ℳ(푋) ∈ 푆 | 퐵푐) +⩽ ℙ(ℳ(푋) ∈ 푆 | 퐵) + 훾 +⩽ 푒휀ℙ(ℳ(푋) ∈ 푆 | 퐵) + 훿 + 훾 +⩽ +푒휀 +ℙ(퐵) · ℙ(ℳ(푋) ∈ 푆) + 훿 + 훾 +⩽ 푒 +휀+log +� +1 +1−훾 +� +· ℙ(ℳ(푋) ∈ 푆) + (훿 + 훾) +⩽ 푒휀+2훾 · ℙ(ℳ(푋) ∈ 푆) + (훿 + 훾) , +where we used that log(1 − 훾) ⩾ −2훾 for 훾 ∈ [0, 1/2]. +□ +E.2 +Privatizing input using the Gaussian Mechanism +In this section, we will proof the following helpful lemma used in the privacy analysis +of our clustering algorithm (Algorithm 6.5). In summary, it says that when restricted to +some set our input has small ℓ2 sensitivity, we can first add Gaussian noise proportional +to this sensitivity and afterwards treat this part of the input as "privatized". In particular, +for the remainder of the privacy analysis we can treat this part as the same on adjacent +inputs. Note that we phrase the lemma in terms of matrix inputs since this is what we use +in our application. Of course, it also holds for more general inputs. +Lemma E.4. Let 푉, 푉′ ∈ ℝ푛×푑, 푚 ∈ [푛] and Δ > 0 be such that there exists a set 푆 of size at least +푛 − 푚 satisfying +∀푖 ∈ 푆. +��푉푖 − 푉′ +푖 +��2 +2 ⩽ Δ2 , +where 푉푖, 푉′ +푖 denote the rows of 푉, 푉′, respectively. Let 풜2 : ℝ푛×푑 → 풪 be an algorithm that +is (휀2, 훿2)-differentially private in the standard sense, i.e,., for all sets 풮 ⊆ 풪 and datasets +푋, 푋′ ∈: ℝ푛×푑 differing only in a single row it holds that +ℙ(풜2(푋) ∈ 푆) ⩽ 푒휀2ℙ(풜2(푋′) ∈ 푆) + 훿2 . +33In particular, this randomness is independent of the input +68 + +Further, let 풜1 : ℝ푛×푑 → ℝ푛×푑 be the Gaussian Mechanism with parameters Δ, 휀1, 훿1. I.e., on +input 푀 it samples W ∼ 푁 +� +0, 2Δ2 · log(2/훿1) +휀2 +1 +�푛×푑 +and outputs 푀 + W. +Then for +휀′ ≔ 휀1 + 푚휀2 , +훿′ ≔ 푒휀1푚푒(푚−1)휀2훿2 + 훿1 . +풜2 ◦ 풜1 is (휀′, 훿′)-differentially private with respect to 푉 and 푉′, i.e., for all sets 풮 ⊆ 풪 it holds +that +ℙ((풜2 ◦ 풜1)(푉) ∈ 푆) ⩽ 푒휀′ℙ((풜2 ◦ 풜1)(푉′) ∈ 푆) + 훿′ . +Proof. Without loss of generality, assume that 푆 = {1, . . . , 푚}. Denote by 푉1, 푉2 the first 푚 +and last 푛 − 푚 rows of 푉 respectively. Analogously for 푉′ +1, 푉′ +2. We will later partitin the +noise W of the Gaussian mechanism in the same way. Further, for a subset 퐴 of ℝ푛×푛 and +푌 ∈ ℝ푚×푛 define +푇퐴,푌 = +� +푋 ∈ ℝ(푛−푚)×푛 +���� +� +푋 +푌 +� +∈ 퐴 +� +⊆ ℝ(푛−푚)×푛 . +Note that +� +푋 +푌 +� +∈ 퐴 if and only if 푋 ∈ 푇퐴,푌. +Let 풮 ⊆ 풪. It now follows that +ℙ풜2,W[(풜2 ◦ 풜1)(푉) ∈ 푆] = +피 +풜2,W +� +ퟙ +� +푉 + W ∈ 풜−1 +2 (푆) +�� += +피 +풜2,W2 +� +피 +W1 +� +ퟙ +�� +푉1 + W1 +푉2 + W2 +� +∈ 풜−1 +2 (푆) +�� ���� W2 +� += +피 +풜2,W2 +� +피 +W1 +� +ퟙ +� +푉1 + W1 ∈ 푇풜−1 +2 (푆),푉2+W2 +�� ���� W2 +� +⩽ 푒휀1 · +피 +풜2,W2 +� +피 +W1 +� +ퟙ +� +푉′ +1 + W1 ∈ 푇풜−1 +2 (푆),푉2+W2 +�� ���� W2 +� ++ 훿1 += 푒휀1 · +피 +풜2,W +� +ퟙ +�� +푉′ +1 + W1 +푉2 + W2 +� +∈ 풜−1 +2 (푆) +�� ++ 훿1 , +where the inequality follows by the guarantees of the Gaussian Mechanism. Further, we +can bound +피 +풜2,W +� +ퟙ +�� +푉′ +1 + W1 +푉2 + W2 +� +∈ 풜−1 +2 (푆) +�� += 피 +W +� +피 +풜2 +� +ퟙ +� +풜2 +� +푉′ +1 + W1 +푉2 + W2 +� +∈ 푆 +� ���� W +�� +⩽ 푒푚휀2 · 피 +W +� +피 +풜2 +� +ퟙ +� +풜2 +� +푉′ +1 + W1 +푉′ +2 + W2 +� +∈ 푆 +� ���� W +�� ++ 푚푒(푚−1)휀2훿2 += 푒푚휀2 · +피 +풜2,W +� +ퟙ +�� +푉′ +1 + W1 +푉′ +2 + W2 +� +∈ 풜−1 +2 (푆) +�� ++ 푚푒(푚−1)휀2훿2 , +69 + +where the inequality follows by the privacy guarantees of 풜2 combined with standard +group privacy arguments. +Putting the above two displays together and plugging in the definition of 휀′, 훿′ we +finally obtain +ℙ풜2,W[(풜2 ◦ 풜1)(푉) ∈ 푆] ⩽ 푒휀′ℙ풜2,W[(풜2 ◦ 풜1)(푉′) ∈ 푆] + 훿′ . +□ +70 + diff --git a/udE3T4oBgHgl3EQf-Auh/content/tmp_files/load_file.txt b/udE3T4oBgHgl3EQf-Auh/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..883769c3a83fb5350f7dd30ab9b1ac713010989b --- /dev/null +++ b/udE3T4oBgHgl3EQf-Auh/content/tmp_files/load_file.txt @@ -0,0 +1,2874 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf,len=2873 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='04822v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='DS] 11 Jan 2023 Private estimation algorithms for stochastic block models and mixture models∗ Hongjie Chen† Vincent Cohen-Addad‡ Tommaso d’Orsi† Alessandro Epasto‡ Jacob Imola§ David Steurer† Stefan Tiegel† January 13, 2023 Abstract We introduce general tools for designing efficient private estimation algorithms, in the high-dimensional settings, whose statistical guarantees almost match those of the best known non-private algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' To illustrate our techniques, we consider two problems: recovery of stochastic block models and learning mixtures of spherical Gaussians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For the former, we present the first efficient (휀, 훿)-differentially private algorithm for both weak recovery and exact recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Previously known algorithms achieving comparable guarantees required quasi-polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For the latter, we design an (휀, 훿)-differentially private algorithm that recovers the centers of the 푘-mixture when the minimum separation is at least 푂(푘1/푡√ 푡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For all choices of 푡, this algorithm requires sample complexity 푛 ⩾ 푘푂(1)푑푂(푡) and time complexity (푛푑)푂(푡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Prior work required minimum separation at least 푂( √ 푘) as well as an explicit upper bound on the Euclidean norm of the centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' ∗This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 815464).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' †ETH Zürich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' ‡Google Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' §UC San Diego.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Contents 1 Introduction 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1 Results .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 4 2 Techniques 7 3 Preliminaries 14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1 Differential privacy .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 18 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2 Sum-of-squares proof .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 21 4 Stability of strongly-convex optimization 21 5 Private recovery for stochastic block models 22 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1 Private weak recovery for stochastic block models .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 23 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2 Private exact recovery for stochastic block models .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 26 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='3 Inefficient recovery using the exponential mechanism .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 31 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4 Lower bound on the parameters for private recovery .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 34 6 Private algorithms for learning mixtures of spherical Gaussians 38 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1 Privacy analysis .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 42 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1 Sensitivity of the matrix W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2 Sensitivity of the resulting vectors .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='3 From low sensitivity to privacy .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 46 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2 Utility analysis .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 49 Bibliography 53 A Concentration inequalities 60 B Linear algebra 60 C Convex optimization 61 D Deferred proofs SBM 62 2 E Deferred proofs for clustering 63 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1 Small Lemmas .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 68 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2 Privatizing input using the Gaussian Mechanism .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 68 3 1 Introduction Computing a model that best matches a dataset is a fundamental question in machine learning and statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Given a set of 푛 samples from a model, how to find the most likely parameters of the model that could have generated this data?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' This basic question has been widely studied for several decades, and recently revisited in the context where the input data has been partially corrupted (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=', where few samples of the data have been adversarially generated—see for instance [LRV16, DKK+19, dKNS20, DKK+22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' This has led to several recent works shedding new lights on classic model estimation problems, such as the Stochastic Block Model (SBM) [GV16, MS16, MPW16, FC20, DdNS22, LM22] and the Gaussian Mixture Model (GMM) [HL18, KSS18, BDH+20, BDJ+22] (see Definitions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Privacy in machine learning and statistical tasks has recently become of critical importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' New regulations, renewed consumer interest as well as privacy leaks, have led the major actors to adopt privacy-preserving solutions for the machine learn- ing [Goo15, App17, USC21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' This new push has resulted in a flurry of activity in al- gorithm design for private machine learning, including very recently for SBMs and GMMs [SNVT22, KSSU19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Despite this activity, it has remain an open challenge to fully understand how privacy requirements impact model estimation problems and in partic- ular their recovery thresholds and the computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' This is the problem we tackle in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' A de facto privacy standard is the differential-privacy framework of Dwork, McSherry, Nissim, and Smith [DMNS06] (more formally, see Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' In this framework, the privacy quality is governed by two parameters, 휀 and 훿, which in essence tell us how the probability of seeing a given output changes (both multiplicatively and additively) between two datasets that differ by any individual data element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' This notion, in essence, quantifies the amount of information leaked by a given algorithm on individual data elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The goal of the algorithm designer is to come up with differentially-private algorithms for 휀 being a small constant and 훿 being of order 1/푛Θ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Very recently, Seif, Nguyen, Vullikanti, and Tandon [SNVT22] were the first to pro- pose differentially-private algorithms for the Stochastic Block Model, with edge-privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Concretely, they propose algorithms achieving exact recovery (exact identification of the planted clustering) for the non-robust setting while ensuring differential-privacy on the edges of the graph (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=', the data elements the privacy properties applied to are the edges of the graph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The approach proposed achieves either a running time of 푛Θ(log 푛) when 휀 is constant, or runs in polynomial time for 휀 being Ω(log 푛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Gaussian Mixture Models have also been studied in the context of differential privacy by Kamath, Sheffet, Singhal, and Ullman [KSSU19] (see also recent work for robust mo- ment estimation in the differential-privacy setting [KMV22, AL22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Similarly, the bounds obtained in this context remain far from the best possible bounds in the non-private regime: for a mixture of 푘 Gaussians, the minimum distance between centers is required 1 to be at least Δ ⩾ √ 푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Moreover, an explicit bound 푅 on the euclidean norm of the centers is needed in input as the sample complexity of these algorithms depends on this bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' In this paper, we tackle both clustering problems (graph clustering with the SBM and metric clustering with the GMM) through a new general privacy-preserving framework that brings us significantly closer to the state-of-the-art of non-private algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Our new perspective on the problems also appear to be easily extendable to robust algorithm design in broader settings;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' as, indeed, our approach uses and analyses algorithms that are known to work in the robust setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' As observed in the past (see Kothari, Manurangsi and Velingker [KMV22]) the two goals of designing privacy-preserving machine learning models and robust model estima- tion are tightly related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The objective is to design algorithms that extract global information without over-relaying on individual data samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The algorithm of [KMV22] hence yields differentially-private robust mean estimation by leveraging an algorithm for robust mean estimation satisfying some basic conditions that enable privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Our work further tightens this connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' At a high level, previous works for robust model estimation followa two-step process: (1)showthat, with high probability, anysubset of size containing at least a (1 − 휀) fraction of the model-generated data is well-behaved (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=', concentration enables model estimation up to the information-theoretic threshold);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' and (2) If (1) holds, then an adversarial perturbation of an 휀 fraction of the data does not alter the quality of the output much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' From afar, this may look like a differentially-private algorithm, however an important requirement for achieving differential-privacy is that the algorithm should not only be private when (1) holds, but it must provide privacy for any possible input graph, not only for typical graphs from the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' This means that (2) must be generalized so that (2) holds even when (1) does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' More concretely, previous approaches for robust model estimation are deterministic (and succeeds with the probability that the output is typical enough) and so clearly cannot achieve differential-privacy off-the-shelf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' It is thus needed to add some noise in the process to enforce privacy, and do so while retaining the best utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Therefore, we must achieve a new sort of robustness: upon small changes of the input, the output of the algorithm does not change significantly, no matter what the input is and whether it is typical for the model or arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' In our paper, we formalize this extended notion of robustness and show a simple sensitivity bound, in the differentially-private sense, on strongly convex functions over constrained sets where both the function and the set depend on the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' This result subsumes some previously known sensitivity bounds in the empirical risk minimization literature (in particular in the output-perturbation ap- proach to ERM, see Section 2 for a comparison), enabling us to apply a small perturbation to the strongly convex functions that are used for model estimation for the SBM and GMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Crucially, we show that such a small perturbation does not alter the utility of the function, and the result yields nearly-optimal recovery thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Our framework, is quite general and we believe may be extended to other problems, the only challenge being to analyse the change in utility induced by the perturbation for the problem at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 2 Before diving into our results, we formally introduce the concrete problem we focus on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Stochastic Block model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The stochastic block model is an extensively studied statistical model for community detection in graphs (see [Abb17] for a survey).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Model 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1 (Stochastic block model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' In its most basic form, the stochastic block model describes the distribution1 of an 푛-vertex graph G ∼ SBM푛(푑, 훾, 푥), where 푥 is a vector of 푛 binary2 labels, 푑 ∈ ℕ, 훾 > 0, and for every pair of distinct vertices 푖, 푗 ∈ [푛] the edge {푖, 푗} is independently added to the graph G with probability (1 + 훾 · 푥푖 · 푥푗) 푑 푛 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Note that for distinct vertices 푖, 푗 ∈ [푛], the edge {푖, 푗} is present in G with probability (1 + 훾) 푑 푛 if the vertices have the same label 푥푖 = 푥푗 and with probability (1 − 훾) 푑 푛 if the vertices have different labels 푥푖 ≠ 푥푗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='3 Given a graph G sampled according to this model, the goal is to recover the (unknown) underlying vector of labels as well as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' In particular, for a chosen algorithm return- ing a partition ˆ푥(G) ∈ {±1}푛, there are two main objective of interest: weak recovery and exact recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The former amounts to finding a partition ˆ푥(G) correlated with the true partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The latter instead corresponds to actually recovering the true partition with high probabil- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' As shown in the following table, by now the statistical and computational landscape of these problems is well understood [DKMZ11, Mas14, MNS15b, MNS18, GV16]: Objective can be achieved (and efficiently so) iff weak recovery ℙG∼SBM푛(푑,훾,푥) � 1 푛 |⟨푥, ˆ푥(G)⟩| ⩾ Ω푑,훾(1) � ⩾ 1 − 표(1) 훾2 · 푑 ⩾ 1 exact recovery ℙG∼SBM푛(푑,훾,푥) � ˆ푥(G) ∈ {푥, −푥} � ⩾ 1 − 표(1) 푑 log 푛 � 1 − � 1 − 훾2 � ⩾ 1 Learning mixtures of spherical Gaussians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The Gaussian Mixture Model we consider is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Model 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2 (Mixtures of spherical Gaussians).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 퐷1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' , 퐷푘 be Gaussian distributions on ℝ푑 with covariance Id and means 휇1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' , 휇푘 satisfying ∀푖 ≠ 푗 , ��휇푖 − 휇푗 �� ⩾ Δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Given a set Y = {y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' , y푛} of 푛 samples from the uniform mixture over 퐷1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' , 퐷푘, estimate 휇1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' , 휇푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 1We use bold characters to denote random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 2More general versions of the stochastic block model allow for more than two labels and general edge probabilities depending on the label assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' However, many of the algorithmic phenomena of the general version can in their essence already be observed for the basic version that we consider in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 3At times we may write 푑푛 , 훾푛 to emphasize that these may be functions of 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We write 표(1), 휔(1) for functions tending to zero (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' infinity) as 푛 grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 3 It is known that when the minimum separation is Δ = 표( � log 푘), superpolynomially many samples are required to estimate the means up to small constant error [RV17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Just above this threshold, at separation 푘푂(1/훾) for any constant 훾, there exist efficient algorithms based on the sum-of-squares hierarchy recovering the means up to accuracy 1/poly(푘) [HL18, KSS18, ST21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' In the regime where Δ = 푂( � log 푘) these algorithms yield the same guarantees but require quasipolynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Recently, [LL22] showed how to efficiently recover the means as long as Δ = 푂 � log(푘) 1 2 +푐 � for any constant 푐 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1 Results Stochastic block model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We present here the first (휀, 훿)-differentially private efficient algorithms for exact recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='3 (Private exact recovery of SBM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 푥 ∈ {±1}푛 be balanced4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For any 훾, 푑, 휀, 훿 > 0 satisfying 푑 log 푛 � 1 − � 1 − 훾2 � ⩾ 4 and 훾푑 log 푛 ⩾ Ω � 1 휀2 · log(1/훿) log 푛 + 1 휀 � , there exists an algorithm that, on input G ∼ SBM푛(푑, 훾, 푥), returns ˆ푥(G) ∈ {푥, −푥} with probability 1 − 표(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Moreover, the algorithm is (휀, 훿)-differentially private5 for any input graph, and runs in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For any constant 휀 > 0, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='3 states that (휀, 훿)-differentially private exact re- covery is possible, in polynomial time, already a constant factor close to the non-private threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Previous results [SNVT22] could only achieve comparable guarantees in time 푂� 푛푂(log 푛)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' It is also important to observe that the theorem provides a trade-off between signal-to-noise ratio of the instance (captured by the expression on the left-hand side with 훾, 푑) and the privacy parameter 휀 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' In particular, we distinguish two regimes: for 푑 = 퐶∗ ·log 푛 one can achieve exact recovery with high probability and privacy parameters 훿 = 푛−푂(퐶∗) , 휀 > 퐶−푂(1) ∗ /훾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For √ 푑 ⩾ 휔(log 푛) one can achieve exact recovery with high probability and privacy parameters 휀 = 표(1), 훿 = 푛−휔(1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Further, we present a second, exponential-time, algorithm based on the exponential mechanism [MT07] which improves over the above in two regards: First, it gives pure privacy guarantees, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=', 훿 = 0, and second, provides strong privacy guarantees for a larger range of graph parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' In fact, we will also prove a lower bound which shows that its privacy guarantees are information theoretically optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='6 All hidden constants are absolute and in particular, do not depend on any graph or privacy parameters unless 4A vector 푥 ∈ {±1}푛 is said to be balanced if �푛 푖=1 푥푖 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 5See Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1 for a precise definition of adjacent graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 6Optimality only holds in the "small error" regime, otherwise it is almost optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' See the lower bound for more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 4 stated otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' In what follows we denote by err( ˆ푥, 푥) the minimum of the hamming distance of ˆ푥 and 푥, and the one of − ˆ푥 and 푥, divided by 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4 (Slightly informal, see Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='17 for full version).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 훾 √ 푑 ⩾ Ω(1), 푥 ∈ {±1}푛 be balanced, and 휁 ⩾ exp� −Ω� 훾2푑�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For any 휀 ⩾ Ω � log(1/휁) 훾푑 � , there exists an algorithm which on input G ∼ SBM푛(훾, 푑, 푥) outputs an estimate ˆ푥(G) ∈ {±1}푛 satisfying err( ˆ푥(G), 푥) ⩽ 휁 with probability at least 1 − 휁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' In addition, the algorithm is 휀-private.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Further, we can achieve error Θ � 1/ � log(1/휁) � with the increased success probability 1 − 푒−푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' A couple of remarks are in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' First, our algorithm works across all degree-regimes in the literature and matches known non-private thresholds and rates up to constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='7 In particular, for 훾2푑 = Θ(1), we achieve weak/partial recovery with either constant or exponentially high success probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Recall that the optimal non-private threshold is 훾2푑 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For the regime, where 훾2푑 = 휔(1), it is known that the optimal error rate is exp� −(1 − 표(1))훾2푑� [ZZ16] even non-privately which we match up to constants - here 표(1) denotes a function that tends to zero as 훾2푑 tends to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Moreover, our algo- rithm achieves exact recovery as soon as 훾2푑 = Ω� log 푛� since then 휁 < 1 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' This also matches known non-private threshholds up to constants [ABH15, MNS15a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We remark that [SNVT22] gave an 휀-DP exponential time algorithm which achieved exact recovery and has inverse polynomial success probability in the utility case as long as 휀 ⩾ Ω � log 푛 훾푑 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We recover this result as a special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='8 In fact, their algorithm is also based on the expo- nential mechanism, but their analysis only applies to the setting of exact recovery, while our result holds much more generally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Another crucial difference is that we show how to privatize a known boosting technique frequently used in the non-private setting, allowing us to achieve error guarantees which are optimal up to constant factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' It is natural to ask whether, for a given set of parameters 훾, 푑, 휁 one could hope to obtain better privacy guarantees than Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Our next result is an information theoretic lower bound which shows that our guarantees are almost tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='5 (Slightly informal, see Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='21 for full version).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Suppose there exists an 휀- differentiallyprivate algorithm such that for any balanced 푥 ∈ {±1}푛, on input G ∼ SBM푛(푑, 훾, 푥), outputs ˆ푥(G) ∈ {±1}푛 satisfying ℙ(err( ˆ푥(G), 푥) < 휁) ⩾ 1 − 휂 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then, 휀 ⩾ Ω �log(1/휁) 훾푑 + log(1/휂) 휁푛훾푑 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1) 7For ease of exposition we did not try to optimize these constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 8With slightly worse constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 5 Notice that this is a lower bound for all desired error rates (weak to exact recovery).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For failure probability 휂 = 휁, the lower bound simplifies to 휀 ⩾ Ω � log(1/휁) 훾푑 � and hence matches Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4 up to constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For exponentially small failure probability, 휂 = 푒−푛, it becomes 휀 ⩾ Ω � 1 휁훾푑 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' To compare, using the substitution � log(1/휁) → 휁, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4 requires 휀 ⩾ Ω � 1 휁2훾푑 � in this regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Further, while formally incomparable, this lower bound also suggests that the guar- antees obtained by our efficient algorithm in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='3 might be close to optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' In particular, setting 훿 = 푛−Θ(1), implies that Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='3 achieves (휀, 푛−Θ(1))-private exact recover, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=', 휁 < 1/푛, whenever9 휀 ⩾ Ω �� log 푛 훾푑 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='5 states that in this exact recovery setting 휀 ⩾ Ω � log 푛 훾푑 � is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We leave it as fascinating open questions to bridge the gaps between upper and lower bounds in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Learning mixtures of spherical Gaussians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Our algorithm for privately learning mix- tures of 푘 spherical Gaussians is the first to break the √ 푘 separation barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='6 (Privately learning mixtures of spherical Gaussians).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Consider an instance of Model 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 푡 > 0 be such that Δ ⩾ 푂 �√ 푡푘1/푡� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For 푛 ⩾ Ω� 푘푂(1) · 푑푂(푡)� , 푘 ⩾ (log 푛)1/5 , there exists an algorithm, running in time (푛푑)푂(푡), that outputs vectors ˆ흁1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' , ˆ흁푘 satisfying max ℓ∈[푘] �� ˆ흁ℓ − 휇휋(ℓ) �� 2 ⩽ 푂(푘−12) , with high probability, for some permutation 휋 : [푘] → [푘] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Moreover, for 휀 ⩾ 푘−10 , 훿 ⩾ 푛−10 , the algorithm is (휀, 훿)-differentially private10 for any input 푌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Prior to this work, known differentially private algorithms [KSSU19] could learn a mixture of 푘-spherical Gaussian only if: (1) they were given a ball of radius 푅 containing all centers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='11 and (2) the minimum separation between centers satisfied Δ ⩾ √ 푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='6 overcomes both obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' First, no explicit upper bounds on the means is required (this also means the sample complexity does not depend on 푅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Second, the theorem drastically improves over the separation requirements of [KSSU19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Further, our algorithms not only work for mixtures of Gaussians but for the significantly more general class of mixtures of Poincaré distributions, for which previous private algorithms are not known to work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Concretely, in the high dimensional regime 푘 ⩾ � log 푑, our algorithm recovers the state- of-the-art guarantees provided by non-private algorithms which are based on the sum-of- squares hierarchy [KSS18, HL18, ST21]:12 9in addition to the condition independent of 휀 10Our notion of adjacent databases here is the obvious one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' See Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 11We remark that in [KSSU19] the sample complexity of the algorithm depends on this radius 푅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 12We remark that [LL22] give a polynomial time algorithm for separation Ω(log(푘)1/2+푐) for constant 푐 > 0 in the non-private setting but for a less general class of mixture distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 6 If Δ ⩾ 푘1/푡∗ for some 푡∗ ∈ ℕ, then by choosing 푡 ⩾ Ω(푡∗) the algorithm recovers the centers, up to a 1/poly(푘) error, in time poly(푘, 푑) and using only poly(푘, 푑) samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' If Δ ⩾ Ω( � log 푘) then choosing 푡 = 푂(log 푘) the algorithm recovers the centers, up to a 1/poly(푘) error, in quasi-polynomial time poly(푘푂(푡), 푑푂(푡2)) and using a quasi- polynomial number of samples poly(푘, 푑푂(푡)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For simplicity of exposition we will limit the presentation to mixtures of spherical Gaus- sians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We reiterate that separation Ω( � log 푘) is information-theoretically necessary for algorithms with polynomial sample complexity [RV17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 2 Techniques We present here our general tools for designing efficient private estimation algorithms in the high-dimensional setting whose statistical guarantees almost match those of the best know non-private algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The algorithms we design have the following structure in common: First, we solve a convex optimization problem with constraints and objective function depending on our input 푌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Second, we round the optimal solution computed in the first step to a solution 푋 for the statistical estimation problem at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We organize our privacy analyses according to this structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' In order to analyze the first step, we prove a simple sensitivity bound for strongly convex optimization problems, which bounds the ℓ2-sensitivity of the optimal solution in terms of a uniform sensitivity bound for the objective function and the feasible region of the optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For bounded problems –such as recovery of stochastic block models– we use this sensitivity bound, in the second step, to show that introducing small additive noise to standard rounding algorithms is enough to achieve privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For unbounded problems –such as learning GMMs– we use this sensitivity bound to show that on adjacent inputs, either most entries of 푋 only change slightly, as in the bounded case, or few entries vary significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We then combine different privacy techniques to hide both type of changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Privacy from sensitivity of strongly convex optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Before illustrating our techniques with some example, it is instructive to explicit our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Here we have a set of inputs 풴 and a family of strongly convex functions ℱ (풴) and convex sets 풦(풴) parametrized by these inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The generic non-private algorithm based on convex optimization we consider works as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Compute ˆ푋 := argmin푋∈풦(푌) 푓푌(푋) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Round ˆ푋 into an integral solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 7 For an estimation problem, a distributional assumption on 풴 is made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then one shows how, for typical inputs Y sampled according to that distribution, the above scheme recovers the desired structured information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We can provide a privatized version of this scheme by arguing that, under reasonable assumptions on ℱ (푌) and 풦(풴), the output of the function argmin푋∈풦(푌) 푓푌(푋) has low ℓ2-sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The consequence of this crucial observation is that one can combine the rounding step 2 with some standard privacy mechanism and achieve differential privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' That is, the second step becomes: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Add random noise N and round ˆ푋 + N into an integral solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Our sensitivity bound is simple, yet it generalizes previously known bounds for strongly convex optimization problems (we provide a detailed comparison later in the section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For adjacent 푌, 푌′ ∈ 풴 , it requires the following ingredients: (i) For each 푋 ∈ 풦(푌) ∩ 풦(푌′) it holds | 푓푌(푋) − 푓푌′(푋)| ⩽ 훼;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' (ii) For each 푋 ∈ 풦(푌) its projection 푋′ onto 풦(푌′)∩풦(푌) satisfies | 푓푌(푋)− 푓푌(푋′)| ⩽ 훼 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Here we think of 훼 as some small quantity (relatively to the problem parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Notice, we may think of (i) as Lipschitz-continuity of the function 푔(푌, 푋) = 푓푌(푋) with respect to 푌 and of (ii) as a bound on the change of the constrained set on adjacent inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' In fact, these assumptions are enough to conclude low ℓ2 sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' If ˆ푋 and ˆ푋′ are the outputs of the first step on inputs 푌, 푌′, then there exists 푋 ∈ 풦(푌) ∩ 풦(푌′) such that | 푓푌( ˆ푋) − 푓푌(푋)| + | 푓푌′( ˆ푋′) − 푓푌′(푋)| ⩽ 푂(훼) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' By strong convexity of 푓푌 , 푓푌′ this implies ��� ˆ푋 − 푋 ��� 2 2 + ��� ˆ푋′ − 푋 ��� 2 2 ⩽ 푂(훼) which ultimately means ∥ ˆ푋 − ˆ푋′∥2 2 ⩽ 푂(훼).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Thus, starting from our assumptions on the point-wise distance of 푓푌 , 푓푌′ we were able to conclude low ℓ2-sensitivity of our output!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' A simple application: weak recovery of stochastic block models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The ideas introduced above, combined with existing algorithms for weak recovery of stochastic block mod- els, immediately imply a private algorithm for the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' To illustrate this, consider Model 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1 with parameters 훾2푑 ⩾ 퐶, for some large enough constant 퐶 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 푥 ∈ {±1}푛 be balanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Here the input 푌 is an 푛-by-푛 matrix corresponding to the rescaled centered adjacency matrix of the graph: 푌푖푗 = � 1 훾푑 � 1 − 푑 푛 � if 푖푗 ∈ 퐸(퐺) − 1 훾푛 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 8 The basic semidefinite program [GV16, MS16] can be recast as the strong constrained optimization question of finding the orthogonal projection of the matrix 푌 onto the set 풦 := {푋 ∈ ℝ푛×푛 | 푋 ⪰ 0 , ∥푋∥∞ ⩽ 1/푛} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' That is: ˆ푋 := argmin푋∈풦 ∥푌 − 푋∥2 F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' It is a standard fact that, if our input was G ∼ SBM푛(푑, 훾, 푥), then with high probability 푋(G) = argmin푋∈풦 푓푌(G)(푋) would have leading eigenvalue, eigenvector pair satisfying 휆1(G) ⩾ 1 − 푂(1/훾2푑) , ⟨푣1(G), 푥/∥푥∥⟩2 ⩾ 1 − 푂� 1/훾2푑� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' This problem fits perfectly the description of the previous paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' In fact, it stands to reason that the closeness of the projections ˆ푋, ˆ푋′ of inputs 푌, 푌′ should be proportional to the distance between 푌 and 푌′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Our sensitivity argument above formalizes this simple intuition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Concretely, observe that the constrained set 풦 is fixed and that for each 푋 ∈ 풦 it holds | 푓푌(푋) − 푓푌′(푋)| ⩽ 푂� ∥푌 − 푌′∥2 F + ∥푌 − 푌′∥1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' It is easy to see that on adjacent input we have ∥푌 − 푌′∥2 F + ∥푌 − 푌′∥1 ⩽ 푂(1/푛훾푑) and thus this immediately yields ∥ ˆ푋 − ˆ푋′∥2 F ⩽ 푂(1/푛훾푑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The rounding step is now straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Using the Gaussian mechanism we return the leading eigenvector of ˆ푋 + N where N ∼ 푁 � 0, 1 푛훾푑 · log(1/훿) 휀2 �푛×푛 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' This matrix has Frobei- nus norm significantly larger than ˆ푋 but its spectral norm is only ∥N∥ ⩽ � 푛 log(1/훿) 휀 � 1 푛훾푑 ⩽ 1 휀 · � log(1/훿) 훾푑 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Thus by standard linear algebra, for typical instances G ∼ SBM푛(푑, 훾, 푥), the leading eigenvector of ˆ푋(G) + N will be highly correlated with the true community vector 푥 whenever the average degree 푑 is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' In conclusion, a simple randomized rounding step is enough!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1 (From weak recovery to exact recovery).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' In the non-private setting, given a weak recovery algorithm for the stochastic block model, one can use this as an initial estimate for a boosting procedure based on majority voting to achieve exact recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We show that this can be done privately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' See Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' An advanced application: learning mixtures of Gaussians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' In the context of stochastic block models our argument greatly benefited from two key properties: first, on adjacent inputs the difference ∥푌 − 푌′∥F was bounded;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' and second, the convex set 풦 was fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' In the context of learning mixtures of spherical Gaussians as in Model 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2, both this properties are not satisfied (notice how one of this second properties would be satisfied assuming bounded centers!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' So additional ingredients are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 9 The first observation, useful to overcome the first obstacle, is that before finding the centers, one can first find the 푛-by-푛 membership matrix 푊(푌) where 푊(푌)푖푗 = 1 if 푖, 푗 where sampled from the same mixture component and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The advantage here is that, on adjacent inputs, ∥푊(푌) − 푊(푌′)∥2 F ⩽ 2푛/푘 and thus one recovers the first property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='13 Here early sum-of-squares algorithms for the problem [HL18, KSS18] turns out to be convenient as they rely on minimizing the function ∥푊 ∥2 F subject to the following system of polynomial inequalities in variables 푧11 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' , , 푧1푘 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' , 푧푛푘, with 푊푖푗 = � ℓ 푧푖ℓ 푧푗ℓ for all 푖, 푗 ∈ [푛] and a parameter 푡 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' \uf8f1\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f2 \uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f3 푧2 푖ℓ = 푧푖ℓ ∀푖 ∈ [푛] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' ℓ ∈ [푘] (indicators) � ℓ∈[푘] 푧푖ℓ ⩽ 1 ∀푖 ∈ [푛] (cluster membership) 푧푖ℓ · 푧푖ℓ′ = 0 ∀푖 ∈ [푛] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' ℓ ∈ [푘] (unique membership) � 푖 푧푖ℓ = 푛/푘 ∀ℓ ∈ [푘] (size of clusters) 휇′ ℓ = 푘 푛 � 푖 푧푖ℓ · 푦푖 ∀ℓ ∈ [푘] (means of clusters) 푘 푛 � 푖 푧푖ℓ ⟨푦푖 − 휇′ ℓ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 푢⟩2푡 ⩽ (2푡)푡 · ∥푢∥푡 2 ∀푢 ∈ ℝ푑 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' ℓ ∈ [푘] (subgaussianity of 푡-moment) \uf8fc\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8fd \uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8fe (풫(푌)) For the scope of this discussion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='14 we may disregard computational issues and assume we have access to an algorithm returning a point from the convex hull 풦(푌) of all solutions to our system of inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='15 Here each indicator variable 푧푖ℓ ∈ {0, 1} is meant to indicate whether sample 푦푖 is believed to be in cluster 퐶ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' In the non-private settings, the idea behind the program is that –for typical Y sampled according to Model 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2 with minimum separation Δ ⩾ 푘1/푡√ 푡– any solution푊(Y) ∈ 풦(Y) is close to the ground truth matrix 푊∗(Y) in Frobenius norm: ∥푊(Y) − 푊∗(Y)∥2 F ⩽ 1/poly(푘) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Each row 푊(Y)푖 may be seen as inducing a uniform distribution over a subset of Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Thus, combining the above bound with the fact that subgaussian distributions at small total variation distance have means that are close, we can conclude the algorithm recovers the centers of the mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' While this program suggests a path to recover the first property, it also possesses a fatal flaw: the projection 푊′ of 푊 ∈ 풦(푌) onto 풦(푌) ∩ 풦(푌′) may be far in the sense that 13Notice for typical inputs Y from Model 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2 one expect ∥푊(Y)∥F ≈ 푛2/푘 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 14While this is far from being true, it turns out that having access to a pseudo-distribution satisfying 풫(푌) is enough for our subsequent argument to work, albeit with some additional technical work required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 15We remark that a priori it is also not clear how to encode the subgaussian constraint in a way that we could recover a degree-푡 pseudo-distribution satisfying 풫(푌) in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' By now this is well understood, we discuss this in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 16More generally, we may think of a vector 푣 ∈ ℝ푛 as the vector inducing the distribution given by 푣/∥푣∥1 onto the set 푌 of 푛 elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 10 |∥푊 ∥2 F − ∥푊′∥2 F| ⩾ Ω(∥푊 ∥2 F + ∥푊′∥2 F) ⩾ Ω(푛2/푘) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The reason behind this phenomenon can be found in the constraint � 푖 푧푖ℓ = 푛/푘 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The set indicated by the vector (푧1ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' , 푧푛ℓ) may be subgaussian in the sense of 풫(푌) for input 푌 but, upon changing a single sample, this may no longer be true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We work around this obstacle in two steps: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We replace the above constraint with � 푖 푧푖ℓ ⩽ 푛/푘 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We compute ˆ푊 := argmin푊 solving 풫(푌)∥퐽 − 푊 ∥2 F , where 퐽 ∈ ℝ푛×푛 is the all-ones matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='17 The catch now is that the program is satisfiable for any input푌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Moreover, we can guarantee property (ii) (required by our sensitivity argument) for 훼 ⩽ 푂(푛/푘), since we can obtain 푊′ ∈ 풦(푌)∩풦(푌′) simply zeroing out the row/column in 푊 corresponding to the sample differing in 푌 and 푌′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then for typical inputs Y, the correlation with the true solution is guaranteed by the new strongly convex objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' From low sensitivity of the indicators to low sensitivity of the estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For adjacent inputs 푌, 푌′ let ˆ푊, ˆ푊′ be respectively the matrices computed by the above strongly convex programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Our discussion implies that, applying our sensitivity bound, we can show ∥ ˆ푊 − ˆ푊′∥2 F ⩽ 푂(푛/푘) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The problem is that simply applying a randomized rounding approach here cannot work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The reason is that even tough the vector ˆ푊푖 induces a subgaussian distribution, the vector ˆ푊푖 + 푣 for 푣 ∈ ℝ푛, might not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Without the subgaussian constraint we cannot provide any meaningful utility bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' In other words, the root of our problem is that there exists heavy-tailed distributions that are arbitrarily close in total variation distance to any given subgaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' On the other hand, our sensitivity bound implies ∥ ˆ푊 − ˆ푊′∥2 1 ⩽ 표(∥ ˆ푊 ∥1) and thus, all but a vanishing fraction of rows 푖 ∈ [푛] must satisfy ∥ ˆ푊푖 − ˆ푊′ 푖 ∥1 ⩽ 표(∥ ˆ푊푖∥1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For each row 푖 , let 휇푖 , 휇′ 푖 be the means of the distributions induced respectively by ˆ푊푖 , ˆ푊′ 푖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We thus find ourselves in the following settings: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For a set of (1 − 표(1)) · 푛 good rows ��휇푖 − 휇′ 푖 �� 2 ⩽ 표(1) , 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For the set ℬ of remaining bad rows, the distance ��휇푖 − 휇′ 푖 �� 2 may be unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We hide differences of the first type as follows: pick a random subsample 퓢 of [푛] of size 푛푐, for some small 푐 > 0, and for each picked row use the Gaussian mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The subsampling step is useful as it allows us to decrease the standard deviation of the entry-wise random noise by a factor 푛1−푐 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We hide differences of the second type as follows: use the classic high dimensional (휀, 훿)-private histogram learner on 퓢 and for the 푘 largest bins of highest count privately return their average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The crux of the argument here is that the cardinality of ℬ ∩ 퓢 is 17We remark that for technical reasons our function in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1 will be slightly different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We do not discuss it here to avoid obfuscating our main message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 11 sufficiently small that the privacy guarantees of the histogram learner can be extended even for inputs that differ in |ℬ ∩ 퓢| many samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Finally, standard composition arguments will guarantee privacy of the whole algo- rithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Comparison with previous works on empirical risk minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Results along the lines of the sensitivity bound described at the beginning of the section (see Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1 for a formal statement) have been extensively used in the context of empirical risk minimization [CMS11, KST12, SCS13, BST14, WYX17, MSVV21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Most results focus on the special case of unconstrained optimization of strongly convex functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' In contrast, our sensitivity bound applies to the significantly more general settings where both the objective functions and the constrained set may depend on the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='18 Most notably for our settings of interest, [CMS11] studied unconstrained optimization of (smooth) strongly convex functions depending on the input, with bounded gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='We recover such a result for 푋′ = 푋 in (ii) In [MSVV21], the authors considered constraint optimization of objective functions where the domain (but not the function) may depend on the input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' They showed how one can achieve differential privacy while optimize the desired objective function by randomly perturbing the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' It is important to remark that, in [MSVV21], the notion of utility is based on the optimization problem (and their guarantees are tight only up to logarithmic factors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' In the settings we consider, even in the special case where 푓 does not depend on the input, this notion of utility may not correspond to the notion of utility required by the estimation problem, and thus, the corresponding guarantees may turn out to be too loose to ensure the desired error bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Exponential time pure-DP algorithm for SBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Our exponential time algorithm is based on the exponential mechanism [MT07].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' In particular, for a given graph 퐺, recall that 푌 = 1 훾푑 � 퐴(퐺) − 푑 푛 퐽� , where 퐴(퐺) is the adjacency matrix of 퐺 and 퐽 the all-ones matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Define the function 푠 : {±1}푛 → ℝ as 푠(푥) = ⟨푥, 푌푥⟩ and Δ = 2 훾푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' In privacy terms, these are called the score function and the sensitivity - the maximum amount 푠 can change on adjacent graphs - which can be readily seen to be 2 훾푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The exponential mechanism then amounts to outputting a sample from the distribution with density 푝(푥) ∝ exp� 휀 2Δ푠(푥)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1) Standard arguments show that this procedure is 휀-private.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Note, that it is well-known that if G ∼ SBM푛(푑, 훾, 푥∗) and 훾2푑 is larger than some universal constant, 푌 is close to 1 푛 푥∗(푥∗)⊤ in cut-norm (or (∞ → 1)-norm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' That is, the quadratic form 푌 − 1 푛 푥∗(푥∗)⊤ is 18The attentive reader may argue that one could cast convex optimization over a constrained domain as unconstrained optimization of a new convex function with the appropriate penalty terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' In practice however, this turns out to be hard to do for constraints such as Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 12 close to zero over the hypercube [GV16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' It follows, that the maximizer of score function 푠 over the hypercube is close to 1 √푛 푥∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' It is not too hard to show that with exponentially high probability (in 푛), this remains true also for samples from the above distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' On an intuitive level this follows since we assign exponentially larger mass to points achieving comparable scores as the maximizer than to points achieving smaller scores (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' While this algorithm matches known non-private thresholds and rates up to constants and has close to optimal privacy guaratnees (see the discusion in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1), there are several obstacles to making it efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We discuss several approaches: First, one could try to sample from the distribution in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1) directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Note that this corresponds to an Ising model over the hypercube with interaction matrix 퐽 ≔ 휀 훿푌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' However, known samplers, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' [EKZ22, KLR22], require strong assumptions on the spectrum of 퐽 which are not satisfied in our setting - in particular, 퐽 could have arbitrarly many eigenvalues of magnitude larger than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' A second approach would be to relax the support of the distribution to all positive semi-definite matrices with diagonal entries equal to 1/푛 - similar to the set 풦 considered for our approximate DP algorithm - and the score function to the inner product of 푌 with such matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Although such "convexification" techniques of the exponential mechanism have recently seen success in the design of pure-DP algorithms [HKM22] and this particular relaxation is known to work in the non-private setting [GV16, FC20], it fails in this case: The volume of the set of matrices achieving large enough score is smaller by a factor of exp� −푛2� than the set of all feasible matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Hence, the exponential boost by the reweighing of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1) is not enough to ensure outputting such a candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' A third strategy would be the following: In the non-private setting, for the restricted regime of 훾2푑 ⩾ 퐶 log 푛, for a large enough constant 퐶 > 0, standard matrix concentration bounds show that PCA can recover the label of a large constant fraction of the vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' However, known lower bounds for pure-DP PCA algorithms [KT13], prevent us from recovering this result in the private setting: In particular, define two matrices to be adjacent if there difference has spectral norm at most 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' In this setting, any 휀-DP algorithm, which outputs a vector achieving constant correlation with the top eigenvector of an 푛 × 푛 input matrix needs to have spectral norm at least Ω� 푛 휀 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Translated to our scaling used above, this would mean ∥푌∥ ⩾ Ω(푛훾푑 휀 ), whereas we have ∥푌∥ ⩽ 푂(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Information theoretic privacy lower bounds for stochastic block models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Our informa- tion theoretic lower bound for stochastic block models is based on the following idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Sup- pose we have an 휀-differentially private exact recovery algorithm of SBM such that, over the randomness of the algorithm and the input G ∼ SBM푛(푑, 훾, 푥), the algorithm outputs ˆ푥(G) ∈ {±푥} with probability at least 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Note for any 푥 ∈ {±1}푛, SBM푛(푑, 훾, 푥) is just a product distribution of �푛 2 � Bernoulli distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Fixing an arbitrary balanced 푦 ∈ {±1}푛, there exist balanced 푦1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' , 푦푛 ∈ {±1}푛 such that Ham(푦, 푦푖) = 2 for 푖 ∈ [푛].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For each 푖 ∈ [푛], one may find a coupling 휔푖 of distributions SBM푛(푑, 훾, 푦) and SBM푛(푑, 훾, 푦푖) such 13 that, if(G, G′) ∼ 휔푖 then G and G′ typicallydifferbyonly 푂(훾푑)edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then bythe assump- tion our algorithm is 휀-differentially private, it follows that, on input G ∼ SBM푛(푑, 훾, 푦), our private algorithm outputs ±푦푖 with probability at least 푒−휀·푂(훾푑)· 2 3 for each 푖 ∈ [푛].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Since the sum of probabilities of disjoint events does not exceed one, we get 푛 · 푒−푂(휀훾푑) · 2 3 ⩽ 1, which implies 휀 ⩾ Ω(log 푛 훾푑 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 3 Preliminaries We use boldface characters for random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We hide multiplicative factors logarithmic in 푛 using the notation ˜푂(·) , ˜Ω(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Similarly, we hide absolute constant multiplicative factors using the standard notation 푂(·) , Ω(·) , Θ(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Often times we use the letter 퐶 do denote universal constants independent of the parameters at play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We write 표(1), 휔(1) for functions tending to zero (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' infinity) as 푛 grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We say that an event happens with high probability if this probability is at least 1 − 표(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Throughout the paper, when we say "an algorithm runs in time 푂(푞)" we mean that the number of basic arithmetic operations involved is 푂(푞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' That is, we ignore bit complexity issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Vectors, matrices, tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We use Id푛 to denote the 푛-by-푛 dimensional identity matrix, 퐽푛 ∈ ℝ푛×푛 the all-ones matrix and 0푛 , 1푛 ∈ ℝ푛 to denote respectively the zero and the all-ones vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' When the context is clear we drop the subscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For matrices 퐴, 퐵 ∈ ℝ푛×푛 we write 퐴 ⪰ 퐵 if 퐴 − 퐵 is positive semidefinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For a matrix 푀, we denote its eigenvalues by 휆1(푀) , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' , 휆푛(푀), we simply write 휆푖 when the context is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We denote by ∥푀∥ the spectral norm of 푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We denote by ℝ푑⊗푡 the set of real-valued order-푡 tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' for a 푑 × 푑 matrix 푀, we denote by 푀⊗푡 the 푡-fold Kronecker product 푀 ⊗ 푀 ⊗ · · · ⊗ 푀 �������������������������������������� 푡 times .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We define the flattening, or vectorization, of 푀 to be the 푑푡-dimensional vector, whose entries are the entries of 푀 appearing in lexicographic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' With a slight abuse of notation we refer to this flattening with 푀, ambiguities will be clarified form context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We denote by 푁 � 0, 휎2�푑⊗푡 the distribution over Gaussian tensors with 푑푡 entries with standard deviation 휎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Given 푢, 푣 ∈ {±1}푛, we use Ham(푢, 푣) := �푛 푖=1 1[푢푖 ≠ 푣푖] to denote their Hamming distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Given a vector 푢 ∈ ℝ푛, we let sign(푢) ∈ {±1}푛 denote its sign vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' A vector 푢 ∈ {±1}푛 is said to be balanced if �푛 푖=1 푢푖 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We consider graphs on 푛 vertices and let 풢푛 be the set of all graphs on 푛 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For a graph 퐺 on 푛 vertices we denote by 퐴(퐺) ∈ ℝ푛×푛 its adjacency matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' When the context is clear we simply write 퐴 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 푉(퐺) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 퐸(퐺)) denote the vertex (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' edge) set of graph 퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Given two graphs 퐺, 퐻 on the same vertex set 푉, let 퐺 \\ 퐻 := (푉, 퐸(퐺) \\ 퐻(퐺)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Given a graph 퐻, 퐻′ ⊆ 퐻 means 퐻′ is a subgraph of 퐻 such that 푉(퐻′) = 푉(퐻) and 퐸(퐻) ⊆ 퐸(퐻).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The Hamming distance between two graphs 퐺, 퐻 is defined to be the size of the symmetric difference between their edge sets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Ham(퐺, 퐻) := |퐸(퐺)△퐸(퐻)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1 Differential privacy In this section we introduce standard notions of differential privacy [DMNS06].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1 (Differential privacy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' An algorithm ℳ : 풴 → 풪 is said to be (휀, 훿)- differentially private for 휀, 훿 > 0 if and only if, for every 푆 ⊆ 풪 and every neighboring datasets 푌, 푌′ ∈ 풴 we have ℙ[ℳ(푌) ∈ 푆] ⩽ 푒휀 · ℙ[ℳ(푌′) ∈ 푆] + 훿 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' To avoid confusion, for each problem we will exactly state the relevant notion of neigh- boring datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Differential privacy is closed under post-processing and composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2 (Post-processing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' If ℳ : 풴 → 풪 is an (휀, 훿)-differentially private algorithm and ℳ′ : 풴 → 풵 is any randomized function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then the algorithm ℳ′(ℳ(푌)) is (휀, 훿)-differentially private.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' In order to talk about composition it is convenient to also consider DP algorithms whose privacy guarantee holds only against subsets of inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='3 (Differential Privacy Under Condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' An algorithm ℳ : 풴 → 풪 is said to be (휀, 훿)-differentially private under condition Ψ (or (휀, 훿)-DP under condition Ψ) for 휀, 훿 > 0 if and only if, for every 푆 ⊆ 풪 and every neighboring datasets 푌, 푌′ ∈ 풴 both satisfying Ψ we have ℙ[ℳ(푌) ∈ 푆] ⩽ 푒휀 · ℙ[ℳ(푌′) ∈ 푆] + 훿 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' It is not hard to see that the following composition theorem holds for privacy under condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4 (Composition for Algorithm with Halting, [KMV22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let ℳ1 : 풴 → 풪1 ∪ {⊥} , ℳ2 : 풪1 × 풴 → 풪2 ∪ {⊥} , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' , ℳ푡 : 풪푡−1 × 풴 → 풪푡 ∪ {⊥} be algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Furthermore, let ℳ denote the algorithm that proceeds as follows (with 풪0 being empty): For 푖 = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' , 푡 compute 표푖 = ℳ푖(표푖−1, 푌) and, if 표푖 = ⊥, halt and output ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Finally, if the algorithm has not halted, then output 표푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Suppose that: For any 1 ⩽ 푖 ⩽ 푡, we say that 푌 satisfies the condition Ψ푖 if running the algorithm on 푌 does not result in halting after applying ℳ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' , ℳ푖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' ℳ1 is (휀1, 훿1)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' ℳ푖 is (휀푖, 훿푖)-DP (with respect to neighboring datasets in the second argument) under condition Ψ푖−1 for all 푖 = {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' , 푡} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then ℳ is (� 푖 휀푖, � 푖 훿푖)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1 Basic differential privacy mechanisms The Gaussian and the Laplace mechanism are among the most widely used mechanisms in differential privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' They work by adding a noise drawn from the Gaussian (respectively Laplace) distribution to the output of the function one wants to privatize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The magnitude of the noise depends on the sensitivity of the function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='5 (Sensitivity of function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 푓 : 풴 → ℝ푑 be a function, its ℓ1-sensitivity and ℓ2-sensitivity are respectively Δ 푓 ,1 := max 푌 ,푌′∈풴 푌 ,푌′ are adjacent ��푓 (푌) − 푓 (푌′) �� 1 Δ 푓 ,2 := max 푌 ,푌′∈풴 푌 ,푌′ are adjacent �� 푓 (푌) − 푓 (푌′) �� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For function with bounded ℓ1-sensitivity the Laplace mechanism is often the tool of choice to achieve privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='6 (Laplace distribution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The Laplace distribution with mean 휇 and parameter 푏 > 0, denoted by Lap(휇, 푏), has PDF 1 2푏 푒−|푥−휇|/푏 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let Lap(푏) denote Lap(0, 푏).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' A standard tail bound concerning the Laplace distribution will be useful throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Fact 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='7 (Laplace tail bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 풙 ∼ Lap(휇, 푏).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then, ℙ � |풙 − 휇| > 푡 � ⩽ 푒−푡/푏 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The Laplace distribution is useful for the following mechanism Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='8 (Laplace mechanism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 푓 : 풴 → ℝ푑 be any function with ℓ1-sensitivity at most Δ 푓 ,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then the algorithm that adds Lap � Δ푓 ,1 휀 �⊗푑 to 푓 is (휀, 0)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' It is also useful to consider the "truncated" version of the Laplace distribution where the noise distribution is shifted and truncated to be non-positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='9 (Truncated Laplace distribution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The (negatively) truncated Laplace distri- bution w with mean 휇 and parameter 푏 on ℝ, denoted by tLap(휇, 푏), is defined as Lap(휇, 푏) conditioned on the value being non-positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='10 (Truncated Laplace mechanism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 푓 : 풴 → ℝ be any function with ℓ1- sensitivity at most Δ 푓 ,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then the algorithm that adds tLap � −Δ 푓 ,1 � 1 + log(1/훿) 휀 � , Δ 푓 ,1/휀 � to 푓 is (휀, 훿)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The following tail bound is useful when reasoning about truncated Laplace random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 16 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='11 (Tail bound truncated Laplace).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Suppose 휇 < 0 and 푏 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 풙 ∼ tLap(휇, 푏).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then,for 푦 < 휇 we have that ℙ � 풙 < 푦 � ⩽ 푒(푦−휇/푏) 2 − 푒휇/푏 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' In constrast, when the function has bounded ℓ2-sensitivity, the Gaussian mechanism provides privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='12 (Gaussian mechanism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 푓 : 풴 → ℝ푑 be any function with ℓ2-sensitivity at most Δ 푓 ,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 0 < 휀 , 훿 ⩽ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then the algorithm that adds 푁 � 0, Δ2 푓 ,2·2 log(2/훿) 휀2 Id � to 푓 is (휀, 훿)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2 Private histograms Here we present a classical private mechanism to learn a high dimensional histogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='13 (High-dimensional private histogram learner, see [KV18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 푞, 푏 , 휀 > 0 and 0 < 훿 < 1/푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let {퐼푖}∞ 푖=−∞ be a partition of ℝ into intervals of length 푏, where 퐼푖 := � 푥 ∈ ℝ �� 푞 + (푖 − 1) · 푏 ⩽ 푥 < 푞 + 푖 · 푏 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Consider the partition of ℝ푑 into sets � 퐵푖1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=',푖푑 �∞ 푖1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=',푖푑=1 where 퐵푖1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=',푖푑 := � 푥 ∈ ℝ푑 �� ∀푗 ∈ [푑] , 푥푗 ∈ 퐼푖푗 � Let 푌 = � 푦1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' , 푦푛 � ⊆ ℝ푑 be a database of 푛 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For each 퐵푖1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=',푖푑, let 푝푖1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=',푖푑 = 1 푛 ��� 푗 ∈ [푛] �� 푦푗 ∈ 퐵푖1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=',푖푑 ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For 푛 ⩾ 8 휀훼 ·log 2 훿훽 , there exists an efficient (휀, 훿)-differentially private algorithm that returns ˆ풑1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=',1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' , ˆ풑푖1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=',푖푑, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' satisfying ℙ � max 푖1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=',푖푑∈ℕ|푝푖1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=',푖푑 − ˆ풑푖1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=',푖푑| ⩾ 훼 � ⩽ 훽 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We consider the following algorithm, applied to each 푖1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' , 푖푑 ∈ ℕ on input 푌: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' If 푝푖1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=',푖푑 = 0 set ˆ푝푖1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=',푖푑 = 0 , otherwise let ˆ풑푖1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=',푖푑 = 푝푖1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=',푖푑 + 흉 where 흉 ∼ Lap� 0, 2 푛휀 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' If ˆ풑푖1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=',푖푑 ⩽ 3 log(2/훿) 휀푛 set ˆ풑푖1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=',푖푑 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' First we argue utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' By construction we get ˆ풑푖1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=',푖푑 = 0 whenever 푝푖1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=',푖푑 = 0, thus we may focus on non-zero 푝푖1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=',푖푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' There are at most 푛 non zero 푝푖1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=',푖푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' By choice of 푛, 훿 and by Fact 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='7 the maximum over 푛 independent trials 흉 ∼ Lap� 0, 2 푛휀 � is bounded by 훼 in absolute value with probability at least 훽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' It remains to argue privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 푌 = � 푦1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' , 푦푛 � , 푌′ = � 푦′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' , 푦′ 푛 � be adjacent databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For 푖1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' , 푖푑 ∈ ℕ, let 푝푖1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=',푖푑 = ��� 푗 ∈ [푛] �� 푦푗 ∈ 퐵푖1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=',푖푑 ��� 17 푝′ 푖1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=',푖푑 = ��� � 푗 ∈ [푛] ��� 푦′ 푗 ∈ 퐵푖1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=',푖푑 ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Since 푌, 푌′ are adjacent there exists only two set of indices ℐ := {푖1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' , 푖푑} and 풥 := � 푗1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' , 푗푑 � such that 푝ℐ ≠ 푝′ ℐ and 푝풥 ≠ 푝′ 풥 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Assume without loss of generality 푝ℐ > 푝′ ℐ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then it must be 푝ℐ = 푝′ ℐ +1/푛 and 푝풥 = 푝′ 풥 −1/푛 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Thus by the standard tail bound on the Laplace distribution in Fact 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='7 and by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='8, we immediately get that the algorithm is (휀, 훿)-differentially private.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2 Sum-of-squares and pseudo-distributions We introduce here the sum-of-squares notion necessary for our private algorithm learning mixtures of Gaussians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We remark that these notions are not needed for Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 푤 = (푤1, 푤2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' , 푤푛) be a tuple of 푛 indeterminates and let ℝ[푤] be the set of polynomials with real coefficients and indeterminates 푤, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' , 푤푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We say that a polynomial 푝 ∈ ℝ[푤] is a sum-of-squares (sos) if there are polynomials 푞1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' , 푞푟 such that 푝 = 푞2 1 + · · ·+ 푞2 푟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1 Pseudo-distributions Pseudo-distributions are generalizations of probability distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We can represent a discrete (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=', finitely supported) probability distribution over ℝ푛 by its probability mass function 퐷 : ℝ푛 → ℝ such that 퐷 ⩾ 0 and � 푤∈supp(퐷) 퐷(푤) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Similarly, we can describe a pseudo-distribution by its mass function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Here, we relax the constraint 퐷 ⩾ 0 and only require that 퐷 passes certain low-degree non-negativity tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Concretely, a level-ℓ pseudo-distribution is a finitely-supported function 퐷 : ℝ푛 → ℝ such that � 푤 퐷(푤) = 1 and � 푤 퐷(푤)푓 (푤)2 ⩾ 0 for every polynomial 푓 of degree at most ℓ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' (Here, the summations are over the support of 퐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=') A straightforward polynomial- interpolation argument shows that every level-∞-pseudo distribution satisfies 퐷 ⩾ 0 and is thus an actual probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We define the pseudo-expectation of a function 푓 on ℝ푑 with respect to a pseudo-distribution 퐷, denoted ˜피퐷(푤) 푓 (푤), as ˜피퐷(푤) 푓 (푤) = � 푤 퐷(푤)푓 (푤) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1) The degree-ℓ moment tensor of a pseudo-distribution 퐷 is the tensor 피퐷(푤)(1, 푤1, 푤2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' , 푤푛)⊗ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' In particular, the moment tensor has an entry corresponding to the pseudo-expectation of all monomials of degree at most ℓ in 푤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The set of all degree-ℓ moment tensors of probability distribution is a convex set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Similarly, the set of all degree-ℓ moment tensors of degree 푑 pseudo-distributions is also convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Key to the algorithmic utility of pseudo-distributions is the fact that while there can be no efficient separation oracle for the convex set of all degree-ℓ moment tensors of an actual probability distribution, there’s a separation oracle running in time 푛푂(ℓ) for the convex set of the degree-ℓ moment tensors of all level-ℓ pseudodistributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 18 Fact 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='14 ([Sho87, Par00, Nes00, Las01]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For any 푛, ℓ ∈ ℕ, the following set has a 푛푂(ℓ)-time weak separation oracle (in the sense of [GLS81]): � ˜피퐷(푤)(1, 푤1, 푤2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' , 푤푛)⊗푑 | degree-d pseudo-distribution 퐷 over ℝ푛� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2) This fact, together with the equivalence of weak separation and optimization [GLS81] allows us to efficiently optimize over pseudo-distributions (approximately)—this algo- rithm is referred to as the sum-of-squares algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The level-ℓ sum-of-squares algorithm optimizes over the space of all level-ℓ pseudo- distributions that satisfy a given set of polynomial constraints—we formally define this next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='15 (Constrained pseudo-distributions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 퐷 be a level-ℓ pseudo-distribution over ℝ푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 풜 = { 푓1 ⩾ 0, 푓2 ⩾ 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' , 푓푚 ⩾ 0} be a system of 푚 polynomial inequality con- straints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We say that 퐷 satisfies the system of constraints 풜 at degree 푟, denoted 퐷 푟 풜, if for ev- ery 푆 ⊆ [푚] and every sum-of-squares polynomial ℎ with deg ℎ + � 푖∈푆 max{deg 푓푖, 푟} ⩽ ℓ, ˜피퐷ℎ · � 푖∈푆 푓푖 ⩾ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We write 퐷 풜 (without specifying the degree) if 퐷 0 풜 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Furthermore, we say that 퐷 푟 풜 holds approximately if the above inequalities are satisfied up to an error of 2−푛ℓ · ∥ℎ∥ · � 푖∈푆∥ 푓푖∥, where ∥·∥ denotes the Euclidean norm19 of the coefficients of a polynomial in the monomial basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We remark that if 퐷 is an actual (discrete) probability distribution, then we have 퐷 풜 if and only if 퐷 is supported on solutions to the constraints 풜.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We say that a system 풜 of polynomial constraints is explicitly bounded if it contains a constraint of the form {∥푤∥2 ⩽ 푀}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The following fact is a consequence of Fact 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='14 and [GLS81], Fact 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='16 (Efficient Optimization over Pseudo-distributions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' There exists an (푛 + 푚)푂(ℓ)- time algorithm that, given any explicitly bounded and satisfiable system20 풜 of 푚 polynomial constraints in 푛 variables, outputs a level-ℓ pseudo-distribution that satisfies 풜 approximately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2 Sum-of-squares proof Let 푓1, 푓2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' , 푓푟 and 푔 be multivariate polynomials in 푤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' A sum-of-squares proof that the constraints { 푓1 ⩾ 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' , 푓푚 ⩾ 0} imply the constraint {푔 ⩾ 0} consists of sum-of-squares polynomials (푝푆)푆⊆[푚] such that 푔 = � 푆⊆[푚] 푝푆 · Π푖∈푆 푓푖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='3) 19The choice of norm is not important here because the factor 2−푛ℓ swamps the effects of choosing another norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 20Here, we assume that the bit complexity of the constraints in 풜 is (푛 + 푚)푂(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 19 We say that this proof has degree ℓ if for every set 푆 ⊆ [푚], the polynomial 푝푆Π푖∈푆 푓푖 has degree at most ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' If there is a degree ℓ SoS proof that { 푓푖 ⩾ 0 | 푖 ⩽ 푟} implies {푔 ⩾ 0}, we write: { 푓푖 ⩾ 0 | 푖 ⩽ 푟} ℓ {푔 ⩾ 0} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4) Sum-of-squares proofs satisfy the following inference rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For all polynomials 푓 , 푔 : ℝ푛 → ℝ and for all functions 퐹: ℝ푛 → ℝ푚, 퐺: ℝ푛 → ℝ푘, 퐻 : ℝ푝 → ℝ푛 such that each of the coordinates of the outputs are polynomials of the inputs, we have: 풜 ℓ { 푓 ⩾ 0, 푔 ⩾ 0} 풜 ℓ { 푓 + 푔 ⩾ 0} , 풜 ℓ { 푓 ⩾ 0}, 풜 ℓ′ {푔 ⩾ 0} 풜 ℓ+ℓ′ { 푓 · 푔 ⩾ 0} (addition and multiplication) 풜 ℓ ℬ, ℬ ℓ′ 퐶 풜 ℓ·ℓ′ 퐶 (transitivity) {퐹 ⩾ 0} ℓ {퐺 ⩾ 0} {퐹(퐻) ⩾ 0} ℓ·deg(퐻) {퐺(퐻) ⩾ 0} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' (substitution) Low-degree sum-of-squaresproofsare sound and complete ifwe take low-level pseudo- distributions as models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Concretely, sum-of-squares proofs allow us to deduce properties of pseudo- distributions that satisfy some constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Fact 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='17 (Soundness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' If 퐷 푟 풜 for a level-ℓ pseudo-distribution 퐷 and there exists a sum-of- squares proof 풜 푟′ ℬ, then 퐷 푟·푟′+푟′ ℬ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' If the pseudo-distribution 퐷 satisfies 풜 only approximately, soundness continues to hold if we require an upper bound on the bit-complexity of the sum-of-squares 풜 푟′ 퐵 (number of bits required to write down the proof).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' In our applications, the bit complexity of all sum of squares proofs will be 푛푂(ℓ) (assum- ing that all numbers in the input have bit complexity 푛푂(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' This bound suffices in order to argue about pseudo-distributions that satisfy polynomial constraints approximately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The following fact shows that every property of low-level pseudo-distributions can be derived by low-degree sum-of-squares proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Fact 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='18 (Completeness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Suppose 푑 ⩾ 푟′ ⩾ 푟 and 풜 is a collection of polynomial constraints with degree at most 푟, and 풜 ⊢ {�푛 푖=1 푤2 푖 ⩽ 퐵} for some finite 퐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let {푔 ⩾ 0} be a polynomial constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' If every degree-푑 pseudo-distribution that satisfies 퐷 푟 풜 also satisfies 퐷 푟′ {푔 ⩾ 0}, then for every 휀 > 0, there is a sum-of-squares proof 풜 푑 {푔 ⩾ −휀}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 20 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='3 Explictly bounded distributions We will consider a subset of subgaussian distributions denoted as certifiably subgaussians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Many subgaussians distributions are known to be certifiably subgaussian (see [KSS18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='19 (Explicitly bounded distribution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 푡 ∈ ℕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' A distribution 퐷 over ℝ푑 with mean 휇 is called 2푡-explicitly 휎-bounded if for each even integer 푠 such that 1 ⩽ 푠 ⩽ 푡 the following equation has a degree 푠 sum-of-squares proof in the vector variable 푢 푢 2푠 � 피 x∼퐷⟨x − 휇, 푢⟩2푠 ⩽ (휎푠)푠 · ∥푢∥2푠 2 � Furthermore, we say that 퐷 is explicitly bounded if it is 2푡-explicitly 휎-bounded for every 푡 ∈ ℕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' A finite set 푋 ⊆ ℝ푑 is said to be 2푡-explicitly 휎-bounded if the uniform distribution on 푋 is 2푡-explicitly 휎-bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Sets that are 2푡-explicitly 휎-bounded with large intersection satisfy certain key proper- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Before introducing them we conveniently present the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='20 (Weight vector inducing distribution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 푌 be a set of size 푛 and let 푝 ∈ [0, 1]푛 be a vector satisfying ��푝 �� 1 = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We say that 푝 induces the distribution 퐷 with support 푌 if ℙy∼퐷 � y = 푦푖 � = 푝푖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='21 ([KSS18, HL18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 푌 ⊆ ℝ푑 be a set of cardinality 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 푝, 푝′ ∈ [0, 1]푛 be weight vectors satisfying ��푝 �� 1 = ��푝′�� 1 = 1 and ��푝 − 푝′�� 1 ⩽ 훽 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Suppose that 푝 (respectively 푝′) induces a 2푡-explicitly 휎1-bounded (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 휎2) distribution over 푌 with mean 휇(푝) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 휇(푝′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' There exists an absolute constant 훽∗ such that, if 훽 ⩽ 훽∗, then for 휎 = 휎1 + 휎2 : ��휇(푝) − 휇(푝′) �� ⩽ 훽1−1/2푡 · 푂 �√ 휎푡 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' In the context of learning Gaussian mixtures, we will make heavy use of the statement below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='22 ([KSS18, HL18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 푌 be a 2푡-explicitly 휎-bounded set of size 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 푝 ∈ ℝ푛 be the weight vector inducing the uniform distribution over 푌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 푝′ ∈ ℝ푛 be a unit vector satisfying ��푝 − 푝′�� 1 ⩽ 훽 for some 훽 ⩽ 훽∗ where 훽∗ is a small constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then 푝′ induces a 2푡-explicitly (휎 + 푂(훽1−1/2푡))-bounded distribution over 푌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 4 Stability of strongly-convex optimization In this section, we prove ℓ2 sensitivity bounds for the minimizers of a general class of (strongly) convex optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' In particular, we show how to translate a uniform point-wise sensitivity bound for the objective functions into a ℓ2 sensitivity bound for the minimizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 21 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1 (Stability of strongly-convex optimization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 풴 be a set of databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 풦(풴) be a family closed convex subsets of ℝ푚 parametrized by 푌 ∈ 풴 and let ℱ (풴) be a family of functions 푓푌 : 풦(푌) → ℝ , parametrized by 푌 ∈ 풴 , such that: (i) for adjacent databases 푌, 푌′ ∈ 풴 , and 푋 ∈ 풦(푌) there exist 푋′ ∈ 풦(푌′) ∩ 풦(푌) satisfying �� 푓푌(푋) − 푓푌′(푋′) �� ⩽ 훼 and ��푓푌′(푋′) − 푓푌(푋′) �� ⩽ 훼 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' (ii) 푓푌 is 휅-strongly convex in 푋 ∈ 풦(푌).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then for 푌, 푌′ ∈ 풴, ˆ푋 := arg min푋∈풦(푌) 푓푌(푋) and ˆ푋′ := arg min푋′∈풦(푌′) 푓푌′(푋′) , it holds ��� ˆ푋 − ˆ푋′��� 2 2 ⩽ 12훼 휅 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 푋′ ∈ 풦(푌)∩풦(푌′) be a point such that ��� 푓푌(푋′) − 푓푌′( ˆ푋′) ��� ⩽ 훼 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' By Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2 it holds ��� ˆ푋 − ˆ푋′��� 2 2 ⩽ 2 ��� ˆ푋 − 푋′��� 2 2 + 2 ���푋′ − ˆ푋′��� 2 2 ⩽ 4 휅 � 푓푌(푋′) − 푓푌( ˆ푋) + 푓푌′(푋′) − 푓푌′( ˆ푋′) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Suppose now w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 푓푌( ˆ푋) ⩾ 푓푌′( ˆ푋′), a symmetric argument works in the other case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then 푓푌′( ˆ푋′) + 훼 ⩾ 푓푌(푋′) ⩾ 푓푌( ˆ푋) ⩾ 푓푌′( ˆ푋′) and 푓푌′( ˆ푋′) + 2훼 ⩾ 푓푌(푋′) + 훼 ⩾ 푓푌′(푋′) ⩾ 푓푌′( ˆ푋′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' It follows as desired 푓푌(푋′) − 푓푌( ˆ푋) + 푓푌′(푋′) − 푓푌′( ˆ푋′) ⩽ 3훼 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' □ 5 Private recovery for stochastic block models In this section, we present how to achieve exact recovery in stochastic block models pri- vately and thus prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' To this end, we first use the stability of strongly convex optimization (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1) to obtain a private weak recovery algorithm in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then we show how to privately boost the weak recovery algorithm to achieve exact recovery in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4, we complement our algorithmic results by providing an almost tight lower bound on the privacy parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We start by defining the relevant notion of adjacent databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 22 Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1 (Adjacent graphs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 퐺 , 퐺′ be graphs with vertex set [푛].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We say that 퐺 , 퐺′ are adjacent if |퐸(퐺)△퐸(퐺′)| = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2 (Parameters as public information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We remark that we assume the parameters 푛, 훾, 푑 to be public information given in input to the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1 Private weak recovery for stochastic block models In this section, we show how to achieve weak recovery privately via stability of strongly convex optimization (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We first introduce one convenient notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The error rate of an estimate ˆ푥 ∈ {±1}푛 of the true partition 푥 ∈ {±1}푛 is defined as err( ˆ푥, 푥) := 1 푛 · min{Ham( ˆ푥, 푥), Ham( ˆ푥, −푥)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='21 Our main result is the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Suppose 훾 √ 푑 ⩾ 12800 , 휀, 훿 ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' There exists an (Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4) such that, for any 푥 ∈ {±1}푛, on input G ∼ SBM푛(훾, 푑, 푥), outputs ˆ푥(G) ∈ {±1}푛 satisfying err( ˆ푥(G), 푥) ⩽ 푂 � 1 훾 √ 푑 + 1 훾푑 · log(2/훿) 휀2 � with probability 1 − exp(−Ω(푛)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Moreover, the algorithm is (휀, 훿)-differentially private for any input graph and runs in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Before presenting the algorithm we introduce some notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Given a graph 퐺, let 푌(퐺) := 1 훾푑(퐴(퐺) − 푑 푛 퐽) where 퐴(퐺) is the adjacency matrix of 퐺 and 퐽 denotes all-one ma- trices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Define 풦 := � 푋 ∈ ℝ푛×푛 �� 푋 ⪰ 0 , 푋푖푖 = 1 푛 ∀푖 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The algorithm starts with projecting matrix 푌(퐺) to set 풦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' To ensure privacy, then it adds Gaussian noise to the projection 푋1 and obtains a private matrix 푋2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The last step applies a standard rounding method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4 (Private weak recovery for SBM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Input: Graph 퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Operations: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Projection: 푋1 ← argmin푋∈풦 ∥푌(퐺) − 푋∥2 퐹.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Noise addition: X2 ← 푋1 + W where W ∼ 풩 � 0, 24 푛훾푑 log(2/훿) 휀2 �푛×푛 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Rounding: Compute the leading eigenvector v of X2 and return sign(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' In the rest of this section, we will show Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4 is private in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='6 and its utility guarantee in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='3 follows directly from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='6 and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 21Note |⟨ ˆ푥, 푥⟩| = (1 − 2 err( ˆ푥, 푥)) · 푛 for any ˆ푥, 푥 ∈ {±1}푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 23 Privacy analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 풴 be the set of all matrices 푌(퐺) = 1 훾푑(퐴(퐺) − 푑 푛 퐽) where 퐺 is a graph on 푛 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We further define 푓 : 풴 → ℝ to be the function 푓 (푌) := min 푋∈풦 ∥푌 − 푋∥2 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1) We first use Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1 to prove that function 푓 is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='5 (Stability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The function 푓 as defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1) has ℓ2-sensitivity Δ 푓 ,2 ⩽ � 24 푛훾푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 푔 : 풴 × 풦 → ℝ be the function 푔(푌, 푋) := ∥푋∥2 F − 2⟨푌, 푋⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1 it suffices to prove that 푔 has ℓ1-sensitivity 4 푛훾푑 with respect to 푌 and that it is 2-strongly con- vex with respect to 푋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The sensitivity bound follows by observing that adjacent databases 푌, 푌′ satisfy ∥푌 − 푌′∥1 ⩽ 2 훾푑 and that any 푋 ∈ 풦 satisfies ∥푋∥∞ ⩽ 1 푛 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Thus it remains to prove strong convexity with respect to 푋 ∈ 풦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 푋, 푋′ ∈ 풦 then ∥푋′∥2 F = ∥푋∥2 F + 2⟨푋′ − 푋, 푋⟩ + ∥푋 − 푋′∥2 F = ∥푋∥2 F + 2⟨푋′ − 푋, 푋 + 푌 − 푌⟩ + ∥푋 − 푋′∥2 F = 푔(푌, 푋) + ⟨푋′ − 푋, ∇푔(푋, 푌)⟩ + 2⟨푋′, 푌⟩ + ∥푋 − 푋′∥2 F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' That is 푔(푌, 푋) is 2-strongly convex with respect to 푋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Note any 푋 ∈ 풦 is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then the result follows by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' □ Then it is easy to show the algorithm is private.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='6 (Privacy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The weak recovery algorithm (Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4) is (휀, 훿)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Since any 푋 ∈ 풦 is symmetric, we only need to add a symmetric noise matrix to obtain privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Combining Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='5 with Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='12, we immediately get that the algorithm is (휀, 훿)-private.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' □ Utility analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Now we show the utility guarantee of our priavte weak recovery algo- rithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='7 (Utility).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For any 푥 ∈ {±1}푛, on input G ∼ SBM푛(훾, 푑, 푥), Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4 efficiently outputs ˆ푥(G) ∈ {±1}푛 satisfying err( ˆ푥(G), 푥) ⩽ 6400 훾 √ 푑 + 7000 훾푑 · log(2/훿) 휀2 , with probability 1 − exp(−Ω(푛)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' To prove Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='7, we need the following lemma which is an adaption of a well- known result in SBM [GV16, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Its proof is deferred to Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 24 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Consider the settings of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' With probability 1 − exp(−Ω(푛)), ����푋1(G) − 1 푛 푥푥⊤ ���� 2 퐹 ⩽ 800 훾 √ 푑 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='8, we have ����푋1(G) − 1 푛 푥푥⊤ ���� ⩽ ����푋1(G) − 1 푛 푥푥⊤ ���� 퐹 ⩽ � 800 훾 √ 푑 =: 푟(훾, 푑) with probability 1 − exp(−Ω(푛)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We condition our following analysis on this event hap- pening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let u be the leading eigenvector of 푋1(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 흀1 and 흀2 be the largest and second largest eigenvalues of 푋1(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' By Weyl’s inequality (Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1) and the assumption 훾 √ 푑 ⩾ 12800, we have 흀1 − 흀2 ⩾ 1 − 2푟(훾, 푑) ⩾ 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let v be the leading eigenvector of 푋1(G) + W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' By Davis-Kahan’s theorem (Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2), we have ∥u − v∥ ⩽ 2∥W∥ 흀1 − 흀2 ⩽ 4∥W∥, ��u − 푥/ √ 푛 �� ⩽ 2 ����푋1(G) − 1 푛 푥푥⊤ ���� ⩽ 2푟(훾, 푑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Putting things together and using Fact A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1, we have ��v − 푥/ √ 푛 �� ⩽ ∥u − v∥ + ��u − 푥/ √ 푛 �� ⩽ 24 √ 6 � 훾푑 � log(2/훿) 휀 + 2푟(훾, 푑) with probability 1 − exp(−Ω(푛)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Observe Ham(sign(푦), 푥) ⩽ ∥푦 − 푥∥2 for any 푦 ∈ ℝ푛 and any 푥 ∈ {±1}푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then with probability 1 − exp(−Ω(푛)), Ham(sign(v), 푥) ⩽ ��√ 푛 · v − 푥 ��2 ⩽ 6400 훾 √ 푑 + 7000 훾푑 · log(2/훿) 휀2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' □ Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='6 and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' □ 25 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2 Private exact recovery for stochastic block models In this section, we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We show how to achieve exact recovery in stochastic block models privately by combining the private weak recovery algorithm we obtained in the previous section and a private majority voting scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Since exact recovery is only possible with logarithmic average degree (just to avoid iso- lated vertices), it is more convenient to work with the following standard parameterization of stochastic block models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 훼 > 훽 > 0 be fixed constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The intra-community edge probability is 훼 · log 푛 푛 , and the inter-community edge probability is 훽 · log 푛 푛 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' In the language of Model 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1, it is SBM푛( 훼+훽 2 log 푛, 훼−훽 훼+훽 , 푥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Our main result is the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='9 (Private exact recovery of SBM, restatement of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 휀, 훿 ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Suppose 훼, 훽 are fixed constants satisfying22 √ 훼 − � 훽 ⩾ 4 and 훼 − 훽 ⩾ Ω � 1 휀2 · log(2/훿) log 푛 + 1 휀 � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2) Then there exists an algorithm (Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='11) such that, for any balanced23 푥 ∈ {±1}푛, on input G ∼ SBM푛( 훼+훽 2 · log 푛, 훼−훽 훼+훽 , 푥), outputs ˆ푥(G) ∈ {푥, −푥} with probability 1 − 표(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Moreover, the algorithm is (휀, 훿)-differentially private for any input graph and runs in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' In a standard regime ofprivacyparameterswhere 휀 ⩽ 푂(1)and 훿 = 1/poly(푛), the private exact recovery threshold Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2) reads √ 훼 − � 훽 ⩾ 4 and 훼 − 훽 ⩾ Ω� 휀−2 + 휀−1� , Recall the non-private exact recovery threshold is √훼 − � 훽 > √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Thus the non-private part in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 4, is close to optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='11 starts with randomly splitting the input graph 퐺 into two subgraphs G1 and G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Setting the graph-splitting probability to 1/2, each subgraph will contain about half of the edges of 퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then we run an (휀, 훿)-DP weak recovery algorithm (Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4) on G1 to get a rough estimate ˜푥(G1) of accuracy around 90%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Finally, we boost the accuracy to 100% by doing majority voting (Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='12) on G2 based on the rough estimate ˜푥(G1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' That is, if a vertex has more neighbors from the opposite community (according to ˜푥(G1)) in G2, then we assign this vertex to the opposite community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' To make the majority voting step private, we add some noise to the vote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 22In the language of Model 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1, for any 푡 we have √훼 − � 훽 ⩾ 푡 if and only if 푑 log 푛 (1 − � 1 − 훾2) ⩾ 푡2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 23Recall a vector 푥 ∈ {±1}푛 is said to be balanced if �푛 푖=1 푥푖 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 26 Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='11 (Private exact recovery for SBM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Input: Graph 퐺 Operations: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Graph-splitting: Initialize G1 to be an empty graph on vertex set 푉(퐺).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Indepen- dently put each edge of 퐺 in G1 with probability 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let G2 = 퐺 \\ G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Rough estimation on G1: Run the (휀, 훿)-DP partial recovery algorithm (Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4) on G1 to get a rough estimate ˜푥(G1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Majority voting on G2: Run the (휀, 0)-DP majority voting algorithm (Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='12) with input (G2, ˜푥(G1)) and get output ˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Return ˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='12 (Private majority voting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Input: Graph 퐺, rough estimate ˜푥 ∈ {±1}푛 Operations: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For each vertex 푣 ∈ 푉(퐺), let Z푣 = S푣 − D푣 where D푣 = � {푢,푣}∈퐸(퐺) 1[ ˜푥푢 ≠ ˜푥푣] , S푣 = � {푢,푣}∈퐸(퐺) 1[ ˜푥푢 = ˜푥푣] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Set ˆx푣 = sign(Z푣 + W푣) · ˜푥(G1)푣 where W푣 ∼ Lap(2/휀).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Return ˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' In the rest of this section, we will show Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='11 is private in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='14 and it recovers the hidden communities exactly with high probability in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='9 follows directly from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='14 and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Privacy analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We first show the differential privcay of the majority voting algorithm (Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='12) with respect to input graph 퐺 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' assuming fixed the input rough esti- mate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='12 is (휀, 0)-DP with respect to input 퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Observing the ℓ1-sensitivity of the degree count function 푍 in step is 2, the (휀, 0)-DP follows directly from Laplace mechanism (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='12) and post-processing (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' □ Then the privacy of the private exact recovery algorithm (Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='11) is a conse- quence of composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 27 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='14 (Privacy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='11 is (휀, 훿)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 풜1 : 풢푛 → {±1}푛 denote the (휀, 훿)-DP recovery algorithm in step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 풜2 : 풢푛 × {±1}푛 → {±1}푛 denote the (휀, 훿)-DP majority voting algorithm in step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 풜 be the composition of 풜1 and 풜2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We first make several notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Given a graph 퐻 and an edge 푒, 퐻푒 is a graph obtained b adding 푒 to 퐻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Given a graph 퐻, G1(퐻) is a random subgraph of 퐻 by keeping each edge of 퐻 with probability 휆 independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Now, fix two adjacent graphs 퐺 and 퐺푒 where edge 푒 appears in 퐺푒 but not in 퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Also, fix two arbitrary possible outputs 푥1, 푥2 ∈ {±1}푛 of algorithm 풜.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='24 It is direct to see, ℙ(풜(퐺) = (푥1, 푥2)) = � 퐻⊆퐺 ℙ(풜1(퐻) = 푥1) ℙ(풜2(퐺 \\ 퐻, 푥1) = 푥2) ℙ(G1(퐺) = 퐻).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='3) Since ℙ(G1(퐺) = 퐻) = ℙ(G1(퐺푒) = 퐻) + ℙ(G1(퐺푒) = 퐻푒) for any 퐻 ⊆ 퐺, we have ℙ(풜(퐺푒) = (푥1, 푥2)) = � 퐻⊆퐺 ℙ(풜1(퐻) = 푥1) ℙ(풜2(퐺푒 \\ 퐻, 푥1) = 푥2) ℙ(G1(퐺푒) = 퐻) + ℙ(풜1(퐻푒) = 푥1) ℙ(풜2(퐺푒 \\ 퐻푒, 푥1) = 푥2) ℙ(G1(퐺푒) = 퐻푒) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4) Since both 풜1 and 풜2 are (휀, 훿)-DP, we have for each 퐻 ⊆ 퐺, ℙ(풜1(퐻푒) = 푥1) ⩽ 푒휀 ℙ(풜1(퐻) = 푥1) + 훿, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='5) ℙ(풜2(퐺푒 \\ 퐻, 푥1) = 푥2) ⩽ 푒휀 ℙ(풜2(퐺 \\ 퐻, 푥1) = 푥2) + 훿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='6) Plugging Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='5) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='6) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4), we obtain ℙ(풜(퐺푒) = (푥1, 푥2)) ⩽ � 퐻⊆퐺 [푒휀 ℙ(풜1(퐻) = 푥1) ℙ(풜2(퐺 \\ 퐻, 푥1) = 푥2) + 훿] ℙ(G1(퐺) = 퐻) = 푒휀 ℙ(풜(퐺) = (푥1, 푥2)) + 훿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Similarly, we can show ℙ(풜(퐺) = (푥1, 푥2)) ⩽ 푒휀 ℙ(풜(퐺푒) = (푥1, 푥2)) + 훿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='7) □ Utility analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We first show the utility guarantee of the priavte majority voting algo- rithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Suppose G is generated by first sampling G ∼ SBM푛( 훼+훽 2 log 푛, 훼−훽 훼+훽 , 푥) for some balanced 푥 and then for each vertex removing at most Δ ⩽ 푂(log2 푛) adjacent edges arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then on input G and a balanced rough estimate ˜푥 satisfying Ham( ˜푥, 푥) ⩽ 푛/16, Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='12 efficiently outputs ˆ푥(G) such that for each vertex 푣, ℙ( ˆ푥(G)푣 ≠ 푥푣) ⩽ exp � − 1 64 · 휀(훼 − 훽) · log 푛 � + 2 · exp � − 1 162 · (훼 − 훽)2 훼 + 훽 log 푛 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 24We can imagine that algorithm 풜 first outputs (푥1, 푥2) and then outputs 푥2 as a post-processing step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 28 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let us fix an arbitrary vertex 푣 and analyze the probability ℙ( ˆ푥(G)푣 ≠ 푥푣).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 푟 := Ham( ˜푥, 푥)/푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then it is not hard to see ℙ( ˆ푥(G)푣 ≠ 푥푣) ⩽ ℙ(B + A′ − A − B′ + W > 0) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='8) where A ∼ Binomial((1/2 − 푟)푛 − Δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 훼 log 푛 푛 ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' corresponding to the number of neighbors that are from the same community and correctly labeled by ˜푥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' B′ ∼ Binomial(푟푛 − Δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 훽 log 푛 푛 ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' corresponding to the number of neighbors that are from the different community but incorrectly labeled by ˜푥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' B ∼ Binomial((1/2 − 푟)푛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 훽 log 푛 푛 ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' corresponding to the number of neighbors that are from the different community and correctly labeled by ˜푥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' A′ ∼ Binomial(푟푛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 훼 log 푛 푛 ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' corresponding to the number of neighbors that are from the same community but incorrectly labeled by ˜푥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' W ∼ Lap(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 2/휀),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The Δ term appearing in both A and B′ corresponds to the worst case where Δ “favorable” edges are removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' If 푟 ⩾ Ω(1), then Δ = 푂(log2 푛) is negligible to 푟푛 = Θ(푛) and we can safely ignore the effect of removing Δ edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' If 푟 = 표(1), then we can safely assume ˜푥 is correct on all vertices and ignore the effect of removing Δ edges as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Thus, we will assume Δ = 0 in the following analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For any 푡, 푡′, we have ℙ(A′ + B − A − B′ + W > 0) ⩽ ℙ(A′ + B + W > 푡) + ℙ(A + B′ ⩽ 푡) ⩽ ℙ(A′ + B ⩾ 푡 − 푡′) + ℙ(W ⩾ 푡′) + ℙ(A + B′ ⩽ 푡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We choose 푡, 푡′ by first picking two constants 푎, 푏 > 0 satisfying 푎 + 푏 < 1 and then solving 피[A′ + B] − 푡 = 푎 · (피[A + B′] − 피[A′ + B]) and 푡′ = (1 − 푎 − 푏) · (피[A + B′] − 피[A′ + B]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' By Fact 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='7, ℙ(W > 푡′) ⩽ exp � −푡′휀 2 � ⩽ exp � −(1/4 − 푟)(1 − 푎 − 푏) 2 휀(훼 − 훽) · log 푛 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' By Fact A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4 and the assumption 푟 ⩽ 1/16, we have ℙ(A + B′ ⩽ 푡) ⩽ exp � −(피[A + B′] − 푡)2 2 피[A + B′] � ⩽ exp � −(1/4 − 푟)2푎2 · (훼 − 훽)2 훼 + 훽 log 푛 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 29 Setting 푏 = 1/2, by Fact A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4 and the assumption 푟 ⩽ 1/16, we have ℙ(A′ + B ⩾ 푡 − 푡′) ⩽ exp � −(푡 − 푡′ − 피[A′ + B])2 푡 − 푡′ + 피[A′ + B] � ⩽ exp � −2(1/4 − 푟)2 7 (훼 − 훽)2 훼 + 훽 log 푛 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Further setting 푎 = 1/3, we have ℙ( ˆ푥(G)푣 ≠ 푥푣) ⩽ exp � −1/4 − 푟 12 휀(훼 − 훽) · log 푛 � + 2 · exp � −(1/4 − 푟)2 9 (훼 − 훽)2 훼 + 훽 log 푛 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Finally, plugging the assumption 푟 ⩽ 1/16 to conclude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' □ Then it is not difficult to show the utility guarantee of our priavte exact recovery algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='16 (Utility).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Suppose 훼, 훽 are fixed constants satisfying √ 훼 − � 훽 ⩾ 4 and 훼 − 훽 ⩾ Ω � 1 휀2 · log(2/훿) log 푛 + 1 휀 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then for any balanced 푥 ∈ {±1}푛, on input G ∼ SBM푛( 훼+훽 2 log 푛, 훼−훽 훼+훽 , 푥), Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='11 efficiently outputs ˆ푥(G) satisfying ˆ푥(G) ∈ {푥, −푥} with probability 1 − 표(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We will show the probability of a fixed vertex being misclassified is at most 표(1/푛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then by union bound, exact recovery can be achieved with probability 1 − 표(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' As the graph-splitting probability is 1/2, G1 follows SBM푛( 훼 2 · log 푛 푛 , 훽 2 · log 푛 푛 , 푥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' By Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='3, the rough estimate ˜푥(G1) satisfies25 err( ˜푥(G1), 푥) ⩽ 푟 := 표(1) + 14000 (훼 − 훽)휀2 · log(2/훿) log 푛 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='9) with probability at least 1 − exp(−Ω(푛)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Without loss of generality, we can assume Ham( ˜푥(G1), 푥) ⩽ 푟푛, since we consider −푥 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' By Fact A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2, the maximum de- gree of G1 is at most Δ := 2 log2 푛 with probability at least 1 − 푛 exp(−(log 푛)2/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' In the following, we condition our analysis on the above two events regarding ˜푥(G1) and G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Now, let us fix a vertex and analyze the probability 푝푒 that it is misclassified after majority voting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' With 퐺1 being fixed, G2 can be thought of as being generated by first sampling G and then removing 퐺1 from G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' To make 푟 ⩽ 1/16, it suffices to ensure 훼 − 훽 > 5002 휀2 · log(2/훿) log 푛 by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='Then by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='15, we have 푝푒 ⩽ exp � − 1 64 · 휀(훼 − 훽) · log 푛 � + 2 · exp � − 1 162 · (훼 − 훽)2 훼 + 훽 log 푛 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 25It is easy to make the output of Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4 balanced at the cost of increasing the error rate by a factor of at most 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 30 To make 푝푒 at most 표(1/푛), it suffices to ensure 1 64 · 휀(훼 − 훽) > 1 and 1 162 · (훼 − 훽)2 훼 + 훽 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Note (훼 − 훽)2/(훼 + 훽) > (√훼 − � 훽)2 for 훼 > 훽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Therefore, as long as √ 훼 − � 훽 ⩾ 4 and 훼 − 훽 ⩾ 5002 휀2 log(2/훿) log 푛 + 64 휀 , Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='11 recovers the hidden communities exactly with probability 1 − 표(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' □ Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='14 and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='3 Inefficient recovery using the exponential mechanism In this section, we will present an inefficient algorithm satisfying pure privacy which succeeds for all ranges of parameters - ranging from weak to exact recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The algorithm is based on the exponential mechanism [MT07] combined with the majority voting scheme introduced in section Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' In particular, we will show Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='17 (Full version of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 훾 √ 푑 ⩾ 12800 and 푥 ∈ {±1}푛 be balanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 휁 ⩾ 2 exp � − 훾2푑 512 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For any 휀 ⩾ 64 log(2/휁) 훾푑 , there exists an algorithm, Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='18, which on input G ∼ SBM푛(훾, 푑, 푥∗) outputs an estimate ˆ푥(G) ∈ {±1}푛 satisfying err( ˆ푥(G), 푥∗) ⩽ 휁 with probability at least 1−휁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' In addition, the algorithm is 휀-private.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Further, by slightly modifying the algorithm, we can achieve error 20/ � log(1/휁) with probability 1 − 푒−푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='26 A couple of remarks are in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' First, our algorithm works across all degree-regimes in the literature and matches known non-private thresholds and rates up to constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We remark that for ease of exposition we did not try to optimize these constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' In particular, for 훾2푑 a constant we achieve weak recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We reiterate, that 훾2푑 > 1 is the optimal non-private threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For the regime, where 훾2푑 = 휔(1), it is known that the optimal error rate is exp� −(1 − 표(1))훾2푑� even non-privately [ZZ16], where 표(1) goes to zero as 훾2푑 tends to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We match this up to constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Moreover, our algorithm achieves exact recovery as soon as 훾2푑 ⩾ 512 log 푛 since then 휁 < 1 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' This also matches known non-private threshholds up to constants [ABH15, MNS15a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Also, our dependence on the privacy parameter 휀 is also optimal as shown by the information-theoretic lower bounds in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 26The first, smaller, error guarantee additionally needs the requirement that 휁 ⩽ exp(−640).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The second one does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 31 We also emphasize, that if we only aim to achieve error on the order of 1 훾 √ 푑 = Θ � 1 � log(1/휁) � , we can achieve exponentially small failure probability in 푛, while keeping the privacy pa- rameter 휀 the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' This can be achieved, by ommitting the boosting step in our algorithm and will be clear from the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We remark that in this case, we can also handle non-balanced communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Again, for an input graph 퐺, consider the matrix 푌(퐺) = 1 훾푑 � 퐴(퐺) − 푑 푛 퐽� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For 푥 ∈ {±1}푛 we define the score function 푠퐺(푥) = ⟨푥, 푌(퐺)푥⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Since the entries of 퐴(퐺) are in [0, 1] and adjacent graphs differ in at most one edge, it follows immediately, that this score function has sensitivity at most Δ = max 퐺∼퐺′ , 푥∈{±1}푛 |푠퐺(푥) − 푠퐺′(푥)| = 2 훾푑 · max 퐺∼퐺′ , 푥∈{±1}푛 |⟨푥, (퐴(퐺) − 퐴(퐺′))푥⟩| ⩽ 2 훾푑 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='18 (Inefficient algorithm for SBM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Input: Graph 퐺, privacy parameter 휀 > 0 Operations: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Graph-splitting: Initialize G1 to be an empty graph on vertex set 푉(퐺).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Indepen- dently assign each edge of 퐺 to G1 with probability 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let G2 = 퐺 \\ G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Rough estimation on G1: Sample ˜푥 from the distribution with density 푝(푥) ∝ exp � 휀 2Δ⟨푥, 푌(G1)푥⟩ � , where Δ = 2 훾푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Majority voting on G2: Run the 휀-DP majority voting algorithm (Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='12) with input (G2, ˜푥(G1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Denote its output by ˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Return ˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We first analyze the privacy guarantees of the above algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='18 is 휀-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For simplicity and clarity of notation, we will show that the algorithm satisfies 2휀-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Clearly, the graph splitting step is 0-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Step 2 corresponds to the exponential mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 32 Since the sensitivity of the score function is at most Δ = 2 훾푑 it follows by the standard analysis of the mechanism that this step is 휀-DP [MT07].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='13, the majority voting step is also 휀-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Hence, the result follows by composition (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' □ Next, we will analyze its utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 훾 √ 푑 ⩾ 12800 and 푥 ∈ {±1}푛 be balanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let exp(−640) ⩾ 휁 ⩾ 2 exp � − 훾2푑 512 � , 휀 ⩾ 64 log(2/휁) 훾푑 , and G ∼ SBM푛(훾, 푑, 푥∗), the output ˆ푥(G) ∈ {±1}푛 of Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='18 satisfies err( ˆ푥(G), 푥∗) ⩽ 휁 with probability at least 1 − 휁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We will first show that the rough estimate ˜푥 obtained in step 2 achieves err( ˜푥, 푥∗) ⩽ 20 � log(1/휁) with probability 푒−푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' This will prove the second part of the theorem - for this we don’t need that 휁 ⩽ exp(−640).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' In fact, arbitrary 휁 works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The final error guarantee will then follow by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' First, notice that similar to the proof of [GV16, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1], using Bernstein’s inequality and a union bound, we can show that (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Fact D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2 for a full proof) max 푥∈{±1}푛 ����⟨푥, � 푌(G) − 1 푛 푥∗(푥∗)⊤ � 푥⟩ ���� ⩽ 100푛 훾 √ 푑 ⩽ 5 � log(1/휁) with probability at least 1 − exp−10푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Recall that 푠G(푥) = ⟨푥, 푌(G)푥⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 훼 = 5 √ log(1/휁).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We call 푥 ∈ {±1}푛 good if 푠G(푥) ⩾ (1 − 3훼)푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' It follows that for good 푥 it holds that 1 푛 · ⟨푥, 푥∗⟩2 ⩾ ⟨푥, 푌(G)푥⟩ − ���� � 푥, � 푌(G) − 1 푛 푥∗(푥∗)⊤ � 푥 ����� ⩾ (1 − 4훼)푛 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Which implies that 2 err(푥, 푥∗) ⩽ 1 − √ 1 − 4훼 = 1 − 1 − 4훼 √ 1 − 4훼 ⩽ 1 − 1 − 4훼 1 − 2훼 = 2훼 1 − 2훼 ⩽ 4훼 , where we used that 훼 ⩽ 1/4 and that √ 1 − 4푥 ⩽ 1 − 2푥 for 푥 ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Hence, we have for good 푥 that err(푥, 푥∗) ⩽ 20 � log(1/휁) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Since 푠G(푥∗) ⩾ (1 − 훼)푛,there is at least one good candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Hence, we can bound the probability that we do not output a good 푥 as exp� 휀 2Δ(1 − 3훼)푛� 푒푛 exp� 휀 2Δ(1 − 훼)푛� 1 = exp �� 1 − 2휀훼 Δ � 푛 � ⩽ 푒−푛 , 33 where we used that 2휀훼 Δ ⩾ 64 log(2/휁) 훾푑 5훾푑 � log(1/휁) ⩾ 320 � log(1/휁) ⩾ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We will use Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='15 to proof the final conclusion of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' In what follows, assume without loss of generality that Ham(푥, 푥∗) < Ham(푥, −푥∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The above discussion implies that Ham(푥, 푥∗) ⩽ 8훼푛 ⩽ 40푛 � log(1/휁) ⩽ 푛 16 , where the last inequality uses 휁 ⩽ 푒−640.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Further, by Fact A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2 it also follows that the maxi- mum degree of G2 is at most 푂 � log2 푛 � (by some margin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Recall that G2 ∼ SBM(푑, 훾, 푥∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' In the parametrization of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='15 this means that 훼 = (1 + 훾)푑 log 푛 , 훽 = (1 − 훾)푑 log 푛 , 훼 − 훽 = 2훾푑 log 푛 , 훼 + 훽 = 2푑 log 푛 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Thus, it follows that the output ˆ푥 of the majority voting step satisfies for every vertex 푣 ℙ( ˆ푥(G)푣 ≠ 푥푣) ⩽ exp � − 1 64 · 휀(훼 − 훽) · log 푛 � + 2 · exp � − 1 162 · (훼 − 훽)2 훼 + 훽 log 푛 � ⩽ exp � − 1 32 · 휀훾푑 � + exp � − 1 162 · 훾2푑 � ⩽ 휁2/4 + 휁2/4 ⩽ 휁2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' By Markov’s Inequality it now follows that ℙ(err( ˆ푥(G), 푥∗) ⩾ 휁) ⩽ 휁 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4 Lower bound on the parameters for private recovery In this section, we prove a tight lower bound for private recovery for stochastic block models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Recall the definition of error rate, err(푢, 푣) := 1 푛 · min{Ham(푢, 푣), Ham(푢, −푣)} for 푢, 푣 ∈ {±1}푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Our main result is the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='21 (Full version ofTheorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Suppose there exists an 휀-differentially private algo- rithm such that for any balanced 푥 ∈ {±1}푛, on input G ∼ SBM푛(푑, 훾, 푥), outputs ˆ푥(G) ∈ {±1}푛 satisfying ℙ(err( ˆ푥(G), 푥) < 휁) ⩾ 1 − 휂, 34 where27 1/푛 ⩽ 휁 ⩽ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='04 and the randomness is over both the algorithm and stochastic block models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then, 푒2휀 − 1 ⩾ Ω �log(1/휁) 훾푑 + log(1/휂) 휁푛훾푑 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='10) Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Both terms in lower bound Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='10) are tight up to constants by the fol- lowing argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Considering typical privacy parameters 휀 ⩽ 1, then 푒2휀 − 1 ≈ 2휀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For exponentially small failure probability, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 휂 = 2−Ω(푛), the lower bound reads 휀 ⩾ Ω( 1 훾푑 · 1 휁), which is achieved by Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='18 without the boosting step - see the discussion af- ter Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For polynomially small failure probability, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='휂 = 1/poly(푛), the lower bound Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='10) reads 휀 ⩾ Ω( 1 훾푑 · log 1 휁), which is achieved by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' By setting 휁 = 1/푛 in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='21, we directly obtain a tight lower bound for private exact recovery as a corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Suppose there exists an 휀-differentially private algorithm such that for any bal- anced 푥 ∈ {±1}푛, on input G ∼ SBM푛(푑, 훾, 푥), outputs ˆ푥(G) ∈ {±1}푛 satisfying ℙ( ˆ푥(G) ∈ {푥, −푥}) ⩾ 1 − 휂, where the randomness is over both the algorithm and stochastic block models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then, 푒2휀 − 1 ⩾ Ω �log(푛) + log 1 휂 훾푑 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='11) Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The lower bound Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='11) for priavte exact recovery is tight up to constants, since there exists an (inefficient) 휀-differentially priavte exact recovery algorithm with 휀 ⩽ 푂(log 푛 훾푑 ) and 휂 = 1/poly(푛) by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='17 and [SNVT22, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' In rest of this section, we will prove Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The proof applies the packing lower bound argument similar to [HKM22, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' To this end, we first show err(·, ·) is a semimetric over {±1}푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' err(·, ·) is a semimetric over {±1}푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Symmetry and non-negativity are obvious from the definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We will show err(·, ·) satisfies triangle inequality via case analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 푢, 푣, 푤 ∈ {±1}푛 be three arbitrary sign vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' By symmetry, we only need to consider the following four cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Case 1: Ham(푢, 푣), Ham(푢, 푤), Ham(푣, 푤) ⩽ 푛/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' This case is reduced to showing Ham- ming distance satisfies triangle inequality, which is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Case 2: Ham(푢, 푣), Ham(푢, 푤) ⩽ 푛/2 and Ham(푣, 푤) ⩾ 푛/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We need to check two subcases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' First, err(푢, 푣) ⩽ err(푢, 푤) + err(푣, 푤) ⇔ Ham(푢, 푣) + Ham(푣, 푤) ⩽ Ham(푢, 푤) + 푛 27Error rate less than 1/푛 already means exact recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Thus it does not make sense to set 휁 to any value strictly smaller than 1/푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The upper bound 휁 ⩽ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='04 is just a technical condition our proof needs for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 35 ⇐ Ham(푢, 푣) + 퐻(푢, 푣) + 퐻(푢, 푤) ⩽ Ham(푢, 푤) + 푛 ⇔ Ham(푢, 푣) ⩽ 푛/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Second, err(푣, 푤) ⩽ err(푢, 푣) + err(푢, 푤) ⇔ 푛 ⩽ Ham(푣, 푤) + Ham(푢, 푣) + Ham(푢, 푤) ⇐ 푛 ⩽ 2 Ham(푣, 푤).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Case 3: Ham(푢, 푣) ⩽ 푛/2 and Ham(푢, 푤), Ham(푣, 푤) ⩾ 푛/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' This case can be reduced to case 1 by considering 푢, 푣, −푤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Case 4: Ham(푢, 푣), Ham(푢, 푤), Ham(푣, 푤) ⩾ 푛/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' This case can be reduced to case 2 by considering −푢, 푣, 푤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' □ Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Suppose there exists an 휀-differentially private algorithm satisfying the theorem’s assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We first make the following notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Given a semimetric 휌 over {±1}푛, a center 푣 ∈ {±1}푛, and a radius 푟 ⩾ 0, define 퐵휌(푣, 푟) := {푤 ∈ {±1}푛 : 1⊤푤 = 0, 휌(푤, 푣) ⩽ 푟}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Pick an arbitrary balanced 푥 ∈ {±1}푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 푀 = {푥1, 푥2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' , 푥푚} be a maximal 2휁- packing of 퐵err(푥, 4휁) in semimetric err(·, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' By maximality of 푀, we have 퐵err(푥, 4휁) ⊆ ∪푚 푖=1퐵err(푥푖, 2휁), which implies |퐵err(푥, 4휁)| ⩽ 푚 � 푖=1 ��퐵err(푥푖, 2휁) �� =⇒ |퐵Ham(푥, 4휁)| ⩽ 푚 � 푖=1 2 · ��퐵Ham(푥푖, 2휁) �� = 2푚 · |퐵Ham(푥, 2휁)| =⇒ 2푚 ⩾ |퐵Ham(푥, 4휁푛)| |퐵Ham(푥, 2휁푛)| = �푛/2 2휁푛 �2 �푛/2 휁푛 �2 ⩾ � 1 4휁 �4휁푛 � 푒 2휁 �2휁푛 = � 1 8푒휁 �2휁푛 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='12) For each 푖 ∈ [푚], define 푌푖 := {푤 ∈ {±1}푛 : err(푤, 푥푖) ⩽ 휁}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then 푌푖’s are pairwise disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For each 푖 ∈ [푚], let 푃푖 be the distribution over 푛-vertex graphs generated by SBM푛(푑, 훾, 푥푖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' By our assumption on the algorithm, we have for any 푖 ∈ [푚] that ℙ G∼푃푖 ( ˆ푥(G) ∈ 푌푖) ⩾ 1 − 휂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Combining the fact that 푌푖’s are pairwise disjoint, we have 푚 � 푖=1 ℙ G∼푃1 ( ˆ푥(G) ∈ 푌푖) = ℙ G∼푃1 � ˆ푥(G) ∈ ∪푚 푖=1푌푖 � ⩽ 1 =⇒ 푚 � 푖=2 ℙ G∼푃1 ( ˆ푥(G) ∈ 푌푖) ⩽ 휂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='13) In the following, we will lower bound ℙG∼푃1( ˆ푥(G) ∈ 푌푖) for each 푖 ∈ [푚] \\ {1} using group privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 36 Note each 푃푖 is a product of �푛 2 � independent Bernoulli distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Thus for any 푖, 푗 ∈ [푚], there exists a coupling 휔푖푗 of 푃푖 and 푃푗 such that, if (G, H) ∼ 휔, then Ham(G, H) ∼ Binomial(푁푖푗, 푝), where 푝 = 2훾푑/푛 and 푁푖푗 = Ham(푥푖, 푥푗) · (푛 − Ham(푥푖, 푥푗)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Applying group privacy, we have for any two graphs 퐺, 퐻 and for any 푆 ⊆ {±1}푛 that28 ℙ( ˆ푥(퐺) ∈ 푆) ⩽ exp(휀 · Ham(퐺, 퐻)) · ℙ( ˆ푥(퐻) ∈ 푆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='14) For each 푖 ∈ [푚], taking expectations on both sides of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='14) with respect to coupling 휔푖1 and setting 푆 = 푌푖, we have 피 (G,H)∼휔푖1 ℙ( ˆ푥(G) ∈ 푌푖) ⩽ 피 (G,H)∼휔푖1 exp(휀 · Ham(G, H)) · ℙ( ˆ푥(H) ∈ 푌푖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='15) The left side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='15) is equal to 피 (G,H)∼휔푖1 ℙ( ˆ푥(G) ∈ 푌푖) = ℙ G∼푃푖( ˆ푥(G) ∈ 푌푖) ⩾ 1 − 휂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Upper bounding the right side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='15) by Cauchy-Schwartz inequality, we have 피 (G,H)∼휔푖1 exp(휀 · Ham(G, H)) · ℙ( ˆ푥(H) ∈ 푌푖) ⩽ � 피 (G,H)∼휔푖1 exp(2휀 · Ham(G, H)) �1/2 � 피 (G,H)∼휔푖1 ℙ( ˆ푥(H) ∈ 푌푖)2 �1/2 = � 피 X∼Binomial(푁푖1,푝) exp(2휀 · X) �1/2 � 피 H∼푃1 ℙ( ˆ푥(H) ∈ 푌푖)2 �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Using the formula for the moment generating function of binomial distributions, we have 피 X∼Binomial(푁푖1,푝) exp(2휀 · X) = (1 − 푝 + 푝 · 푒2휀)푁푖1, and it is easy to see 피 H∼푃1 ℙ( ˆ푥(H) ∈ 푌푖)2 = 피 H∼푃1 (피 1[ ˆ푥(H) ∈ 푌푖])2 ⩽ ℙ H∼푃1 ( ˆ푥(H) ∈ 푌푖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Putting things together, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='15) implies for each 푖 ∈ [푚] that ℙ H∼푃1 ( ˆ푥(H) ∈ 푌푖) ⩾ (1 − 휂)2 (1 − 푝 + 푝 · 푒2휀)푁푖1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='16) Since 푥푖 ∈ 퐵err(푥, 4휁) for 푖 ∈ [푚], by assuming 휁 ⩽ 1/16, we have 푁푖1 = Ham(푥푖, 푥1) · (푛 − Ham(푥1, 푥푖)) ⩽ 8휁푛(푛 − 8휁푛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='17) 28In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='14), the randomness only comes from the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 37 Recalling 푝 = 2훾푑/푛 and combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='12), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='13), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='16) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='17), we have (푚 − 1) · (1 − 휂)2 (1 − 푝 + 푝 · 푒2휀)8휁푛(푛−8휁푛) ⩽ 휂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' By taking logarithm on both sides, using 푡 ⩾ log(1 + 푡) for any 푡 > −1, and assuming 휁 ⩽ 1/(8푒), we have 푒2휀 − 1 ≳ log 1 8푒휁 훾푑 + log 1 휂 휁푛훾푑 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' □ 6 Private algorithms for learning mixtures of spherical Gaussians In this section we present a private algorithm for recovering the centers of a mixtures of 푘 Gaussians (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Model 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 풴 ⊆ � ℝ푑�⊗푛 be the collection of sets of 푛 points in ℝ푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We consider the following notion of adjacency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1 (Adjacent databases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We say that 푌, 푌′ ∈ 풴 are adjacent if |푌 ∩ 푌′| ⩾ 푛 −1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2 (Problem parametersaspublicinformation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We considerthe parameters 푛, 푘, Δ to be public information given as input to the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Next we present the main theorem of the section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='3 (Privately learning spherical mixtures of Gaussians).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Consider an instance of Model 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 푡 ∈ ℕ be such that Δ ⩾ 푂 �√ 푡푘1/푡� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For 푛 ⩾ Ω� 푘푂(1) · 푑푂(푡)� , 푘 ⩾ (log 푛)1/5 , there exists an algorithm, running in time (푛푑)푂(푡), that outputs vectors ˆ흁1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' , ˆ흁ℓ satisfying max ℓ∈[푘] �� ˆ흁ℓ − 휇휋(ℓ) �� 2 ⩽ 푂(푘−12) , with high probability, for some permutation 휋 : [푘] → [푘] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='29 Moreover, for 휀 ⩾ 푘−10 , 훿 ⩾ 푛−10 , the algorithm is (휀, 훿)-differentially private for any input 푌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We remark that our algorithm not only works for mixtures of Gaussians but for all mixtures of 2푡-explicitly bounded distributions (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Our algorithm is based on the sum-of-squares hierarchy and at the heart lies the following sum-of-squares program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The indeterminates 푧11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' , 푧1푘, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' , 푧푛푘 and vector- valued indeterminates 휇′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' , 휇′ 푘, will be central to the proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 푛, 푘, 푡 be fixed parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 29We remark that we chose constants to optimize readibility and not the smallest possible ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 38 \uf8f1\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f2 \uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f3 푧2 푖ℓ = 푧푖ℓ ∀푖 ∈ [푛] , ℓ ∈ [푘] (indicators) � ℓ∈[푘] 푧푖ℓ ⩽ 1 ∀푖 ∈ [푛] (cluster mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=') 푧푖ℓ · 푧푖ℓ′ = 0 ∀푖 ∈ [푛] , ℓ ∈ [푘] (uniq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=') � 푖 푧푖ℓ ⩽ 푛/푘 ∀ℓ ∈ [푘] (size of clusters) 휇′ ℓ = 푘 푛 � 푖 푧푖ℓ · 푦푖 ∀ℓ ∈ [푘] (means of clusters) ∀푣 ∈ ℝ푑 : 푘 푛 푛 � 푖=1 푧푖ℓ ⟨푦푖 − 휇′ ℓ , 푣⟩2푠 + ��푄푣⊗푠��2 = (2푠)푠 · ∥푣∥2푠 2 ∀푠 ⩽ 푡, ℓ ∈ [푘] (푡 moment) \uf8fc\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8fd \uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8fe (풫푛,푘,푡(푌)) We remark that the moment constraint encodes the 2푡-explicit 2-boundedness con- straint introduced in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Note that in the form stated above there are infinitely many constraints, one for each vector 푣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' This is just for notational convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' This con- straint postulates equality of two polynomials in 푣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Formally, this can also be encoded by requiring there coefficients to agree and hence eliminating the variable 푣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' It is not hard to see that this can be done adding only polynomially many constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Further, the matrix variable 푄 represents the SOS proof of the 2푡-explicit 2-boundedness constraint and we can hence deduce that for all 0 ⩽ 푠 ⩽ 푡 풫 2푠 푣 � 푘 푛 푛 � 푖=1 푧푖ℓ ⟨푦푖 − 휇′ ℓ , 푣⟩2푠 ⩽ (2푠)푠∥푠∥2푠 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Before presenting the algorithm we will introduce some additional notation which will be convenient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We assume 푡, 푛, 푘 to be fixed throughout the section and drop the cor- responding subscripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For 푌 ∈ 풴, let 풵(푌) be the set of degree-10푡 pseudo-distributions satisfying 풫(푌).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For each 휁 ∈ 풵(푌) define 푊(휁) as the 푛-by-푛 matrix satisfying 푊(휁)푖푗 = ˜피휁 \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 � ℓ∈[푘] 푧푖ℓ · 푧푗ℓ \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We let 풲(푌) := {푊(휁) | 휁 ∈ 풵(푌)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Recall that 퐽 denotes the all-ones matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We define the function 푔 : ℝ푛×푛 → ℝ as 푔(푊) = ∥푊 ∥2 F − (10)10푘300⟨퐽, 푊⟩ and let 푊(ˆ휁(푌)) ≔ argmin푊∈풲(푌) 푔(푊) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We also consider the following function 39 Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4 (Soft thresholding function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We denote by 휙 : [0, 1] → [0, 1] the function 휙(푥) = \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2 \uf8f4\uf8f4\uf8f4\uf8f3 0 if 푥 ⩽ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='8 , 1 if 푥 ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='9 , 푥−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='9−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='8 otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Notice that 휙(·) is 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='9−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='8 = 10 Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Next we introduce our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Notice the algorithm relies on certain private subroutines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We describe them later in the section to improve the presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 40 Algorithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='5 (Private algorithm for learning mixtures of Gaussians).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Input: Set of 푛 points 푌 ⊆ ℝ푑 , 휀 , 훿 > 0 , 푘, 푡 ∈ ℕ , 푑∗ = 100 log 푛 , 푏 = 푘−15 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Compute 푊 = 푊(ˆ휁(푌)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Pick 흉 ∼ tLap � −푛1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='6� 1 + log(1/훿) 휀 � , 푛1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='6 휀 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' If |흉| ⩾ 푛1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='7 or ��휙(푊) �� 1 ⩽ 푛2 푘 · � 1 − 1 푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1 − 1 푘100 � + 흉 reject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For all 푖 ∈ [푛] , compute the 푛-dimensional vector 휈(푖) = � 0 if ��휙(푊푖) �� 1 = 0 ��휙(푊푖) ��−1 1 � 푗 휙(푊푖푗) · 푦푗 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Pick a set 퓢 of 푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='01 indices 푖 ∈ [푛] uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For each 푖 ∈ 퓢 let ¯흂(푖) = 휈(푖) + w where w ∼ 푁 � 0, 푛−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='18 · log(2/훿) 휀2 Id � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Pick 횽 ∼ 푁 � 0, 1 푑∗ �푑∗×푑 , q 푢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='푎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='푟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' ∼ [0, 푏] and run the histogram learner of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='13 with input 횽 ¯흂(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' , 횽 ¯흂(푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='01) and parameters q, 푏, 훼 = 푘−10, 훽 = 푛−10, 훿∗ = 훿 푛 , 휀∗ = 휀 · 10푘50 푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='01 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' , B푘 be the resulting 푑∗-dimensional bins with highest counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Break ties randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Reject if min푖∈[푘] ��� 푗 �� 횽 ¯흂(푗) ∈ B푖 ��� < 푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='01 2푘 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For each 푙 ∈ [푘] output ˆ흁푙 ≔ 1 ��� 푗 �� 횽 ¯흂(푗) ∈ B푖 ��� · �� � � 횽 ¯흂(푗)∈B푙 ¯흂(푗)�� � + w′ , where w′ ∼ 푁 � 0, 푁 � 0, 32 · 푘−120 · log(2푘푛/훿) 휀2 Id �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For convenience, we introduce some preliminary facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='6 (Good 푌).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let Y be sampled according to Model 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We say that Y is good if: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' for each ℓ ∈ [푘], there are at least 푛 푘 − 푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='6 and most 푛 푘 + 푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='6 points sampled from 퐷ℓ in Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let Yℓ ⊆ Y be such set of points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 41 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Each Yℓ is 2푡-explicitly 2-bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' It turns out that typical instances Y are indeed good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' [HL18, KSS18] Consider the settings of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then Y is good with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Further, in this case the sets 풵(푌) and 풲(푌) are non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1 Privacy analysis In this section we show that our clustering algorithm is private.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='8 (Differential privacy of the algorithm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Consider the settings of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then Algorithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='5 is (휀, 훿)-differentially private.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We split our analysis in multiple steps and combine them at the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' On a high level, we will argue that on adjacent inputs 푌, 푌′ many of the vectors 휈(푖) by the algorithm are close to each other and a small part can be very far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We can then show that we can mask this small difference using the Gaussian mechanism and afterwards treat this subset of the vectors as privatized (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then we can combine this with known histogram learners to deal with the small set of 휈(푖)’s that is far from each other on adjacent inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1 Sensitivity of the matrix W Here we use Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1 to reason about the sensitivity of 휙(푊(ˆ휁(푌))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For adjacent datasets 푌, 푌′ ∈ 풴 we let ˆ휁 , ˆ휁′ be the pseudo-distribution corresponding to 푊(ˆ휁(푌)) and 푊(ˆ휁(푌′)) computed in step 1 of the algorithm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We prove the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='9 (ℓ1-sensitivity of 휙(푊)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Consider the settings of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 푊, 푊′ be re- spectively be the matrices computed in step 1 by Algorithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='5 on adjacent inputs 푌, 푌′ ∈ 풴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then ��휙(푊) − 휙(푊′) �� 1 ⩽ 푛1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For all but 푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='8 rows 푖 of 휙(푊), 휙(푊′), it holds ��휙(푊)푖 − 휙(푊′)푖 �� 1 ⩽ 푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The second inequality is an immediate consequence of the first via Markov’s in- equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Thus it suffices to prove the first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Since 휙(·) is 10-Lipschitz, we immediately obtain the result if ���푊(ˆ휁(푌)) − 푊(ˆ휁(푌′)) ��� 1 ⩽ 푛1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='55 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 42 Thus we focus on this inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' To prove it, we verify the two conditions of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' First notice that 푔 is 2-strongly convex with respect to its input 푊.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Indeed for 푊, 푊′ ∈ 풲(푌), since ∀푖, 푗 ∈ [푛] , 푊푖푗 ⩾ 0 it holds that ∥푊′∥2 F = ∥푊 ∥2 F + ∥푊 − 푊′∥2 F + 2⟨푊′ − 푊, 푊⟩ = ∥푊 ∥2 F + ∥푊 − 푊′∥2 F + 2⟨푊′ − 푊, 푊⟩ + ⟨푊′ − 푊, (10)10푘300(퐽 − 퐽)⟩ = 푔(푊) + ∥푊 − 푊′∥2 F + ⟨푊′ − 푊, ∇푔(푊)⟩ + ⟨푊′, (10)10푘300퐽⟩ , where we used that ∇푔(푊) = 2푊 − (10)10푘300퐽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Thus it remain to prove (i) of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let ˆ휁 ∈ 풵(푌) , ˆ휁′ ∈ 풵(푌′) be the pseudo-distributions such that 푊푌(ˆ휁) = 푊 and 푊푌(ˆ휁′) = 푊′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We claim that there always exists 휁adj ∈ 풵(푌) ∩ 풵(푌′) such that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' |푔(푊(휁)) − 푔(푊(휁adj)| ⩽ 2푛 푘 · � (10)10푘300 + 1� ⩽ 3 · (10)10푘300푛 , 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' |푔푌′(푊(휁adj)) − 푔(푊(휁adj)| = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Note that in this case the second point is always true since 푔 doesn’t depend on 푌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Together with Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1 these two inequalities will imply that ���푊(ˆ휁(푌)) − 푊(ˆ휁(푌′)) ��� 2 F ⩽ 18 · (10)10푘300푛 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' By assumption on 푛, an application of Cauchy-Schwarz will give us the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' So, let 푖 be the index at which 푌, 푌′ differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We construct 휁adj as follows: for all polyno- mials 푝 of degree at most 10푡 we let ˜피휁adj � 푝 � = � ˜피휁 � 푝 � if 푝 does not contain variables 푧푖ℓ for any ℓ ∈ [푘] 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' By construction 휁adj ∈ 풵(푌)∩풵(푌′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Moreover, 푊(휁), 푊(휁adj) differ in at most 2푛/푘 entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Since all entries of the two matrices are in [0, 1], the first inequality follows by definition of the objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2 Sensitivity of the resulting vectors In this section we argue that if the algorithm does not reject in step 3 then the vectors 휈(푖) are stable on adjacent inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Concretely our statement goes as follows: Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='10 (Stability of the 휈(푖)’s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Consider the settings of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Suppose Algorithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='5 does not reject in step 3, on adjacent inputs 푌 , 푌′ ∈ 풴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then for all but 6푛 푘50 indices 푖 ∈ [푛], it holds: ���휈(푖) 푌 − 휈(푖) 푌′ ��� 2 ⩽ 푂� 푛−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='10 crucially relies on the next statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 43 Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='11 (Covariance bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Consider the settings of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 푊 be the matrix computed by Algorithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='5 on input 푌 ∈ 풴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For 푖 ∈ [푛], if ��휙(푊푖) �� 1 ⩾ 푛 푘 · � 1 − 10 푘50 � then 휈(푖) induces a 2-explicitly 40-bounded distribution over 푌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' First, by assumption notice that there must be at least 푛 푘 · � 1 − 10 푘50 � entries of 휙(푊푖) larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We denote the set of 푗 ∈ [푛] such that 푊푖푗 ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='8 by 풢 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 휁 ∈ 풵(푌) be the degree 10푡 pseudo-distribution so that 푊 = 푊(휁(푌)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Since 휁 satisfies 풫(푌), for ℓ ∈ [푘] it follows from the moment bound constraint for 푠 = 1 that for all unit vectors 푢 it holds that 풫 4 \uf8f1\uf8f4\uf8f4\uf8f2 \uf8f4\uf8f4\uf8f3 0 ⩽ 푘 푛 푛 � 푗=1 푧푗ℓ ⟨y푗 − 휇′ 푙, 푢⟩2 ⩽ 2 \uf8fc\uf8f4\uf8f4\uf8fd \uf8f4\uf8f4\uf8fe , Using the SOS triangle inequality (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Fact E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2) 2 푎,푏 (푎 + 푏)2 ⩽ 2(푎2 + 푏2) it now follows that 0 ⪯ ˜피휁 \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 푘2 푛2 � 푗 ,푗′∈[푛] 푧푗ℓ 푧푗′ℓ · � 푦푗 − 푦푗′�⊗2 \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb ⪯ 8Id and thus 0 ⪯ ˜피휁 \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 푘2 푛2 � ℓ∈[푘] � 푗 ,푗′∈[푛] 푧푖ℓ 푧푗ℓ 푧푗′ℓ · � 푦푗 − 푦푗′�⊗2 \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb ⪯ 8Id .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Furthermore using 풫(푌) 2 {푧푖ℓ 푧푖ℓ′ = 0} for ℓ ≠ ℓ′ we have ˜피휁 \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 � ℓ∈[푘] � 푗 ,푗′∈[푛] 푧푖ℓ 푧푗ℓ 푧푗′ℓ \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb = ˜피휁 \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 �� � � ℓ∈[푘] ,푗∈[푛] 푧푖ℓ 푧푗ℓ�� � �� � � ℓ′∈[푘] ,푗′∈[푛] 푧푖ℓ′푧푗′ℓ′�� � \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Now, for fixed 푗 , 푗′ ∈ [푛], using � 푎2 = 푎 , 푏2 = 푏 � 푂(1) � 1 + 푎푏 − 푎 − 푏 = 1 − 푎푏 − (푎 − 푏)2 ⩾ 0 � with 푎 = � ℓ∈[푘] 푧푖ℓ 푧푗ℓ and 푏 = � ℓ′∈[푘] 푧푖ℓ′푧푗′ℓ′ we get ˜피휁 \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 �� � � ℓ∈[푘] 푧푖ℓ 푧푗ℓ�� � �� � � ℓ′∈[푘] 푧푖ℓ′푧푗′ℓ′�� � \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb ⩾ ˜피휁 \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 � ℓ∈[푘] 푧푖ℓ 푧푗ℓ + � ℓ′∈[푘] 푧푖ℓ′푧푗′ℓ′ \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb − 1 = 푊푖푗 + 푊푖푗′ − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Now if 푗, 푗′ ∈ 풢 we must have � ℓ∈[푘] ˜피휁 � 푧푖ℓ 푧푗ℓ 푧푗′ℓ � = ˜피휁 \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 �� � � ℓ∈[푘] 푧푖ℓ 푧푗ℓ�� � �� � � ℓ′∈[푘] 푧푖ℓ′푧푗′ℓ′�� � \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 44 Since 휙(푊푖푗) ⩽ 1 by definition and ��휙(푊푖) �� 1 ⩾ 푛 푘 · � 1 − 10 푘50 � , we conclude ��휙(푊푖) �� 1 −2 \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 � 푗 ,푗′∈[푛] 휙(푊푖푗)휙(푊푖푗′)� 푦푗 − 푦푗′�⊗2 \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb ⪯ 5 · 푘2 푛2 � 푗 ,푗′∈[푛] ,ℓ∈[푘] ˜피휁 � 푧푖ℓ 푧푗ℓ 푧푗′ℓ � � 푦푗 − 푦푗′�⊗2 ⪯ 40Id .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' □ We can now prove Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 푊, 푊′ be the matrices computed by Algorithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='5 in step 1 on input 푌, 푌′, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 풢 ⊆ [푛] be the set of indices 푖 such that ��휙(푊)푖 − 휙(푊′)푖 �� 1 ⩽ 푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Notice that |풢| ⩾ 푛 − 푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='8 by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Since on input 푌 the algorithm did not reject in step 3 we must have ��휙(푊) �� 1 ⩾ 푛2 푘 · � 1 − 1 푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1 − 1 푘100 � − 푛1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='7 ⩾ 푛2 푘 · � 1 − 2 푘100 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 푔푊 be the number of indices 푖 ∈ 풢 such that ��휙(푊)푖 �� 1 ⩾ 푛 푘 · � 1 − 1 푘50 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' It holds that 푛2 푘 · � 1 − 2 푘100 � ⩽ 푔푊 · 푛 푘 + (푛 − |풢|) · 푛 푘 + � |퐺| − 푔푤 � 푛 푘 · � 1 − 1 푘50 � ⩽ 푔푊 · 푛 푘 · 1 푘50 + 푛1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='8 푘 + 푛2 푘 · � 1 − 1 푘50 � ⩽ 푔푊 · 푛 푘 · 1 푘50 + 푛2 푘 · � 1 + 1 푘100 − 1 푘50 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Rearring now yields 푔푊 ⩾ 푛 · � 1 − 3 푘50 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Similarly, let 푔푊′ be the number of indices 푖 ∈ 풢 such that ��휙(푊′)푖 �� 1 ⩾ 푛 푘 · � 1 − 1 푘50 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' By an analogous argument it follows that 푔푊′ ⩾ 푛 · � 1 − 3 푘50 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Thus, by the pigeonhole principle there are at least 푔푊 ⩾ 푛 · � 1 − 6 푘50 � indices 푖 such that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' ��휙(푊)푖 �� 1 ⩾ 푛 푘 � 1 − 1 푘50 � , 45 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' ��휙(푊′)푖 �� 1 ⩾ 푛 푘 � 1 − 1 푘50 � , 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' ��휙(푊)푖 − 휙(푊′)푖 �� 1 ⩽ 푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Combining these with Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='11 we may also add 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' the distribution induced by ��휙(푊푖) ��−1 1 휙(푊푖) is 2-explicitly 40-bounded, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' the distribution induced by ��휙(푊′ 푖 ) ��−1 1 휙(푊′ 푖 ) is 2-explicitly 40-bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Using that for non-zero vectors 푥, 푦 it holds that ��� 푥 ∥푥∥ − 푦 ∥푦∥ ��� ⩽ 2 ∥푥∥ ��푥 − 푦 �� points 1 to 3 above imply that ��� ��휙(푊푖) ��−1 1 휙(푊푖) − ��휙(푊′ 푖 ) ��−1 1 휙(푊′ 푖 ) ��� 1 ⩽ 2푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='8 푛 푘 · � 1 − 1 푘50 � = 푂� 푛−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Hence, applying Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='21 with 푡 = 1 it follows that ���휈(푖) 푌 − 휈(푖) 푌′ ��� 2 ⩽ 푂� 푛−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='3 From low sensitivity to privacy In this section we argue privacy of the whole algorithm, proving Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Before doing that we observe that low-sensitivity is preserved with high probability under subsampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Fact 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='12 (Stability of 퓢).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Consider the settings of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Suppose Algorithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='5 does not reject in step 3, on adjacent inputs 푌 , 푌′ ∈ 풴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' With probability at least 1 − 푒−푛Ω(1) over the random choices of 퓢, for all but 10푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='01 푘50 indices 푖 ∈ 퓢, it holds: ���휈(푖) 푌 − 휈(푖) 푌′ ��� 2 ⩽ 푂� 푛−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' There are at most 6푛 푘50 such indices in [푛] by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' By Chernoff’s bound, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Fact A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4, the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' □ Finally, we prove our main privacy lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For simplicity, we will prove that the algorithm is (5휀, 5훿)-private.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 푌, 푌′ ∈ 풴 be adjacent inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='10 and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='9 the test in step 3 of Algorithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='5 is (휀, 훿)-private.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Thus suppose now the algorithm did not reject in step 3 on inputs푌, 푌′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' By composition (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4) it is enough to show that the rest of the algorithm is (휀, 훿)-private with respect to 푌, 푌′ under this condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Next, let 휈(1) 푌 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' , 휈(푛) 푌 and 휈(1) 푌′ , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' , 휈(푛) 푌′ be the vectors 46 computed in step 4 of the algorithm and 풮 be the random set of indices computed in step 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='30 By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='10 and Fact 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='12 with probability 1 − 푒−푛Ω(1) over the random choices of 퓢 we get that for all but 10푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='01 푘50 indices 푖 ∈ 퓢, it holds that ���휈(푖) 푌 − 휈(푖) 푌′ ��� 2 ⩽ 푂� 푛−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Denote this set of indices by 풢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Note, that we may incorporate the failure probability 푒−푛Ω(1) ⩽ min{휀/2, 훿/2} into the final privacy parameters using Fact E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Denote by V, V′ the |퓢|-by-푑 matrices respectively with rows 휈(푖1) 푌 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' , 휈 (푖|퓢|) 푌 and 휈(푖1) 푌′ , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' , 휈 (푖|퓢|) 푌′ , where 푖1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' , 푖|퓢| are the indices in 퓢 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Recall, that |풢| rows of V and V′ differ by at most 푂� 푛−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1� in ℓ2-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Thus, by the Gaussian mechanism used in step 6 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='12) and Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4 it is enough to show that step 7 to step 9 of the algo- rithm are private with respect to pairs of inputs 푉 and 푉′ differing in at most 1 row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='31 In particular, suppose these steps are (휀1, 훿1)-private.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then, for 푚 = 푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='01 − |풢| ⩽ 10푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='01 푘50 , by Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4 it follows that step 6 to step 9 are (휀′, 훿′)-differentially private with 휀′ ≔ 휀 + 푚휀1 , 훿′ ≔ 푒휀푚푒(푚−1)휀1훿1 + 훿 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Consider steps 7 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Recall, that in step 7 we invoke the histogram learner with parameters 푏 = 푘−15, q 푢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='푎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='푟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' ∼ [0, 푏], 훼 = 푘−10, 훽 = 푛−10, 훿∗ = 훿 푛 , 휀∗ = 휀 · 10푘50 푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='01 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Hence, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='13 this step is (휀∗, 훿∗)-private since 8 휀∗훼 · log � 2 훿∗훽 � ⩽ 200 · 푘10 · 푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='01 10 · 푘50 · 휀 log 푛 = 20 · 푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='01 푘40 · 휀 log 푛 ⩽ 푛 , for 휀 ⩾ 푘−10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Step 8 is private by post-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Next, we argue that step 9 is private by showing that the average over the bins has small ℓ2-sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4 we can consider the bins B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' , B푘 computed in the previous step as fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Further, we can assume that the algorithm did not reject in step 8, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=', that each bin contains at least 푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='01 2푘 points of 푉 and 푉′ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' As a consequence, every bin contains at least two (projections of) points of the input 푉 or 푉′ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' In particular, it contains at least one (projection of a) point which is present in both 푉 and 푉′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Fix a bin B푙 and let ¯휈∗ be such that it is both in 푉 and 푉′ and 횽¯휈∗ ∈ B푙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Also, define 푆푙 ≔ ��� � 푗 ��� 횽¯휈(푗) 푌 ∈ B푖 ���� , 30Note that since this does not depend on 푌 or 푌′, respectively, we can assume this to be the same in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Formally, this can be shown, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=', via a direct calculation or using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 31Note that for the remainder of the analysis, these do not correspond to V and V′, since those differ in 푚 rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4 handles this difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 47 푆′ 푙 ≔ ��� � 푗 ��� 횽¯휈(푗) 푌′ ∈ B푖 ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Assume 푉 and 푉′ differ on index 푗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We consider two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' First, assume that 횽¯휈(푗) 푌 and 횽¯휈(푗) 푌′ both lie in B푙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' In this case, 푆푙 = 푆′ 푙 and using Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='5 it follows that with probability 푛−100 ⩽ min{휀/2, 훿/2} it holds that ���¯휈(푗) 푌 − ¯휈(푗) 푌′ ��� 2 ⩽ ���¯휈(푗) 푌 − ¯휈∗��� 2 + ���¯휈∗ − ¯휈(푗) 푌′ ��� ⩽ 10 · ����횽¯휈(푗) 푌 − 횽¯휈∗��� 2 + ���횽¯휈(푗) 푌′ − 횽¯휈∗��� 2 � ⩽ 20 · √ 푑∗ · 푏 ⩽ 200 · 푘−12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' And hence we can bound ������� 1 푆푙 ��� � � 횽¯휈(푗) 푌 ∈B푙 ¯휈(푗) 푌 ��� � − 1 푆′ 푙 ��� � � 횽¯휈(푗) 푌′∈B푙 ¯휈(푗) 푌′ ��� � ������� 2 ⩽ ���¯휈(푗) 푌 − ¯휈(푗) 푌′ ��� 2 푆푙 ⩽ 400 · 푘−11 푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='01 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Next, assume that 횽¯휈(푗) 푌 ∉ B푙 and 횽¯휈(푗) 푌′ ∈ B푙 (the other case works symetrically).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' It follows ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='that 푆푙 = 푆′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='푙 − 1 and we can bound ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='푆푙 ��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='횽¯휈(푗) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='푌 ∈B푙 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='¯휈(푗) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='푌 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='푆′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='푙 ��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='횽¯휈(푗) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='푌′∈B푙 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='¯휈(푗) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='푌′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='푆푙 · 푆′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='푙 ������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='푆′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='푙 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='횽¯휈(푗) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='푌 ∈B푙 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='¯휈(푗) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='푌 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='− � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='푆′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='푙 − 1���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='횽¯휈(푗) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='푌′∈B푙 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='¯휈(푗) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='푌′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='푆푙 · 푆′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='푙 ������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='푆′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='푙 · ¯휈(푗) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='푌′ + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='횽 ¯휈(푗) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='푌′∈B푙 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='¯휈(푗) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='푌′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='= 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='푆푙 ������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='¯휈(푗) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='푌′ − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='푆′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='푙 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='횽¯휈(푗) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='푌′∈B푙 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='¯휈(푗) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='푌′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='⩽ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='푑∗ · 푏 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='푆푙 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='⩽ 20 · 푘−11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='01 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Hence, the ℓ2-sensitivity is at most Δ ≔ 400·푘−11 푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='01 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Since 2Δ2 · log(2/(훿∗/푘)) (휀∗/푘)2 = 32 · 푘−120 · log(2푘푛/훿) 휀2 and w′ ∼ 푁 � 0, 32 · 푘−120 · log(2푘푛/훿) 휀2 Id � it follows that outputing ˆ흁푙 is (휀∗/푘, 훿∗/푘)-DP by the Gaussian Mechanism that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4 it follows step 9 is (휀∗, 훿∗)-private.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Hence, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4 it follows that step 7 to step 9 are (2휀∗, 2훿∗)-differentially private.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Using 푚 ⩽ 10푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='01 푘10 it now follows by Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4 that step 6 to step 9 are (휀′, 훿′)-private for 휀′ = 휀 + 2푚휀∗ ⩽ 3휀 , 48 훿′ = 2푒휀푚푒(푚−1)2휀∗훿∗ + 훿 ⩽ 2푚푒3휀 · 훿 푛 + 훿 ⩽ 3훿 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Thus, combined with the private check and Fact E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='3 in step 3 the whole algorithm is (5휀, 5훿)-private.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2 Utility analysis In this section we reason about the utility of Algorithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='5 and prove Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We first introduce some notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='13 (True solution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let Y be an input sampled from Model 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Denote by 푊∗(Y) ∈ 풲(Y) the matrix induced by the true solution (or ground truth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=', let 푊∗(Y)푖푗 = � 1 if 푖 , 푗 were both sampled from the same component of the mixture, 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Whenever the context is clear, we simply write W∗ to ease the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' First, we show that in the utility case step 3 of Algorithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='5 rejects only with low probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='14 (Algorithm does not reject on good inputs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Consider the settings of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Suppose Y is a good set as per Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then ���푊(ˆ휁(Y)) ��� 1 ⩾ 푛2 푘 · � 1 − 푛−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4 − 1 (10)10푘300 � and Algorithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='5 rejects with probability at most exp� −Ω� 푛1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='7�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Since Y is good, there exists W∗ ∈ 풲(Y), corresponding to the indicator matrix of the true solution, such that 푔(W∗) = ∥W∗∥2 F − 1010푘300⟨퐽, W∗⟩ ⩽ 푛2 푘 + 푛1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='6 − (10)10푘300 � 푛2 푘 − 푛1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='6 � = 푛2 푘 � 1 + 푘 푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4 − (10)10푘300 � 1 − 푘 푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Since 푔(푊(ˆ휁(Y))) ⩽ 푔(W∗) it follows that (10)10푘300⟨퐽, 푊(ˆ휁(Y))⟩ ⩾ |푔(푊(ˆ휁(Y)))| ⩾ 푛2 푘 � (10)10푘300 � 1 − 푘 푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4 � − 1 − 푘 푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Since, ���푊(ˆ휁(Y)) ��� 1 ⩾ ⟨퐽, 푊(ˆ휁(Y))⟩ the first claim follows rearranging the terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' This means that the algorithm rejects only if |흉| ⩾ 푛1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Recall that 흉 ∼ tLap � −푛1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='6� 1 + log(1/훿) 휀 � , 푛1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='6 휀 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Hence,by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='11 it follows that ℙ� |흉| ⩾ 푛1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='7� ⩽ exp� −푛1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='7 + 휀 + log(1/훿)� 2 − exp� −휀 − log(1/훿)� = exp� −Ω� 푛1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='7�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' □ 49 The next step shows that on a good input Y the matrix 휙(푊(ˆ휁(Y))) is close to the true solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='15 (Closeness to true solution on good inputs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Consider the settings of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Suppose Y is a good set as per Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 푊(Y) ∈ 풲(Y) be the matrix computed by Algorithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Suppose the algorithm does not reject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then ��휙(푊(Y)) − W∗�� 1 ⩽ 푛2 푘 · 3 푘98 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The proof is similar to the classical utility analysis of the sum-of-squares program found, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=', in [HL18, FKP+19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We defer it to Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Together, the above results imply that the vectors 휈(푖) computed by the algorithm are close to the true centers of the mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='16 (Closeness to true centers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Consider the settings of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Suppose Y is a good set as per Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let W ∈ 풲(Y) be the matrix computed by Algorithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Suppose the algorithm does not reject in step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then for each ℓ ∈ [푘], there exists 푛 푘 · � 1 − 2 푘47 � indices 푖 ∈ [푛], such that ��휈(푖)(W) − 휇ℓ �� 2 ⩽ 푂� 푘−25� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We aim to show that for most indices 푖 ∈ [푛] the vectors ��휙(W푖) ��−1 1 휙(W푖) and ��W∗ 푖 ��−1 1 W∗ 푖 induce a 2-explicitly 40-bounded distribution over Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' If additionally the two vectors are close in ℓ1-norm, the result will follow by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Note that ��W∗ 푖 ��−1 1 W∗ 푖 induces a 2-explicitly 40-bounded distribution by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' By Markov’s inequality and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='15 there can be at most 푛/푘48 indices 푗 ∈ [푛] such that ���휙(W)푗 − W∗ 푗 ��� 1 ⩾ 푛 푘 · 3 푘50 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Consider all remaining indices 푖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' It follows that ��휙(W푖) �� 1 ⩾ ��W∗ 푖 �� 1 − ��휙(W)푖 − W∗ 푖 �� 1 ⩾ 푛 푘 · � 1 − 푘 푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4 − 3 푘50 � ⩾ 푛 푘 · � 1 − 10 푘50 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Hence, by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='11 the distribution induced by ��휙(W푖) ��−1 1 휙(W푖) is 2-explicitly 40- bounded distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Further, using ��W∗ 푖 �� 1 ⩾ 푛 푘 � 1 − 푘 푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4 � we can bound ��� ��휙(W푖) ��−1 1 휙(W푖) − ��W∗ 푖 ��−1 1 W∗ 푖 ��� 1 = ��휙(W푖) ��−1 1 ��W∗ 푖 ��−1 1 · ����W∗ 푖 �� 1휙(W푖) − ��휙(W푖) �� 1W∗ 푖 �� 1 ⩽ ��휙(W푖) ��−1 1 ��W∗ 푖 ��−1 1 · �����휙(W푖) �� 1 − ��W∗ 푖 �� 1 �� · ��휙(W푖) �� 1 + ��휙(W푖) �� 1 · ��휙(W푖) − W∗ 푖 �� 1 � ⩽ ��W∗ 푖 ��−1 1 · 2 ��휙(W푖) − W∗ 푖 �� 1 ⩽ 6 푘50 · � 1 − 푘 푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4 � ⩽ 7 푘50 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 50 Hence, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='21 for each 푙 ∈ [푘] there are at least 푛 푘 − 푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='6 − 푛 푘48 ⩾ 푛 푘 · � 1 − 2 푘47 � indices 푖 such that ������ 휈(푖)(W) − ��W∗ 푖 ��−1 1 푛 � 푗=1 W∗ 푖,푗y푗 ������ 2 ⩽ 푂� 푘−25� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The result now follows by standard concentration bounds applied to the distribution induced by ��W∗ 푖 ��−1 1 W∗ 푖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' □ An immediate consequence of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='16 is that the vectors ¯흂(푖) inherits the good properties of the vectors 휈(푖) with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='17 (Closeness to true centers after sub-sampling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Consider the settings of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Suppose Y is a good set as per Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let W ∈ 풲(Y) be the matrix computed by Algorithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Suppose the algorithm does not reject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then with high probability for each ℓ ∈ [푘], there exists 푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='01 푘 � 1 − 150 푘47 � indices 푖 ∈ 퓢, such that �� ¯흂(푖) − 휇ℓ �� 2 ⩽ 푂� 푘−25� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For each ℓ ∈ [푘], denote by 풯ℓ the set of indices in [푛] satisfying ��휈(푖)(W) − 휇ℓ �� 2 ⩽ 푂� 푘−25� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='16 we know that 풯ℓ has size at least 푛 푘 · � 1 − 2 푘47 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Further, let 풮 be the set of indices selected by the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' By Chernoff’s bound Fact A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4 with probability 1−푒−푛Ω(1) , we have |퓢 ∩ 풯ℓ | ⩾ 푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='01 푘 � 1 − 150 푘47 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Taking a union bound over all ℓ ∈ [푘] we get that with probability 1 − 푒−푛Ω(1) , for each ℓ ∈ [푘], there exists 푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='01 푘 � 1 − 150 푘47 � indices 푖 ∈ 퓢 such that ��휈(푖)(W) − 휇ℓ �� 2 ⩽ 푂� 푘−25� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Now, we obtain the corollary observing (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Fact A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1 with 푚 = 1) that with probability at least 1 − 푒−푛Ω(1), for all 푖 ∈ 퓢 �� ¯흂(푖) − 휈(푖)(W) �� 2 = ∥w∥2 ⩽ 푛−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='05 · � log(2/훿) 휀 √ 푑 ⩽ 푛−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='04 ⩽ 푂� 푘−25� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' □ For each ℓ, denote by 퓖ℓ ⊆ 퓢 the set of indices 푖 ∈ 퓢 satisfying �� ¯흂(푖) − 휇ℓ �� 2 ⩽ 푂� 푘−25� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 퓖 := � ℓ∈[푘] 퓖ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We now have all the tools to prove utility of Algorithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We achieve this by showing thst with high probability, each bin returned by the algorithm at step 7 satisfies 퓖ℓ′ ⊆ Bℓ for some ℓ , ℓ′ ∈ [푘] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Choosing the bins small enough will yield the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 51 Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='18 (Closeness of estimates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Consider the settings of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Suppose Y is a good set as per Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let W ∈ 풲(Y) be the matrix computed by Algorithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Suppose the algorithm does not reject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then with high probability, there exists a permutation 휋 : [푘] → [푘] such that max ℓ∈[푘] ��휇ℓ − ˆ흁휋(ℓ) �� 2 ⩽ 푂� 푘−20� Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Consider distinct ℓ, ℓ′ ∈ [푘].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' By Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='17 for each ¯흂(푖) , ¯흂(푗) ∈ 퓖ℓ it holds that �� ¯흂(푖) − ¯흂(푗)�� 2 ⩽ 퐶 · 푘−25 , for some universal constant 퐶 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Moreover, by assumption on 휇ℓ, 휇ℓ′ for each ¯흂(푖) ∈ 퓖ℓ and ¯흂(푗) ∈ 퓖ℓ′ �� ¯흂(푖) − ¯흂(푗)�� 2 ⩾ Δ − 푂� 푘−25� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Thus, by Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='5 with probability at least 1 − 푒Ω(푑∗) ⩾ 1 − 푛−100 it holds that or each ¯흂(푖) , ¯흂(푗) ∈ 퓖ℓ and ¯흂푟 ∈ 풢ℓ′ with ℓ′ ≠ ℓ , ��횽 ¯흂(푖) − 횽 ¯흂(푗)�� 2 ⩽ 퐶∗ · 푘−25 and ��횽 ¯흂(푖) − 횽 ¯흂(푟)�� 2 ⩾ Δ − 퐶∗ · 푘−25 for some other universal constant 퐶∗ > 퐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 푄횽(퓖ℓ) ⊆ ℝ푑∗ be a ball of radius 퐶∗ · � 푘−25� such that ∀푖 ∈ 퓖ℓ it holds 횽 ¯흂(푖) ∈ 푄횽(퓖ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' That is, 푄횽(퓖ℓ) contains the projection of all points in 퓖ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Recall that 푑∗ = 100 log(푛) ⩽ 100푘5 and 푏 = 푘−15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 퓑 = {B푖}∞ 푖=1 be the sequence of bins computed by the histogram learner of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='13 for ℝ푑∗ at step 7 of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' By choice of 푏, and since q is chosen uniformly at random in [0, 푏], the probability that there exists a bin B ∈ 퓑 containing 푄횽(퓖ℓ) is at least 1 − 푑∗ · 퐶∗ 푏 · � 푘−25� ⩾ 1 − 100퐶∗ 푏 푘−20 ⩾ 1 − 푂� 푘−5� , where we used that 푑∗ = 100 log 푛 ⩽ 100푘5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' A simple union bound over ℓ ∈ [푘] yields that with high probability for all ℓ ∈ [푘] , there exists B ∈ 퓑 such that 푄횽(퓖ℓ) ⊆ B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For simplicity, denote such bin by Bℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We continue our analysis conditioning on the above events, happening with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' First, notice that for all 푙 ∈ [푘] max 푢,푢′∈Bℓ ∥푢 − 푢′∥2 2 ⩽ 푑∗ · 푏2 ⩽ 100푘−25 ⩽ Δ − 퐶∗푘−25 푘10 , and thus there cannot be ℓ, ℓ′ ∈ [푘] such that 푄횽(퓖ℓ) ⊆ Bℓ and 푄횽(퓖′ ℓ) ⊆ Bℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Moreover, by Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='17 and min ℓ∈[푘]|퓖ℓ | ⩾ 푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='01 푘 � 1 − 150 푘47 � , 52 and hence |퓢 \\ 퓖| ⩽ 푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='01 · 150 푘47 = 푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='01 푘 150 푘46 it must be that step 7 returned bins B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' , B푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' This also implies that the algorithm does not reject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Further, by Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='5 for all ¯흂(푖), ¯흂(푗) such that 횽 ¯흂(푖), 횽 ¯흂(푗) ∈ B푙 it holds that �� ¯흂(푖) − ¯흂(푗)�� 2 ⩽ 퐶∗ · ��횽 ¯흂(푖) − 횽 ¯흂(푗)�� 2 ⩽ 퐶∗ · √ 푑∗ · 푏 ⩽ 푂� 푘−12� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' And hence, by triangle inequality, we get �� ¯흂(푖) − 휇푙 �� 2 ⩽ 푂� 푘−12� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Finally, recall that for each ℓ ∈ [푘], ˆ흁푙 ≔ 1 ��� 푗 �� 횽 ¯흂(푗) ∈ B푖 ��� · �� � � 횽 ¯흂(푗)∈B푙 ¯흂(푗)�� � + w′ , where w′ ∼ 푁 � 0, 푁 � 0, 32 · 푘−120 · log(2푘푛/훿) 휀2 Id �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Since by choice of 푛, 푘, 휀 it holds that 32 · 푘−120 · log(2푘푛/훿) 휀2 ⩽ 푂� 푘−90� , we get with probability at least 1 − 푒−푘Ω(1) for each ℓ ∈ [푘], by Fact A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1, with 푚 = 1, and a union bound that ∥w′∥ ⩽ 푂� 푘−20� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Since all ¯흂(푖) such that 횽 ¯흂(푖) ∈ B푙 are at most 푂� 푘−12� far from 휇푙, also their average is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We conclude that �� ˆ흁ℓ − 휇푙 �� 2 ⩽ 푂(푘−12) + ∥w∥2 ⩽ 푂(푘−12) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' □ Now Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='3 is a trivial consequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The error guarantees and privacy guarantees immediately follows combining Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='8, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='15, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='14 and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='18.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 1 [Wai19] Martin J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Wainwright, High-dimensional statistics: A non-asymptotic viewpoint, Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge Uni- versity Press, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 63 58 [WYX17] Di Wang, Minwei Ye, and Jinhui Xu, Differentially private empirical risk minimiza- tion revisited: Faster and more general, Advances in Neural Information Processing Systems 30 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 12 [ZZ16] Anderson Y Zhang and Harrison H Zhou, Minimax rates of community detection in stochastic block models, The Annals of Statistics 44 (2016), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 5, 2252–2280.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 5, 31 59 A Concentration inequalities We introduce here several useful and standard concentration inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Fact A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1 (Concentration of spectral norm of Gaussian matrices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let W ∼ 풩(0, 1)푚×푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then for any 푡, we have ℙ �√ 푚 − √ 푛 − 푡 ⩽ 휎min(W) ⩽ 휎max(W) ⩽ √ 푚 + √ 푛 + 푡 � ⩾ 1 − 2 exp � −푡2 2 � , where 휎min(·) and 휎max(·) denote the minimum and the maximum singular values of a matrix, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let W′ be an 푛-by-푛 symmetric matrix with independent entries sampled from 푁(0, 휎2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then ∥W′∥ ⩽ 3휎√푛 with probability at least 1 − exp(−Ω(푛)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Fact A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2 (Maximum degree of Erdős-Rényi graphs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 퐺 be an Erdős-Rényi graph on 푛 vertices with edge probability 푝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then with probability at least 1 − 푛 exp(−푛푝/3), any vertex in 퐺 has degree at most 2푛푝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Fact A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='3 (Gaussian concentration bounds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let X ∼ 풩(0, 휎2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then for any 푡 ⩾ 0, max{ℙ(X ⩾ 푡), ℙ(X ⩽ −푡)} ⩽ exp � − 푡2 2휎2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Fact A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4 (Chernoff bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' , Xn be independent random variables taking values in {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let X := �푛 푖=1 Xi and let 휇 := 피 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then for any 훿 > 0, ℙ� X ⩽ (1 − 훿)휇� ⩽ exp � −훿2휇 2 � , ℙ� X ⩾ (1 + 훿)휇� ⩽ exp � − 훿2휇 2 + 훿 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='5 ([Joh84]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let Φ be a 푑-by-푛 Gaussian matrix, with each entry independently chosen from 푁(0, 1/푑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then, for every vector 푢 ∈ ℝ푛 and every 훼 ∈ (0, 1) ℙ(∥Φ푢∥ = (1 ± 훼)∥푢∥) ⩾ 1 − 푒−Ω(훼2푑) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' B Linear algebra Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1 (Weyl’s inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 퐴 and 퐵 be symmetric matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 푅 = 퐴 − 퐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 훼1 ⩾ · · · ⩾ 훼푛 be the eigenvalues of 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 훽1 ⩾ · · · ⩾ 훽푛 be the eigenvalues of 퐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then for each 푖 ∈ [푛], ��훼푖 − 훽푖 �� ⩽ ∥푅∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 60 Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2 (Davis-Kahan’s theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 퐴 and 퐵 be symmetric matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 푅 = 퐴 − 퐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 훼1 ⩾ · · · ⩾ 훼푛 be the eigenvalues of 퐴 with corresponding eigenvectors 푣1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' , 푣푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 훽1 ⩾ · · · ⩾ 훽푛 be the eigenvalues of 퐵 with corresponding eigenvectors 푢1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' , 푢푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 휃푖 be the angle between ±푣푖 and ±푢푖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then for each 푖 ∈ [푛], sin(2휃푖) ⩽ 2∥푅∥ min푗≠푖 ��훼푖 − 훼푗 ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' C Convex optimization Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 푓 : ℝ푚 → ℝ be a convex function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 풦 ⊆ ℝ푚 be a convex set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then 푦∗ ∈ 풦 is a minimizer of 푓 over 풦 if and only if there exists a subgradient 푔 ∈ 휕 푓 (푦∗) such that � 푦 − 푦∗, 푔 � ⩾ 0 ∀푦 ∈ 풦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Define indicator function 퐼풦(푦) = � 0, 푦 ∈ 풦, ∞, 푦 ∉ 풦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then for 푦 ∈ 풦, one has 휕퐼풦(푦) = � 푔 ∈ ℝ푚 : � 푔, 푦 − 푦′� ⩾ 0 ∀푦′ ∈ 풦 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Note 푦∗ is a minimizer of 푓 over 풦, if and only if 푦∗ is a minimizer of 푓 + 퐼풦 over ℝ푚, if and only if 0푚 ∈ 휕(푓 + 퐼풦)(푦∗) = 휕 푓 (푦∗) + 휕퐼풦(푦∗), if and only if there exists 푔 ∈ 휕 푓 (푦∗) such that ⟨푔, 푦 − 푦∗⟩ ⩾ 0 for any 푦 ∈ 풦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' □ Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2 (Pythagorean theorem from strong convexity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 푓 : ℝ푚 → ℝ be a convex function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 풦 ⊆ ℝ푚 be a convex set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Suppose 푓 is 휅-strongly convex over 풦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 푥∗ ∈ 풦 be a minimizer of 푓 over 풦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then for any 푥 ∈ 풦, one has ∥푥 − 푥∗∥2 ⩽ 2 휅(푓 (푥) − 푓 (푥∗)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' By strong convexity, for any subgradient 푔 ∈ 휕 푓 (푥∗) one has 푓 (푥) ⩾ 푓 (푥∗) + � 푥 − 푥∗, 푔 � + 휅 2 ∥푥 − 푥∗∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' By Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1, ⟨푥 − 푥∗, 푔⟩ ⩾ 0 for some 푔 ∈ 휕 푓 (푥∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' □ 61 D Deferred proofs SBM We prove Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='8 restated below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1 (Restatement of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Consider the settings of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' With probability 1 − exp(−Ω(푛)) over G ∼ SBM푛(훾, 푑, 푥), ���� ˆ푋(푌(G)) − 1 푛 푥푥⊤ ���� 2 퐹 ⩽ 800 훾 √ 푑 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Recall 풦 = {푋 ∈ ℝ푛×푛 : 푋 ⪰ 0, 푋푖푖 = 1/푛 ∀푖}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 푋∗ := 1 푛 푥푥⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Since ˆ푋 = ˆ푋(푌(G)) is a minimizer of min푋∈풦 ∥푌(G) − 푋∥2 퐹 and 푋∗ ∈ 풦, we have ��� ˆ푋 − 푌(G) ��� 2 퐹 ⩽ ∥푋∗ − 푌(G)∥2 퐹 ⇐⇒ ��� ˆ푋 − 푋∗��� 2 퐹 ⩽ 2 � ˆ푋 − 푋∗, 푌(G) − 푋∗� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The infinity-to-one norm of a matrix 푀 ∈ ℝ푚×푛 is defined as ∥푀∥∞→1 := max{⟨푢, 푀푣⟩ : 푢 ∈ {±1}푚, 푣 ∈ {±1}푛}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' By [GV16, Fact 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2], every 푍 ∈ 풦 satisfies |⟨푍, 푌(G) − 푋∗⟩| ⩽ 퐾퐺 푛 · ∥푌(G) − 푋∗∥∞→1, where 퐾퐺 ⩽ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='783 is Grothendieck’s constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Similar to the proof of [GV16, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1], using Bernstein’s inequality and union bound, we can show (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Fact D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2) ∥푌(G) − 푋∗∥∞→1 ⩽ 100푛 훾 √ 푑 with probability 1 − exp(−Ω(푛)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Putting things together, we have ���� ˆ푋(푌(G)) − 1 푛 푥푥⊤ ���� 2 퐹 ⩽ 400 · 퐾퐺 훾 √ 푑 , with probability 1 − exp(−Ω(푛)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' □ Fact D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 훾 > 0, 푑 ∈ ℕ, 푥∗ ∈ {±1}푛, and G ∼ SBM(훾, 푑, 푥∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 푌(G) = 1 훾푑 � 퐴(G) − 푑 푛 퐽� , where 퐴((퐺)) is the adjacency matrix of (퐺) with entries 푑/푛 on the diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then max 푥∈{±1}푛 ��푥⊤� 푌(G) − 1 푛 푥∗(푥∗)⊤� 푥 �� ⩽ 100푛 훾 √ 푑 with probability at least 1 − 푒−10푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 62 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' The result will follow using Bernstein’s Inequality and a union bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Define 푬 ≔ 푌(G) − 1 푛 푥∗(푥∗)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Fix 푥 ∈ {±1}푛 and for 1 ⩽ 푖 < 푗 ⩽ 푛, let 풁푖,푗 ≔ 푬푖,푗푥푖푥푗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then 푥⊤푬푥 = 2 � 1⩽푖<푗⩽푛 풁푖,푗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Note that 피 풁푖,푗 = 0 , ��풁푖,푗 �� ⩽ 1 훾푛 · �푛 푑 − 1 � + 1 훾푑푛 ⩽ 1 훾푑 , 피 풁2 푖,푗 = Var � 풀(G)푖,푗 � ⩽ 피풀(G)2 푖,푗 ⩽ (1 + 훾) 푑 푛 · 1 훾2푛2 ��푛 푑 − 1 �2 − 1 훾2푛2 � + 1 훾2푛2 ⩽ (1 + 훾) 1 푑훾2푛 + 1 훾2푛2 ⩽ 3 훾2푑푛 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' By Bernstein’s Inequality (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' [Wai19, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='14]) it follows that ℙ�� � � 푖<푗 풁푖,푗 ⩾ 50푛 훾 √ 푑 �� � ⩽ ℙ�� � � 푖<푗 풁푖,푗 ⩾ 푛2 2 · 100푛 훾 √ 푑 �� � ⩽ 2 exp�� � − 104 훾2푑 3 훾2푑푛 + 100 3훾2푑3/2푛 �� � = 2 exp � − 104푛 3 + 100 √ 푑 � ⩽ exp(−50푛) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Hence, by a union bound over all 푥 ∈ {±1}푛 it follows that max 푥∈{±1}푛 ��푥⊤� 푌(G) − 1 푛 푥∗(푥∗)⊤� 푥 �� ⩽ 100푛 훾 √ 푑 with probability at least 1 − 푒−10푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' □ E Deferred proofs for clustering In this section, we will prove Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='15 restated below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Lemma (Restatement of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Consider the settings of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Suppose Y is a good set as per Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 푊(Y) ∈ 풲(Y) be the matrix computed by Algorithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Suppose the algorithm does not reject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then ��휙(푊(Y)) − W∗�� 1 ⩽ 푛2 푘 · 3 푘98 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We will need the following fact about our clustering program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Similar facts where used, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=', in [HL18, FKP+19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' One difference for us is that we don’t have a constraint on the lower bound on the cluster size indicated by our SOS variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' However, since we maximize a variant of the ℓ1 norm of the second moment matrix of the pseudo-distribution this will make up for this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 63 Fact E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Consider the same setting as in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 0 < 훿 ⩽ 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='5·1010 · 1 푘201 and denote by C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' , C푘 ⊆ [푛] the indices belonging to each true cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then 푊(Y) satisfies the following three properties: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For all 푖, 푗 ∈ [푛] it holds that 0 ⩽ W푖,푗 ⩽ 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' for all 푖 ∈ [푛] it holds that �푛 푗=1 W푖,푗 ⩽ 푛 푘 and for at least (1 − 1 1000푘100)푛 indices 푖 ∈ [푛] it holds that �푛 푗=1 W푖,푗 ⩾ (1 − 1 (10)6푘200) · 푛 푘 , 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' for all 푟 ∈ [푘] it holds that � 푖∈C푟 ,푗∉C푟 W푖,푗 ⩽ 훿 · 푛2 푘 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We will prove Fact E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1 at the end of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' With this in hand, we can proof Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For brevity, we write W = 푊(Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Since 휙(W∗) = W∗ and 휙 is 10- Lipschitz we can also bound ��휙(W) − W∗�� 1 ⩽ 10 · ∥W − W∗∥1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 훿 ⩽ 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='5·1010 · 1 푘201 and again let C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' , C푘 ⊆ [푛] denote the indices belonging to each true cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Note that by assumption that Y is a good sample it holds for each 푟 ∈ [푘] that 푛 푘 − 푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='6 ⩽ |C푟| ⩽ 푛 푘 + 푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 푟, 푟′ ∈ [푘].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We can write ∥W − W∗∥1 = 푘 � 푟=1 � 푖,푗∈C푟 ��W푖,푗 − 1 �� + 푘 � 푟=1 � 푖∈C푟,푗∉C푟 ��W푖,푗 − 0 �� (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1) Note that we can bound the second sum by 푘 · 훿 푛2 푘 using Item 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Further, in what follows consider only indices 푖 such that �푛 푗=1 W푖,푗 ⩾ (1 − 1 (10)6푘200 ) · 푛 푘 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' By Item 2 we can bound the contribution of the other indices by 1 1000푘100 푛 · �푛 푘 + 푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='6� ⩽ 2 1000푘100 · 푛2 푘 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Focusing only on such indices, for the first sum in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1), fix 푟 ∈ [푘].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We will aim to show that most entries of W are large if and only if the corresponding entry of W∗ is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' By Item 3 and Markov’s Inequality, it follows that for at least a (1 − 1 1000푘100)-fraction of the indices 푖 ∈ C푟 it holds that � 푗∉C푟 W푖,푗 ⩽ 1000푘100 · 훿 푛2 푘·|C푟| ⩽ 1000푘100훿 · 푛 1−푘·푛−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4 ⩽ 2000푘101훿 · 푛 푘 , where we used that |C푟| ⩾ 푛 푘 − 푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Call such indices good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Notice that for good indices it follows using Item 2 that � 푗∈C푟 W푖,푗 ⩾ 푛 푘 · (1 − 1 (10)6푘200 − 2000푘101훿) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 64 Denote by 퐺 the number of 푗 ∈ C푟 such that W푖,푗 ⩾ 1 − 1 1000푘100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Using the previous display and that W푖,푗 ⩽ 1 we obtain 푛 푘 · � 1 − 1 (10)6푘200 − 2000푘101훿 � ⩽ � 푗∈C푟 W푖,푗 ⩽ 퐺 · 1 + (|C푟| − 퐺) · (1 − 1 1000푘100) ⩽ 퐺 · 1 1000푘100 + 푛 푘 · (1 + 1 푘푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4) · (1 − 1 1000푘100) ⩽ 퐺 · 1 1000푘100 + 푛 푘 · (1 + 1 푘푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4) , where we also used |C푟| ⩽ 푛 푘 + 푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Rearranging now yields 퐺 ⩾ 푛 푘 · � 1 − 1 1000푘100 − 103푘99 푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4 − 2 · 106푘101훿 � ⩾ 푛 푘 · � 1 − 2 1000푘100 − 2 · 106푘101훿 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We can now bound � 푖,푗∈C푟 ��W푖,푗 − 1 �� = � 푖,푗∈C푟 ,푖 is good ��W푖,푗 − 1 �� + � 푖,푗∈C푟 ,푖 is not good ��W푖,푗 − 1 �� ⩽ |C푟| · � (|C푟| − 퐺) · 1 + |C푟| · 1 1000푘100 � + 1 1000푘100 · |C푟|2 ⩽ |C푟|2(1 + 1 500푘100) − 퐺 · |C푟| ⩽ 푛2 푘2 (1 + 푘 푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4)2(1 + 1 500푘100) − 푛2 푘2 (1 − 2 1000푘100 − 2 · 106푘101훿)(1 − 푘 푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4 ) ⩽ 푛2 푘2 · (30 · 106푘101훿 + 11 500푘100) ⩽ 푛2 푘 · (30 · 106푘100훿 + 11 500푘101) ⩽ 푛2 푘 · 3 125푘101 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Putting everything together, it follows that ��휙(W) − W∗��2 F ⩽ ��휙(W) − W∗�� 1 ⩽ 10 · 푛2 푘 � 훿푘 + 2 1000푘100 + 3 125푘100 � ⩽ 푛2 푘 · 4 푘100 ⩽ 푛2 푘 · 3 푘98 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' □ It remains to verify Fact E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Proof of Fact E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 풫 = 풫푛,푘,푡(Y) be the system of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' (풫푛,푘,푡(푌)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Recall that W푖,푗 = ˜피 � 푙∈[푘] 푧푖,푙푧푗,푙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Since 풫 4 \uf8f1\uf8f4\uf8f4\uf8f2 \uf8f4\uf8f4\uf8f3 0 ⩽ � 푙∈[푘] 푧푖,푙푧푗,푙 ⩽ � 푙∈[푘] 푧푖,푙 ⩽ 1 \uf8fc\uf8f4\uf8f4\uf8fd \uf8f4\uf8f4\uf8fe , it follows that 0 ⩽ W푖,푗 ⩽ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Further, for each 푖 ∈ [푛] it holds that 풫 4 \uf8f1\uf8f4\uf8f4\uf8f2 \uf8f4\uf8f4\uf8f3 � 푗∈[푛],푙∈[푘] 푧푗,푙푧푖,푙 ⩽ 푛 푘 � 푙∈[푘] 푧푖,푙 ⩽ 푛 푘 \uf8fc\uf8f4\uf8f4\uf8fd \uf8f4\uf8f4\uf8fe 65 implying that � 푗∈[푛] W푖,푗 ⩽ 푛 푘 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Further, by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='14 ∥W∥1 ⩾ 푛2 푘 · � 1 − 푛−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4 − 1 (10)10푘300 � ⩾ 푛2 푘 · � 1 − 1 (10)9푘300 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Denote by W푖 the 푖-th row of W and by 퐿 the number of rows which have ℓ1 norm at least (1 − 1 (10)6푘200 ) · 푛 푘 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Since for all 푖 it holds that ∥W푖∥1 ⩽ 푛 푘 it follows that 푛2 푘 · � 1 − 1 (10)9푘300 � ⩽ � 푖∈[푛] ∥W푖∥1 ⩽ 퐿 · 푛 푘 + (푛 − 퐿) · � 1 − 1 (10)6푘200 � 푛 푘 = 퐿 · 1 (10)6푘200 · 푛 푘 + 푛2 푘 · � 1 − 1 (10)6푘200 � Rearranging then yields 퐿 ⩾ (1 − 1 1000푘100) · 푛 which proofs Item 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' It remains to verify Item 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Fix 푟, 푙 ∈ [푘] and define 푧푙(C푟) = 푘 푛 � 푖∈C푟 푧푖,푙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 푡 > 0 be an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We aim to show that for all unit vectors 푣 it holds that 풫 10푡 � 푧푙(C푟) · 1 Δ2푡 � 푟′≠푟 푧푙(C푟′)⟨휇푟 − 휇푟′, 푣⟩2푡 ⩽ 훿 푘 � , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2) where Δ is the minimal separation between the true means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Before proving this, let us examine how we can use this fact to prove Item 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Note, that for all 푟 ≠ 푟′ it holds that � 푠,푢∈[푘] � 휇푟 − 휇푟′, 휇푠−휇푢 ∥휇푠−휇푢∥ �2푡 ⩾ Δ2푡 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Hence, if the above SOS proof indeed exists, we obtain � 푖∈C푟 ,푗∉C푟 W푖,푗 = 푘 � 푙=1 ˜피 � 푖∈C푟 ,푗∉C푟 푧푖,푙푧푗,푙 = 푛2 푘2 ˜피푧푙(C푟) · � 푟′≠푟 푧푙(C푟′) ⩽ 푛2 Δ2푡푘2 � 푠,푢∈[푘] ˜피푧푙(C푟) · � 푟′≠푟 푧푙(C푟) � 휇푟 − 휇푟′, 휇푠−휇푢 ∥휇푠−휇푢∥ �2푡 ⩽ 훿 푘 푘2 · 푛2 푘2 = 훿 · 푛2 푘 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' In the remainder of this proof we will prove Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We will use the following SOS version of the triangle Inequality (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Fact E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2) 2푡 푥,푦 (푥 + 푦)2푡 ⩽ 22푡−1(푥2푡 + 푦2푡) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Recall that 휇′ 푙 = 푘 푛 �푛 푖=1 푧푖,푙 푦푖 and denote by 휇휋(푖) the true mean corresponding to the 푖-th sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 푣 be an arbitrary unit vector, it follows that 풫 10푡 {푧푙(C푟) · 1 Δ2푡 � 푟′≠푟 푧푙(C푟′)⟨휇푟 − 휇푟′, 푣⟩2푡 66 ⩽ 푧푙(C푟) · 22푡−1 Δ2푡 � 푟′≠푟 푧푙(C푟′)� ⟨휇푟 − 휇′ 푙, 푣⟩2푡 + ⟨휇푟′ − 휇′ 푙, 푣⟩2푡� ⩽ 22푡−1 Δ2푡 푘 � 푟=1 푧푙(C푟)⟨휇푟 − 휇′ 푙, 푣⟩2푡 = 22푡−1 Δ2푡 · 푘 푛 푛 � 푖=1 푧푖,푙⟨휇휋(푖) − 휇′ 푙, 푣⟩2푡} , where we used that 풫 1 �푘 푟=1 푧푙(C푟) ⩽ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Using the SOS triangle inequality again and that 풫 2 푧푖,푙 ⩽ 1 we obtain 풫 10푡 {푧푙(C푟) · 1 Δ2푡 � 푟′≠푟 푧푙(C푟′)⟨휇푟 − 휇푟′, 푣⟩2푡 ⩽ 24푡−1 Δ2푡 · � 푘 · 1 푛 푛 � 푖=1 ⟨y푖 − 휇휋(푖), 푣⟩2푡 + 푘 푛 푛 � 푖=1 푧푖,푙⟨y푖 − 휇′ 푙, 푣⟩2푡 � } .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We start by bounding the first sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Recall that by assumption the uniform distribution over each true cluster is 2푡-explicitly 2-bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' It follows that 2푡 { 1 푛 푛 � 푖=1 ⟨y푖 − 휇휋(푖), 푣⟩2푡 = 1 푘 푘 � 푟=1 푘 푛 � 푖∈C푟 ⟨y푖 − 휇푟, 푣⟩2푡 ⩽ 1 푘 푘 � 푟=1 푘 푛 · |C푟| · (2푡)푡 · ∥푣∥2푡 2 (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='3) ⩽ � 1 + 푘 푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4 � (2푡)푡 ⩽ 2(2푡)푡} , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4) where we used that |C푟| ⩽ 푛 푘 + 푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' To bound the second sum, we will use the moment bound constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' In particular, we know that 풫 10푡 � 푘 푛 푛 � 푖=1 푧푖,푙⟨y푖 − 휇′ 푙, 푣⟩2푡 ⩽ (2푡)푡 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='5) Combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='5) now yields 풫 10푡 � 푧푙(C푟) · 1 Δ2푡 � 푟′≠푟 푧푙(C푟′)⟨휇푟 − 휇푟′, 푣⟩2푡 ⩽ 푘22푡+1(2푡)푡 Δ2푡 ⩽ 푘 � 8푡 Δ2 �푡� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Note that by assumption Δ ⩾ 푂( √ 푡푘1/푡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Overloading notation, we can choose the 푡 parameter in the SOS proof to be 202 times the 푡 parameter in the lower bound in the separation to obtain32 � 푖∈C푟 ,푗∉C푟 W푖,푗 ⩽ 훿 · 푛2 푘 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' □ 32Note that this influences the exponent in the running time and sample complexity only by a constant factor and hence doesn’t violate the assumptions of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 67 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='1 Small Lemmas Fact E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2 (Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2 in [KSS18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' For all integers 푡 > 0 it holds that 2푡 푥,푦 (푥 + 푦)2푡 ⩽ 22푡−1(푥2푡 + 푦2푡) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Fact E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 휀, 훿 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let ℳ : 풴 → 풪 be a randomized algorithm that, for every pair of adjacent inputs, with probability at least 1 − 훾 ⩾ 1/2 over the internal randomness of 풴33 satisfies (휀, 훿)-privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then ℳ is (휀 + 2훾, 훿 + 훾)-private.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 푋, 푋′ be adjacent input and let 퐵 be the event under which ℳ is (휀, 훿)-private.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' By assumption, we know that ℙ(퐵) ⩾ 1 − 훾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 푆 ∈ 풪, it follows that ℙ(ℳ(푋) ∈ 푆) = ℙ(퐵) · ℙ(ℳ(푋) ∈ 푆 | 퐵) + ℙ(퐵푐) · ℙ(ℳ(푋) ∈ 푆 | 퐵푐) ⩽ ℙ(ℳ(푋) ∈ 푆 | 퐵) + 훾 ⩽ 푒휀ℙ(ℳ(푋) ∈ 푆 | 퐵) + 훿 + 훾 ⩽ 푒휀 ℙ(퐵) · ℙ(ℳ(푋) ∈ 푆) + 훿 + 훾 ⩽ 푒 휀+log � 1 1−훾 � ℙ(ℳ(푋) ∈ 푆) + (훿 + 훾) ⩽ 푒휀+2훾 · ℙ(ℳ(푋) ∈ 푆) + (훿 + 훾) , where we used that log(1 − 훾) ⩾ −2훾 for 훾 ∈ [0, 1/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' □ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='2 Privatizing input using the Gaussian Mechanism In this section, we will proof the following helpful lemma used in the privacy analysis of our clustering algorithm (Algorithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' In summary, it says that when restricted to some set our input has small ℓ2 sensitivity, we can first add Gaussian noise proportional to this sensitivity and afterwards treat this part of the input as "privatized".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' In particular, for the remainder of the privacy analysis we can treat this part as the same on adjacent inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Note that we phrase the lemma in terms of matrix inputs since this is what we use in our application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Of course, it also holds for more general inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 푉, 푉′ ∈ ℝ푛×푑, 푚 ∈ [푛] and Δ > 0 be such that there exists a set 푆 of size at least 푛 − 푚 satisfying ∀푖 ∈ 푆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' ��푉푖 − 푉′ 푖 ��2 2 ⩽ Δ2 , where 푉푖, 푉′ 푖 denote the rows of 푉, 푉′, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 풜2 : ℝ푛×푑 → 풪 be an algorithm that is (휀2, 훿2)-differentially private in the standard sense, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='e,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=', for all sets 풮 ⊆ 풪 and datasets 푋, 푋′ ∈: ℝ푛×푑 differing only in a single row it holds that ℙ(풜2(푋) ∈ 푆) ⩽ 푒휀2ℙ(풜2(푋′) ∈ 푆) + 훿2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 33In particular, this randomness is independent of the input 68 Further, let 풜1 : ℝ푛×푑 → ℝ푛×푑 be the Gaussian Mechanism with parameters Δ, 휀1, 훿1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=', on input 푀 it samples W ∼ 푁 � 0, 2Δ2 · log(2/훿1) 휀2 1 �푛×푑 and outputs 푀 + W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Then for 휀′ ≔ 휀1 + 푚휀2 , 훿′ ≔ 푒휀1푚푒(푚−1)휀2훿2 + 훿1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' 풜2 ◦ 풜1 is (휀′, 훿′)-differentially private with respect to 푉 and 푉′, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=', for all sets 풮 ⊆ 풪 it holds that ℙ((풜2 ◦ 풜1)(푉) ∈ 푆) ⩽ 푒휀′ℙ((풜2 ◦ 풜1)(푉′) ∈ 푆) + 훿′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Without loss of generality, assume that 푆 = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' , 푚}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Denote by 푉1, 푉2 the first 푚 and last 푛 − 푚 rows of 푉 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Analogously for 푉′ 1, 푉′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' We will later partitin the noise W of the Gaussian mechanism in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Further, for a subset 퐴 of ℝ푛×푛 and 푌 ∈ ℝ푚×푛 define 푇퐴,푌 = � 푋 ∈ ℝ(푛−푚)×푛 ���� � 푋 푌 � ∈ 퐴 � ⊆ ℝ(푛−푚)×푛 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Note that � 푋 푌 � ∈ 퐴 if and only if 푋 ∈ 푇퐴,푌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Let 풮 ⊆ 풪.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' It now follows that ℙ풜2,W[(풜2 ◦ 풜1)(푉) ∈ 푆] = 피 풜2,W � ퟙ � 푉 + W ∈ 풜−1 2 (푆) �� = 피 풜2,W2 � 피 W1 � ퟙ �� 푉1 + W1 푉2 + W2 � ∈ 풜−1 2 (푆) �� ���� W2 � = 피 풜2,W2 � 피 W1 � ퟙ � 푉1 + W1 ∈ 푇풜−1 2 (푆),푉2+W2 �� ���� W2 � ⩽ 푒휀1 · 피 풜2,W2 � 피 W1 � ퟙ � 푉′ 1 + W1 ∈ 푇풜−1 2 (푆),푉2+W2 �� ���� W2 � + 훿1 = 푒휀1 · 피 풜2,W � ퟙ �� 푉′ 1 + W1 푉2 + W2 � ∈ 풜−1 2 (푆) �� + 훿1 , where the inequality follows by the guarantees of the Gaussian Mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Further, we can bound 피 풜2,W � ퟙ �� 푉′ 1 + W1 푉2 + W2 � ∈ 풜−1 2 (푆) �� = 피 W � 피 풜2 � ퟙ � 풜2 � 푉′ 1 + W1 푉2 + W2 � ∈ 푆 � ���� W �� ⩽ 푒푚휀2 · 피 W � 피 풜2 � ퟙ � 풜2 � 푉′ 1 + W1 푉′ 2 + W2 � ∈ 푆 � ���� W �� + 푚푒(푚−1)휀2훿2 = 푒푚휀2 · 피 풜2,W � ퟙ �� 푉′ 1 + W1 푉′ 2 + W2 � ∈ 풜−1 2 (푆) �� + 푚푒(푚−1)휀2훿2 , 69 where the inequality follows by the privacy guarantees of 풜2 combined with standard group privacy arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' Putting the above two displays together and plugging in the definition of 휀′, 훿′ we finally obtain ℙ풜2,W[(풜2 ◦ 풜1)(푉) ∈ 푆] ⩽ 푒휀′ℙ풜2,W[(풜2 ◦ 풜1)(푉′) ∈ 푆] + 훿′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} +page_content=' □ 70' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE3T4oBgHgl3EQf-Auh/content/2301.04822v1.pdf'} diff --git a/v9E2T4oBgHgl3EQfggeM/content/tmp_files/2301.03938v1.pdf.txt b/v9E2T4oBgHgl3EQfggeM/content/tmp_files/2301.03938v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7d42ad1d9c2331dd6295379dfd0d2e78d733140e --- /dev/null +++ b/v9E2T4oBgHgl3EQfggeM/content/tmp_files/2301.03938v1.pdf.txt @@ -0,0 +1,3295 @@ +1 +Consensus based phase connectivity identification for +distribution network with limited observability +Md Umar Hashmi1 +, David Brummund2, Rickard Lundholm1, Arpan Koirala1 +, +and Dirk Van Hertem1 +Abstract +The mitigation of distribution network (DN) unbalance and the use of single-phase flexibility for congestion +mitigation requires accurate phase connection information, which is often not available. For a large DN, the naïve +phase identification proposed in the majority of the prior works using a single voltage reference does not scale well +for a multi-feeder DN. We present a consensus algorithm-based phase identification mechanism which uses multiple +three-phase reference points to improve the prediction of phases. Due to the absence of real measurements for a real- +suburban German DN, the algorithms are developed and evaluated over synthetic data using a digital twin. To utilize +strongly correlated measurements, the DN is clustered into zones. We observe those reference measurements located +in the same zone as the single-phase consumer leads to accurate prediction of DN phases. Four consensus algorithms +are developed and compared. Using numerical results, we recommend the most robust phase identification mechanism. +In our evaluation, measurement error, and the impact of the neutral conductor are also assessed. We assume limited +DN observability and apply our findings to a German DN without smart meters, but only less than 8% of nodes have +measurement boxes along with single-phase consumers with a home energy management system. Voltage time series +for 1 month (hourly sampled) is utilized. The numerical results indicate that for 1% accuracy class measurement, the +phase connectivity of 308 out of 313 single-phase consumers in a German DN can be identified. Further, we also +propose metrics quantifying the goodness of the phase identification. The phase identification framework based on +consensus algorithms for DN zones is scalable for large DN and robust towards measurement errors as the estimation +is not dependent on a single measurement point. +Index Terms +Data-driven, distribution network, machine learning, phase identification, voltage time series +Corresponding author email: mdumar.hashmi@kuleuven.be +1Md Umar Hashmi, Rickard Lundholm, Arpan Koirala and Dirk Van Hertem are with KU Leuven, division Electa & EnergyVille, Genk, +Belgium +2David Brummund is with MITNETZ STROM, Germany +arXiv:2301.03938v1 [eess.SY] 10 Jan 2023 + +2 +CONTENTS +I +Introduction +4 +I-A +Observations of this paper +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +6 +II +Low observability in DN: the German case +8 +II-A +Metering of German DN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +8 +II-B +Roadmap of meter rollout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +8 +II-C +Demo network for EUniversal +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +10 +II-C1 +Network features and meter placement . . . . . . . . . . . . . . . . . . . . +10 +II-D +Need for phase information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +10 +III +Synthetic data generation +13 +III-A +DGS parser for network JSON . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +13 +III-A1 +Bus information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +13 +III-A2 +Branch information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +13 +III-A3 +Device information +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +13 +III-A4 +Switches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +13 +III-B +Mitnetz Strom DN and metadata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +14 +III-C +Randomized phase mapping +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +14 +III-D +Neutral modelling for four-wire DN . . . . . . . . . . . . . . . . . . . . . . . . . . . +16 +III-E +Metering noise injection model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +17 +III-F +Zonal clustering of Mitnetz Strom DN . . . . . . . . . . . . . . . . . . . . . . . . . . +18 +III-G +Power flows for synthetic data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +18 +IV +Phase identification methodology +19 +IV-A +Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +19 +IV-B +Correlation based metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +20 +IV-B1 +J1: correlation of voltage time series . . . . . . . . . . . . . . . . . . . . . +21 +IV-B2 +J2: Salient features with voltage difference time series . . . . . . . . . . . +21 +IV-B3 +J3: Salient features with voltage magnitude time series . . . . . . . . . . . +22 +V +Consensus algorithm +23 +V-A +Naïve phase identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +23 +V-B +Majority rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +23 + +3 +V-C +Weighted measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +24 +V-C1 +Correlation as a measure . . . . . . . . . . . . . . . . . . . . . . . . . . . +24 +V-C2 +Absolute value of correlation as a measure +. . . . . . . . . . . . . . . . . +24 +V-C3 +Maximum value of correlation as a measure . . . . . . . . . . . . . . . . . +24 +V-D +Metrics for phase identification models +. . . . . . . . . . . . . . . . . . . . . . . . . +25 +V-D1 +Modelling accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +25 +V-D2 +Confidence factor for phase identification . . . . . . . . . . . . . . . . . . +25 +V-D3 +Standard deviation with measurement error . . . . . . . . . . . . . . . . . +26 +VI +Numerical results +27 +VI-A +Clustering of distribution network +. . . . . . . . . . . . . . . . . . . . . . . . . . . . +27 +VI-B +Performance of naïve model +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +30 +VI-C +Case study 1: Comparing phase identification models . . . . . . . . . . . . . . . . . . +32 +VI-D +Case study 2: Effect of the proximity of measurement on PCI . . . . . . . . . . . . . +36 +VI-E +Case study 3: Impact of measurement error . . . . . . . . . . . . . . . . . . . . . . . +39 +VI-F +Case study 4: Effect of size of neutral conductor . . . . . . . . . . . . . . . . . . . . +39 +VII +Conclusions +41 +References +43 + +4 +I. INTRODUCTION +The low voltage distribution network (DN) in Europe consists predominantly of single phase (1-φ) loads, +inverter interfaced PV, storage, and electric vehicle charging infrastructure. Often the phase connectivity +of such resources is not accurately known. This lack of DN observability will restrict the monitoring and +control of DN imbalances. Existing DN consists of multiple measurements at the feeder level and end of +the feeder, which provides utilities with some degree of observability. The goal of the paper is to develop a +scalable and robust phase connectivity identification (PCI) framework that considers multiple measurement +points for improving the PCI. +Parameter used +64% +29% +7% +Voltage +Power +Voltage and Power +Tool used +42% +27% +19% +12% +Statistical or ML +Correlationship +Clustering +Optimization +(b) +(a) +Fig. 1: Classifying of phase connectivity identification literature based on parameter(s) and tool(s) used. +Phase identification methodologies can be broadly classified as intrusive and non-intrusive. As the name +suggests, intrusive methods require manual identification of phases and are often labor-intensive and/or +hardware-based [28]. On the other hand, non-intrusive methods are often data-driven. A brief summary of +non-intrusive phase identification methods is detailed in Table I. These data-driven methods can be classified +based on the parameter used and tool used for phase identification. For the literature summarized in Table +I, the classification is presented in Fig. 1. From Fig. 1, we observe that 64% of existing works utilize +voltage magnitude time series and approximately 27% utilize correlation as a tool for PCI. In this work, we +also utilize these widely used techniques for developing a consensus-based phase identification framework +using voltage time series and correlation as the tool. Voltage time series-based phase identification is more + +5 +TABLE I: Literature review on non-intrusive phase identification +Ref +Measurement dependency +Proposed solution +Remarks +Methodology +Input +[1] +AMI voltage time series, partially +incorrect phase label information +Spectral clustering with a sliding window; does not require +substation measurement +91% accuracy, Google street view analysed for +phase identification +Clustering / Unsupervised +ML +voltage +[2] +Voltage +magnitude +(denoted +as +|V |) +Spectral clustering is utilized and MILP model is used for +unbalance mitigation. +156 user DN in China is used for validation. +Majority rule is applied to over predictions over +new data. +Clustering / Unsupervised +ML +voltage +[3] +Voltage magnitude +k-means clustering with Gaussian Mixture Model algorithm +for phase id +91% accurate; with salient features accuracy is +100% +Clustering / Unsupervised +ML +voltage +[4] +Voltage magnitude +k-means clustering is used. Use multiple references with +Majority rule based estimation. +90% accuracy +Clustering / Unsupervised +ML +voltage +[5] +Voltage magnitude +k-medoids clustering is with denoised data +Singular value decomposition is used for denois- +ing +Clustering / Unsupervised +ML +voltage +[6] +Voltage magnitude +principal components are used to extract feature vectors over +which constrained k-means clustering is applied +90% accuracy +Clustering / Unsupervised +ML +voltage +[7] +Voltage magnitude +k-means clustering with principal component analysis +Phase identification is applied for multiple days +separately and a majority rule is applied +Clustering / Unsupervised +ML +voltage +[8] +Active power measurements, sub- +station as reference +extract distinct features from load profiles and correlate with +phase load; limitation: high granularity data needed +93% accuracy with 10% SMs in DN; results +compared with [9] +Correlation +power +[10] +Voltage magnitude +Correlation based; the salient features of the time series are +extracted. +Large enough data sheet leads to 100% accuracy +for 75 consumer DN +Correlation +voltage +[11] +Voltage magnitude +Relies on graph theory and the notion of maximum spanning +tree. Correlation based PCI for a four wire DN. +Closer the measurement points are geographi- +cally, the stronger the correlation between the +voltages +Correlation +voltage +[12] +Voltage magnitude +Difference matrix is created +82% accuracy +Correlation +voltage +[13] +Voltage magnitude time series +Correlation between voltage measurements of SMs with +constrained k-means +substation voltage is used as reference for phase +identification +Correlation with unsuper- +vised ML +voltage +[14] +Active power and voltage time se- +ries data +Correlationship with clustering is performed. Ensemble +learning combines voltage and power-based estimation re- +sults. +Impact of different SM accuracy class is evalu- +ated +Correlation with unsuper- +vised ML +voltage +and +power +[9] +Active power time series at the +transformer and consumers +integer programming along with branch and bound search +algorithm. Access sensitivity of the ratio of measurement +points and total number of consumers +MILP based solution depends on the principle of +conservation of energy. +Optimization +power +[15] +P, Q and |V | measurements +MILP with Bender’s decomposition is used. Accuracy of +phase id is governed by number of data points, SM class, +data resolution +For large EU feeder, the runtime with 5% SM +error is 39.2 hours. Difficult to scale. +Optimization +power (P & Q) +and voltage +[16] +P, Q, |V | measurements +Utilize state-estimation with MILP, also considers errors in +layout information +Data needs are smaller compared to statistical and +ML-based techniques +Optimization +power +and +voltage +[17] +Active power time series +LASSO based data driven approach. Also considers SM +accuracy class +97% accuracy with 60% SMs; LASSO immune +to noise unlike [9] +Statistical or ML +power +[18]– +[20] +Energy measurement time series +data-driven approach with Principal component analysis & +graph theory interpretations +Also considers noisy data +Statistical or ML +power +[21] +Voltage magnitude time series +Develops multi-dimensional calibration in phase id based on +voltage characteristics in LVDN +Observe that voltage characteristics are more +robust under incomplete data +Statistical or ML +voltage +[22] +Voltage magnitude +Linear regression and voltage drop relationship for phase +identification +Observe close measurement are strongly cor- +related +Statistical or ML +voltage +[19] +Voltage phasor time series mea- +sured using microPMUs +Phase id analyses cross correlations over voltage magnitudes +and detects the phase angle difference between reference and +test nodes +multi-phase connections are also considered. Fine +resolution of 120 samples per sec is used +Statistical or ML +voltage phasor +[23] +P, |V | time series +Using statistical analysis of AMI data over a day for DN with +PV. Regression model is used to model substation voltage +using nodal P, V and substation power. +Explore data needs, granularity and impact of PV +penetration levels +Statistical or ML +voltage +and +power +[24] +Voltage magnitude and phase info +of a small representative set +Train an ML model of constrained function of voltage time +series. Since manual measurements are needed thus may not +be scalable +5% selected representative set leads to accuracy +of 91.9% +Statistical or ML +voltage +[25] +Voltage phasor time series +Use sequence component for phase identification +Also utilize SM data for learning the topology of +the DN +Statistical or ML +voltage phasor +[26] +Voltage magnitude +Supervised machine learning with theory of information loss +Accuracy up to 97% +Supervised ML +[27] +Voltage phasor time series +Topology and phase identification using linearized model of +three-phase unbalanced DN. +120 Hz PMU measurements are used. Data col- +lected for 1 second to 1 minute is used for +estimation +Statistical or ML +voltage +phasors + +6 +robust to limited observability in a DN, also observed in [29]. The power-based methods rely on the law of +conservation of energy and require a high degree of observability in a DN. +Motivation: With the increasing generation from renewable energy sources and the growing addition of +flexible loads like electric vehicles, heat pumps, etc. congestions, voltage violations and phase imbalances +in the grid will likely become more frequent. Therefore, suitable corrective measures must be found and +implemented. An important basis for mitigation measures is the detection of congestions and thus, firstly, +improving the network model of the low-voltage grid, which is still largely unmonitored in many parts of +the world today. Accurate network topology is assumed to be known in many works [30], [31]. However, +this assumption is not accurate in the case of EUniversal’s demo networks for the German DN. +A. Observations of this paper +The contributions and observations of the paper are as follows: +• A tailor-made solution for the German DSO, Mitnetz Strom, is proposed for phase identification which +considers multiple measurements in a DN zone for improving the PCI accuracy, this is detailed in Fig. +2. The proposed framework can also be applied for other DNs with limited DN observability. +• Twelve phase identification models are benchmarked over the naïve model. The naïve model considers +only one 3−φ measurement reference in a DN for phase estimation. The proposed phase identification +models build a consensus among multiple measurements for robust phase estimation. +• Metrics are proposed for evaluating phase identification models using (a1) accuracy of estimation, (a2) +confidence factor, and (a3) sensitivity towards measurement errors. Metrics (a2) and (a3) provide a +qualitative metric for evaluating estimation accuracy. +• A detailed description is provided for synthetic data generation, which utilizes an example suburban LV +DN grid model of Mitnetz Strom. This is also crucial for DNs with limited or no historical measurement +data. +• Four case studies are performed in the numerical results. The performance of the naïve model is used +for benchmarking and comparing the proposed phase identification algorithms. +– Firstly, the proposed phase identification models are compared for the read German DN +– Secondly, we quantify the impact of measurement proximity on phase identification metrics. We +observe that for a DN partitioned into zones, the estimation quality deteriorates as the measurement +reference selected is farther away from the zone where the consumer is located. +– Thirdly, the impact of measurement errors on PCI is assessed. For 1% accuracy class measurements, +an estimation accuracy exceeding 98.6% is achieved for a DN with 646 nodes. + +7 +– Most European DNs are four-wire system. In our last case study, we quantify the impact of the +neutral conductor model on phase estimation accuracy. We observe that if the neutral conductor +is not modeled, then a pessimistic estimation is achieved. Based on the knowledge of the authors, +the neutral conductor impact assessment is done for the first time. +Phase mapping +(true phase +connectivity) +Network +Layout +Multi-period +power flow +Consumer +load profile +selector +Meta data for DN +Synthetic data +Inject measurement +errors +Assess +estimation +accuracy +Consensus +based phase +identification +3-ϕ Reference +measurements +Estimation accuracy +Confidence of est. +Sensitivity to +measurement +errors +Evaluation metrics +Digital twin for +synthetic data +generation (Section 3) +Consensus based +Phase identification +(Section 4 and 5) +A +A +Zonal +Clustering +Fig. 2: Consensus-based phase identification, synthetic data generation and metrics used +The paper is organized as follows. Section II presents the German DN case of low observability and the +need for enhanced phase information for future DN operation. Section III outlines the different modeling +steps used for generating synthetic data used for phase connectivity identification. Section IV presents the +methodology, and Section V presents the consensus algorithms used for phase identification. Section VI +presents the four numerical case studies, and section VII concludes the paper. + +8 +II. LOW OBSERVABILITY IN DN: THE GERMAN CASE +The low voltage grid serves households and small consumers connecting at 230 V or 400 V [32]. German +authorities have decided to implement an optional smart meter (SM) roll-out as the information security +standards need to be adjusted. Presently, less than 5% of residential customers are equipped with SMs [33], +[34]. Thus, the smart meter infrastructure is not widespread at a low-voltage level. The German Energy +Industry Act requires that customers with a yearly consumption of over 6000 kWh are provided with smart +measurement systems (when technically possible) [35], [36]. The requirement also applies to generators with +an installed capacity above 7 kW. However, these requirements leave the majority of German households +unaffected. The lack of sufficiently granular metering equipment at the household level is currently a barrier +for implementing imbalance sensitive flexibility activation for solving DN issues. +A. Metering of German DN +Mitnetz Strom is one of the largest regional distribution system operators in Eastern Germany and is +responsible for supplying electricity to 2.2 million electricity consumers. The grid area of Mitnetz Strom +covers an area of 30,804 km2 and is characterized by rural conditions with a high share of renewables. +The installed capacity of renewable energy reached an all-time high of more than 10,000 MW (more than +64,0000 plants) in 2021. This development was spurred primarily by rapid growth in solar energy, as the +number of photovoltaic installations increased by more than 17 percent. [37]. +In Germany, the metering and DSO roles are decoupled. The Meter Point Operator (MPO) is responsible +for the installation, operation, data gathering, and maintenance of energy meters. Note, in many locations, +system operators also perform as MPO. However, the electricity consumer could opt for an independent +MPO. This is in accordance with the § 43 German MsbG (measuring point operation law). In principle, also +a third party can be commissioned as a meter operator with the operation of the metering point on the free +market [38]. Furthermore, there is presently no general legal obligation to share grid-relevant information +obtained from smart meters with the DSO. Due to these constraints, DSO needs to request for receiving +historical data thus cannot be utilized for short-term grid operation, congestion mitigation, etc (due to delays +in DSO making a request and receiving the measurement data). Thus, observability in DNs are limited not +only by SM penetration level but also by data-sharing policies targeting system operators. +B. Roadmap of meter rollout +Fig. 3 shows the meter rollout phases in German. It was expected that by 2032, all German consumers +are to be equipped with modern metering devices. (§ 29 para. 3 p.1 MsbG) compared to other countries, + +9 +thus it will still take several years before the DSO has sufficient data from SMs at its disposal. Complicating +this schedule are also the legal issues concerning safety and privacy of smart meter operation and usage1. +The continued operation and installation of smart metering systems as defined by the Act by the MPOs +is thus still possible. However, there is no longer an obligation to install them [39]. This makes the need +for DN parameter estimation and mechanisms to enhance observability even more crucial for ensuring the +operational integrity of the network. +Fig. 3: Rollout Plan for smart meters in Germany by 2032 [40] +Mitnetz Strom has invested a total of 19 million euros in 2022 in the conversion to digital local network +stations. The plan is to install a total of 226 digiONS in the year 2022. By 2026, up to 30 percent of the +transformer stations and cable distributors in the network area are to be digitally equipped or retrofitted +with the corresponding metering technology. Measured secondary substations are an important component +in the digital transition. They ensure better controllability and transparency of medium and low-voltage +grids, which directly benefits the implementation of the energy transition and security of supply. Increasing +feed-in from renewable energies, rising demand for charging power for electro-mobility, extreme weather +conditions that endanger the energy supply, especially in areas with overhead lines - the reasons for the +digital monitoring and control of electricity grids are many and have one goal: security of supply. +1On 20 May 2022, the Federal Office for Information Security (BSI) withdrew the general ruling of 7 February 2020 on the determination +of technical feasibility pursuant § 30 MsbG (so-called market declaration on the rollout of smart metering systems) with effect for the past. +In addition, the BSI issued a general ruling pursuant to §19 (6) MsbG in which it determined that the use and installation of smart metering +systems available on the market do not pose any significant risks. + +2016 +2017 +2018 +2019 +2020 +2021 +2022 +2023 +2024 +2025 +20262027 +2028 +2029 +2030 +2031 +2032 +>100,000kWhp.a.and interruptiblesystems(914aEnWG) +>10,000kWhp.a. +PilotPhase +>6.000kwhp.a. +RESandCHP>100kW +RESandCHP>7kW +Optional>=6,000kWhp.a. +Optionalgeneration>1-7KW(fornewinstallations) +<=6,o00kWhp.a.andRES+CHP>=7kW (fornewbuildings/renovationimmediately)10 +C. Demo network for EUniversal +In EUniversal, Mitnetz Strom is testing the use of flexibility services and markets and is leading the +German demonstration together with the parent company E.ON SE [41]. The German demonstration tries to +combine principles of the German mandatory process Redispatch 2.0 [42] and a market-based approach to +mitigate grid constraints in a cascaded operation across multiple voltage levels. The goal is to provide DSOs +with access to flexibility from grid customers across the LV/MV level for their active system management. +To this end, the EUniversal consortium is testing various optimization algorithms with the aim of minimizing +activation costs while ensuring the secure operation of the grid and is developing concepts for grid state +estimation of smart grids, of which the first interim results and experiences will be presented. In the German +demo of EUniversal, Mitnetz Strom and its partners are investigating the use of flexibility markets in low +voltage grids for congestion management and voltage maintenance. An attempt is being made to develop +an iterative procedure that will prevent new congestion from occurring when flexibility is activated. +1) Network features and meter placement: The network considered for numerical evaluation is a typical +low-voltage network in Mitnetz’s network area in a small town in Eastern Germany. There are already +some flexible plants, but the penetration with them is not yet significant. Due to application cases, certain +locations in the LV grid are particularly interesting when it comes to equipping with measurement technology. +In particular, the end of the feeder is often an important indicator for the evaluation of potential voltage +band violations, while the current at the beginning of the feeder is important for the thermal constraints +in the network. Unfortunately, these points are often not available in practice due to ownership issues and +the partially pronounced building development in the localities. Therefore, cable distribution cabinets were +selected and equipped with measurements. For EUniversal devices are a bundle of voltage and current sensors +(Rogowksi coils), gateway, and power supply. +D. Need for phase information +LV DN topology identification is essential for efficient network operation, monitoring, and control. This +also assists in planning the phases for new resources connected to the DN. Next, there is a real issue Mitnetz +Strom faced due to the connection of 8 out of 9 electric vehicle chargers was made using a single-phase +(1 − φ). This led to thermal limit violations in that particular phase. With the topology identification, the +phase connections were optimized. According to DIN ISO 50160, [43], limits of voltage deviations are +defined up to ±10%. LV networks are asymmetrically biased by 1-φ loads. Unbalanced new loads, such as +electric fans, heat pumps, and unbalanced charging could amplify this effect. Unbalanced loaded DNs lose +a substantial amount of their power transmission capability. In a research project for an automatic phase + +11 +switch in EV charging showed the effects in charging for 1 − φ or two-phase connected EVs or hybrids in +the grid. These findings presented here are also applicable to 1 − φ heaters, inverter interfaced PV, storage, +and other unbalanced loads. +Fig. 4: Line loading for unplanned EV charger placement +Fig. 5: Line loading after phase redistribution +In the further evaluation of this field test, it is shown that with the average existing charging capacity, +the penetration rate with EVs until a line limit is violated increased from 16% to 47% with symmetrical + +CurrentMeasurements EVs withoutphase selection +40,00 +35,00 +30,00 +25,00 +/[A] +20,00 +15,00 +10,00 +5,00 +0,00 +Time of day [h] +L1 [A] , +1.2[A1 +L3[ACurrent Measurements EVs with phase selection +15,00 +10,00 +5,00 +0,00 +Time of day [h] +L1 [A] —L2[A] —L3 [A]12 +utilization of DN with approximately balanced phases. In the case of flexibility markets, this means the +possibility of using flexibility decreases for ensuring grid integrity. This underlines the importance of having +accurate knowledge of the phase connections, as flexibility activation should not further increase imbalance. +In theory, flexibility activation could also limit DN imbalance. Mitnetz Strom and most DSO’s follow a +passive way of identifying the phase connections. The accurate topology identification is performed manually +in case of a follow-up on customer complaints on power quality. This motivates us to propose a scalable +and robust phase identification mechanism using historical measurement data. + +13 +III. SYNTHETIC DATA GENERATION +In this section, we detail the steps we took for generating synthetic data for the DSO grid layout provided +in DigSilent format. A digital twin is used for the process of phase and load placement. Further, the neutral +conductor modeling, noise injection, and zonal clustering model used in this work are elaborated in this +section. +A. DGS parser for network JSON +A parser has been created to convert the DGS (DIgSilent) [44] file format into a JSON file that is readable +to the PowerModels script. The parser is derived from the GridCal python package [45]. The main difference +between the two formats is that the DGS data format contains different classes with different information in +a hierarchical structure, whereas the JSON file just requires information on the buses, branches, and devices +in the grid. +1) Bus information: The DGS format relies on cubicle information, which can be seen as a connection +point for the different elements in the grid. The cubicle information are the unique IDs given to each +connection point, which are converted to simple grid ids starting from 1 and ending in the number of buses +in the grid. The original ID is saved to allow for cross-checking data and for linking devices to the correct +bus. +2) Branch information: The DGS branch format relies on cable type information, but the JSON file +requires the values directly in per-unit (pu). Therefore, the r and x parameters (amongst other) needs to be +converted from their ohmic values to pu of the total cable length. This requires a multiplication of the base +impedance and the total length of the cable. +3) Device information: The DGS file format can contain detailed information on different loads and +static/synchronous generators. The JSON file format only has information on the devices. Therefore, the +parser extracts the relevant P and Q data from each device in the DGS file and compiles it as a separate +entity in the device file. Additional information included is bus ID, PV size, and connected phase. +4) Switches: A crucial element of the parser is removing the switches that connect branches to substations +and cabins. The switches are removed as they add numerical complexities when calculating the admittance +matrix in PowerModels [46]. Setting the r and x values to zero can lead to infinity values during the +admittance calculation, but setting it to a very low value leads to inefficiencies when running the power +flow. Removal of the switches not only removes numerical complexities but, since switches make up 7% of +the branches in the system, by removing the extra nodes the computation time of the simulation significantly +increases (in the order of a 10%). + +14 +Once the switches have been removed, the IDs should be renumbered. This also serves the purpose of +removing empty nodes in the grid, which has a positive effect on the computation time of the simulation +as the admittance matrix only contains non-zero components, which helps reduce its size. This also tidies +up the network data, making it easier to view from a simulation perspective. +B. Mitnetz Strom DN and metadata +Along with the grid data, metadata on the loads in the grids are also provided in a separate file. +This metadata contains details associated with different consumer devices in the grid. The information +includes a node number to connect it to the grid data, load type (e.g. household, PV, CHP, etc.), the annual +energy consumption, single or three-phase connection, and the available active and reactive power output (if +applicable). The information is compiled and appended to the device’s JSON file. Fig. 6 shows the spread +of the annual cumulative energy consumption of loads connected to the test DN used in this work. +Note that the exact phase connection of single-phase loads is not known. The goal of this paper is to +present a framework for identifying the phase connection of such single-phase consumers. +Fig. 6: Metadata for annual kWh consumption of 331 consumers in the test LV DN considered. Note that 94.5% of consumers +have an annual consumption of lower than 6000 kWh for the test DN. +C. Randomized phase mapping +DSOs actively try to balance the phases so that the load distribution is fairly balanced. Madeira island +case study in [47] and DSO questionnaire in [48] details the phase assignment procedure of a DSO. + +Number of consumers in LV feeo +10 +8 +6 +4 +2 +0 +0 +1000 +2000 +3000 +4000 +5000 +6000 +7000 +8000 +AnnualConsumptioninkWh9000 +1000014 +1215 +In order to generate synthetic data, randomized load mapping is used. For a single phase load with +different levels of annual kWh consumption, a phase is randomly selected from phases A, B, and C with +equal likelihood. +Fig. 7: Phase load distribution of three-phase distribution network for 100000 phase mapping scenarios. +The randomized phase mapping is evaluated based on the sum of the absolute error in phase load (AEPL), +which is given as +AEPL = 1 +3 +1 +D +D +� +i=1 +� +φ∈{A,B,C} +|LD +φ − ¯LD|, +where D denotes the number of Monte Carlo phase mapping scenarios, and ¯LD denotes the mean load +in all the phases and is given as ¯LD = � +φ∈{A,B,C} LD +φ /3. +Using 100000 Monte Carlo simulations for phase mapping, we observe that randomized phase mapping +performs fairly well, with a maximum and mean per phase load deviation of 30% and 9.5% with respect to +the mean load met by the three phases in the worst case (Fig. 8). Fig. 8 shows the distribution of the phase +load errors while performing randomized phase mapping. +Thus, in this work, we utilize randomized phase mapping for data generation for evaluating our proposed +probabilistic phase identification mechanism. + +Histogram for phase loads +5000 +phase A +2500 +5000 +phase B +2500 +of instances +5000 +phase C +Number +2500 +150000 +200000 +250000 +300000 +350000 +400000 +45000016 +Fig. 8: Sum of the absolute value of the difference of phase load and mean load of all the phases. +D. Neutral modelling for four-wire DN +European LV DN is usually different from the North-American one with a larger size distribution trans- +former with multiple low voltage feeders supplying a large number of consumers per transformer. The +German low voltage feeders normally follow a four-wire three-phase configuration with single-grounded +neutral [49]. Such systems are usually reduced to three-wire equivalent using Kron’s reduction [50]. In +Kron’s reduction, it is assumed that the neutral is grounded multiple times and for a perfectly grounded +neutral2, the neutral voltage equals zero. However, for the DN considered in this paper, this assumption is not +true as the neutral is isolated from consumer grounding and is grounded only at the sub-station (See Fig. 9). +The inclusion of sparsely grounded neutral in modeling can be done by taking an exact four-wire model with +four-wire power flow solvers or reducing it to a three-wire equivalent and solving by using the three-wire +solvers. In [51] a new reduction method is proposed for sparsely grounded European LV feeders so that +the impact of neutral is represented as equivalent as a four-wire model without the necessity of carrying +around extra variables and measurements. In such reduction, the 4×4 impedance matrix is transformed to +3×3 matrix is given in (1). +2A perfectly grounded neutral refers to grounding resistance of zero ohms. Typically the grounding resistance is ≈ 5 ohms which leads to a +small voltage drop. In this work, we assume perfectly grounded neutral. + +6000 +5000 +4000 +umber +3000 +N +2000 +1000 +0 +10 +152025 +35 +40 +Absolute error in %17 +b +zl,aa +zl,bb +Ilij,a +Ilij,b +Ilij,c +Ilij,n +Ui,a +zl,cc +zl,nn +zl,ab +zl,bc +zl,cn +Ui,b +Ui,c +Ui,n +Uj,a +Uj,b +Uj,c +Uk,c +Uj,n +Uk,n +Uk,b +Uk,a +isolated neutral +c +a +k +j +i +n +g +substation grounding +Fig. 9: Isolated neutral model of distribution network +Impedance matrix = +� +���� +zs +l,aa − zs +l,na − zs +l,an + zs +l,nn +zs +l,ab − zs +l,nb − zs +l,an + zs +l,nn +zs +l,ac − zs +l,nc − zs +l,an + zs +l,nn +zs +l,ba − zs +l,na − zs +l,bn + zs +l,nn +zs +l,bb − zs +l,nb − zs +l,bn + zs +l,nn +zs +l,bc − zs +l,nc − zs +l,bn + zs +l,nn +zs +l,ca − zs +l,na − zs +l,cn + zs +l,nn +zs +l,cb − zs +l,nb − zs +l,cn + zs +l,nn +zs +l,cc − zs +l,nc − zs +l,cn + zs +l,nn +� +���� +(1) +In (1), zs +l,aa is self-impedance of the phase a of the branch l, zs +l,nn is self-impedance of the neutral +of the branch l, and so on. Similarly, zs +l,ab is the mutual-impedance between phase a and b of branch l. +This transformation is exact and eliminates the error introduced by Kron’s reduction in three-phase DN +modeling for sparsely grounded system [51], [52]. Furthermore, a minor boost in computation time is +achieved compared to the exact four-wire model as the necessity of carrying extra variables for neutral +voltage is removed. This reduction is more relevant as the measured voltages in German demo-grid are also +phase-to-neutral. Interested readers are guided to [51] for details about the transformation. +E. Metering noise injection model +Prior works [14], [15], [17], [20] consider smart meter measurement error based on the accuracy class +of metering infrastructure. Frequently, the measurement error is considered using Gaussian noise. Further, +the measurement accuracy is considered to hold true for three sigma of the times, which corresponds to +99.7% of total instances. The standard deviation of the Gaussian noise is related to the tolerance τ of the +measuring device. The noisy measurement is given as +ˆZ = Z × Norm(1, τ/3), +(2) + +18 +where Z denotes the true measurement, Norm(µ, σ) denotes a sample of a normal distribution with mean +µ and standard deviation σ. In order to evaluate the impact of measurement noise, 1000 Monte Carlo +simulations are considered. +F. Zonal clustering of Mitnetz Strom DN +Identifying the zones of an LV DN can be helpful to the DSO in planning the flexibility needs of a network. +Due to the large numbers of DN feeders, a standardized approach to divide zones based on electrical and/or +geographical distances is deemed essential [53]. In this section, the summary of the clustering framework +to identify the best-suited LV DN zonal partition using electrical distance as a measure is presented, which +is explained in detail in [53]. This zonal partition method uses an incidence matrix-based measure, which +can be obtained with the help of spectral decomposition of the admittance matrix. The adequate number of +zones is obtained based on the maximization of silhouette score while considering the desired number of +clusters. +The zonal partition divides nodes N into c ∈ {1, ..., C} clusters. The spectral clustering proposed in [54], +[55] for the creation of zones or network reduction is used for zonal partition. A double stochastic matrix +is formed, which is a special type of Markov matrix where not only each row but also each column +add to 1. For this transformed matrix, all eigenvalues are real and smaller than or equal to 1, with one +eigenvalue exactly equal to 1 [56]. For identifying C partitions in a graph, the C highest eigenvalues and +corresponding orthonormal eigenvectors are identified. The eigenvector matrix of the order N × C is used +for DN partitioning, in effect reduces the dimensionality of the problem. k-means clustering is used to +partition the spectral data points. The goodness of a cluster is measured using the mean silhouette index +of the network cluster. The silhouette coefficient of a node is a confidence indicator of its association in a +group [54], [57]. +G. Power flows for synthetic data +Power flow equations translate the load information of consumers to the nodal voltage and nodal currents +when the network topology and impedances are known using the first equations. Three-phase unbalanced +power flow equations were used to create the pseudo-measurement point based on the given load data and +network topology. Open-source power flow solver of PowerModelsDistribution.jl was used for +creating such pseudo measurement points [58]. + +19 +IV. PHASE IDENTIFICATION METHODOLOGY +Using the synthetic data generated in the previous section, we develop correlation based voltage matrices +used for consensus algorithm base PCI algorithms in the next section. +A. Notation +A three-phase distribution network (DN) consists of phases denoted as φ ∈ {A, B, C}. The DN consists +of branches, nodes, loads, and generators. A DN is represented as a directed graph by < N, E >, where N +denotes the set of nodes in all the phases and E denotes the set of branches connecting a pair of nodes. +For each node i in the phase φ at any time t, have two variables: (i) voltage magnitude denoted as Vφ,i,t +and phase angle denoted as θφ,i,t. The voltage phasor at a node and phase is governed by power injections. +The branch denoted as (i, j) ∈ E is characterized by line admittance denoted as Yφ,ij. The line admittance +governs power flow and line losses. Nd ⊂ N denotes the nodes with loads connected. For these nodes, the +active and reactive power is given as P d +φ,i,t and Qd +φ,i,t. Ng ⊂ N denotes the nodes with generators connected, +have active and reactive power generation denoted as P g +φ,i,t and Qg +φ,i,t. The time t is sampled hourly, and its +range is given as t ∈ {1, .., T}. +Distribution network zone for cluster C +# of 1-phase loads Lc +# of measurements Mc +Fig. 10: A distribution network cluster with measurement points and single-phase loads with unknown phase connectivity. +The DN is clustered into c ∈ {1, ..., C} clusters. A cluster c consists of Mc number of three-phase reference +measurement points, Lc number of single-phase consumers and Nc are the number of nodes present in that +cluster. +We assume that all the reference measurements are aligned and there are no synchronization +delays considered in this work. A stylized representation of a zone with 1 − φ consumers and 3 − φ + +000020 +reference measurement points are shown in Fig. 10. DN clustering is performed such that Ni ⊂ N and +Ni ∩ Nj = ∅, ∀i ̸= j. The set of nodes where Mc measurements are placed is denoted as iM +c . The set of +nodes where Lc single phase consumers are located is denoted as iL +c . For a vector parameter K, ¯K is the +mean value, and |K| denotes its absolute value. For a vector K, C(K) denotes its cardinality. 1(condition) +returns 1 if the condition is true. +B. Correlation based metrics +In this work, we utilized voltage magnitude time series as a parameter for estimating phases of the single- +phase consumers in each of the clusters. Each of the measurement points is utilized for estimating the phases. +Thus, a unique reference matrix is created using the phase voltage magnitudes at the measurement node. +For cluster c, measurement iM +c (j) where j ∈ {1, .., Mc}, the reference voltage matrix is given as +V ref +iM +c (j) = +� +���������� +VA,iM +c (j),1 +VB,iM +c (j),1 +VC,iM +c (j),1 +VA,iM +c (j),2 +VB,iM +c (j),2 +VC,iM +c (j),2 +: +: +: +: +: +: +VA,iM +c (j),T +VB,iM +c (j),T +VC,iM +c (j),T +� +���������� +(3) +The dimension of V ref +iM +c (j) is T × 3. Note time t is hourly, therefore, C(t ∈ {1, .., T}) = T. +The single phase consumer nodal voltage time series in cluster c forms a column of the matrix denoted +as V L +c , and given as +V L +c = +� +���������� +Vφ,iL +c (1),1 +Vφ,iL +c (2),1 +.. +Vφ,iL +c (Nc),1 +Vφ,iL +c (1),2 +Vφ,iL +c (2),2 +.. +Vφ,iL +c (Nc),2 +: +: +.. +: +: +: +.. +: +Vφ,iL +c (1),T +Vφ,iL +c (2),T +.. +Vφ,iL +c (Nc),T +� +���������� +(4) +The dimension of V L +c is T ×Nc. The connection phase of 1−φ consumers are assumed to be not known. +For the calculation of correlation between reference 3−φ voltage and 1−φ consumer voltage time series, +Pearson correlation3 is utilized. +Three models are presented for generating metrics for phase identification using a consensus algorithm. +These three metrics are described next. +3The Pearson correlation between two vectors X and Y is given as +ρ(X, Y ) = +�(X − ¯ +X)(Y − ¯Y ) +��(X − ¯ +X)2 �(Y − ¯Y )2 +(5) + +21 +1) J1: correlation of voltage time series: The correlation matrix for measurement iM +c (j) is denoted as +ρc,iM +c (j) +J1 += +� +������� +ρ(ViL +c (1), VA,iM +c (j)), +ρ(ViL +c (1), VB,iM +c (j)) +ρ(ViL +c (1), VC,iM +c (j)) +: +: +: +: +: +: +ρ(ViL +c (Nc), VA,iM +c (j)), +ρ(ViL +c (Nc), VB,iM +c (j)) +ρ(ViL +c (Nc), VC,iM +c (j)) +� +������� +(6) +Note that the voltage time series is denoted by dropping t in the notation. For instance, ViL +c (w) denotes the +voltage time series for t ∈ {1, .., T} for single phase consumer located at node id iL +c (w) : w ∈ {1, .., Nc}. +For single-phase consumers with unknown phases, the phase notation is also dropped to avoid confusion. +2) J2: Salient features with voltage difference time series: Salient features in voltage time series could +help in improving phase identification. The use of salient features has been explored in [3], [8], [12]. In this +work, we utilize the difference matrix and a zonal voltage fluctuation threshold for identifying the salient +features. The difference matrix for the reference voltage matrix for cluster c and measurement iM +c (j) is given +as +∆V ref +iM +c (j) = +� +Vφ,iM +c (j),t+1 − Vφ,iM +c (j),t ∀ t, ∀ φ +� +. +(7) +The dimension of ∆V ref +iM +c (j) is (T − 1) × 3. +βc denotes the voltage change threshold for cluster c. +The salient features are extracted using the ∆V ref +iM +c (j) matrix as +isalient +c,iM +c (j) = arg 1 +� +|∆V ref +iM +c (j)(t)| > βc ∀t ∈ {1, .., T − 1} +� +. +(8) +The voltage difference matrix for the connected load matrix is denoted as +∆V L +c = diff(V L +c ), +(9) +where diff operator finds the difference of adjacent rows. The dimension of ∆V L +c matrix is (T − 1) × Nc. +The new reference and load matrix extracts the rows with salient features in the reference matrix and are +given as +∆V ref, J2 +iM +c (j) = +� +∆V ref +iM +c (j)(isalient +c,iM +c (j), :) +� +, +(10) +∆V L, J2 +c += +� +∆V L +c (isalient +c,iM +c (j), :) +� +, +(11) +The correlation matrix with salient features using the voltage difference as a metric is given as +ρc,iM +c (j) +J2 += +� +������� +ρ(∆V L, J2 +c +(1), ∆V ref, J2 +A,iM +c (j)) +.. +ρ(∆V L, J2 +c +(1), ∆V ref, J2 +C,iM +c (j)) +: +: +: +: +: +: +ρ(∆V L, J2 +c +(Nc), ∆V ref, J2 +A,iM +c (j)) +.. +ρ(∆V L, J2 +c +(Nc), ∆V ref, J2 +C,iM +c (j)) +� +������� +(12) + +22 +3) J3: Salient features with voltage magnitude time series: Previously, we used the voltage difference as +a metric for identifying the salient features. The salient features when projected onto the voltage magnitude +would require the previous time stamp to capture the voltage change trajectory. This trajectory captured will +improve the correlation-based metric we are utilizing for phase identification. The new salient feature matrix +is given as +isal, plus +c,iM +c (j) = unique( +� +isalient +c,iM +c (j), isalient +c,iM +c (j) + 1 +� +), +(13) +with unique operator finding unique time stamps, considering there could be repetitions that will be +eliminated. +The new reference and load voltage matrix are given as +V ref, J3 +iM +c (j) = +� +V ref +iM +c (j)(isal, plus +c,iM +c (j), :) +� +, +(14) +V L, J3 +c += +� +∆V L +c (isal, plus +c,iM +c (j), :) +� +. +(15) +The correlation matrix for model J3 is given as +ρc,iM +c (j) +J3 += +� +������ +ρ(V L, J3 +c +(1), V ref, J3 +A,iM +c (j)), +.. +ρ(ViL +c (1), V ref, J3 +C,iM +c (j)) +: +: +: +: +: +: +ρ(V L, J3 +c +(Nc), V ref, J3 +A,iM +c (j)), +.. +ρ(ViL +c (Nc), V ref, J3 +C,iM +c (j)) +� +������ +(16) +The dimension of ρc,iM +c (j) +J1 +, ρc,iM +c (j) +J2 +and ρc,iM +c (j) +J3 +equals Nc × 3. + +23 +V. CONSENSUS ALGORITHM +In Section IV, we calculated correlation matrices using the voltage time series for measurement reference +located at node iM +c (j), j ∈ {1, .., Mc}. Thus, with multiple measurement points in a cluster, independent +phase identifications can be performed. These estimations can be taken into consideration using consensus +algorithms to be presented in this section. +A consensus algorithm is a strategy that a group of agents use to agree with each other on what’s true. In +a multi-sensor PCI scenario, there is just one true phase placement (ground truth), which is given as P true +c +. +Each of the measurements used as a reference for models J1, J2, and J3 as metrics are used for estimating +the true phases of single-phase consumers in cluster c. The advantage of the consensus algorithm is that no +one measurement point limits the PCI accuracy. One of the most widely used consensus algorithms is in +blockchain technology. Consensus algorithms are widely used in state estimation [59]–[61]. In this work, +we use consensus for phase identification in a distribution network. +A. Naïve phase identification +The naïve phase identification, denoted as S0, uses only one of the 3 − φ reference measurement points, +typically the substation measurement. +P est,Jx,S0 +c += arg max ρc,isel +Jx , +(17) +where isel denotes the node id for the reference measurement point. Most literature on phase identification +uses this naïve model with substation time series measurement as the reference, see Tab. I. +B. Majority rule +The majority rule, denoted as S1, is one of the most commonly used consensus algorithm. It identifies the +most agreed-upon estimation. Earlier works such as [62], [63] detail the applications of the majority rule in +building consensus among agents (sensors). +We note that for a cluster c, measurement point located at node iM +c (j), and metric Jx, we can calculate +ρc,iM +c (j) +Jx +. This correlation matrix is used for calculating the estimated phases, as +P est,Jx +c,iM +c (j) = arg max ρc,iM +c (j) +Jx +, +P est,Jx,S1 +c += fS1 +� +P est,Jx +c,iM +c (j) ∀j ∈ {1, .., Mc} +� +. +(18) +The function fS1 calculates the majority among the estimated phases. Consider there are 7 measurement +points in a cluster. For a node, consider 3 of the estimations that predict phase B, and 2 for phases A and +C respectively. In this case, the majority rule predicts the phase to be estimated as phase B. + +24 +C. Weighted measure +Previously, for a majority rule-based consensus algorithm, we assumed all agents to be of equal importance +(or weights). However, if we use the physical laws governing the system, we can calculate the weights for +different agents. In phase identification, earlier works point out that measurement points in geographical +proximity will have a greater voltage correlation among similar phases [11], [22], see Tab. I. In this work, +we use the correlation value as a weighing factor for calculating the estimated phase. A correlation value +of 1 implies 100% correlation. +1) Correlation as a measure: +¯ρc +Jx = +Mc +� +k=1 +3 +� +φ=1 +ρc,k +Jx, +(19) +where ¯ρc +Jx is Nc × 1 vector. The normalized correlation coefficients are given as +GJx,S2 +c += +�Mc +k=1 ρc,k +Jx +¯ρc +Jx +, +(20) +where GJx,S2 +c +is Nc × 3 matrix. The estimated phases are given as +P est,Jx,S2 +c += arg max GJx,S2 +c +. +(21) +2) Absolute value of correlation as a measure: +¯ρc +Jx,abs = +Mc +� +k=1 +3 +� +φ=1 +|ρc,k +Jx|, +(22) +where ¯ρc +Jx,abs is Nc × 1 vector. The normalized correlation coefficients are given as +GJx,S3 +c += +�Mc +k=1 |ρc,k +Jx| +¯ρc +Jx,abs +, +(23) +where GJx,S3 +c +is Nc × 3 matrix. The estimated phases are given as +P est,Jx,S3 +c += arg max GJx,S3 +c +. +(24) +3) Maximum value of correlation as a measure: +¯ρc +Jx,max = +max +k=1,..,Mc max +φ=1,2,3 |ρc,k +Jx|, +(25) +where ¯ρc +Jx,max is Nc × 1 vector. The normalized correlation coefficients are given as +GJx,S4 +c += maxk=1,..,Mc |ρc,k +Jx| +¯ρc +Jx,max +, +(26) +where GJx,S4 +c +is Nc × 3 matrix. The estimated phases are given as +P est,Jx,S4 +c += arg max GJx,S4 +c +. +(27) +Note that (20), (23) and (26) denotes element wise division of vector of length Nc. + +25 +D. Metrics for phase identification models +1) Modelling accuracy: Consider, the true phase information in a cluster is given as P true +c +. The estimation +accuracy is denoted as +Estimation accuracy = +number of correct phase estimation +total number of single phase consumers. +The phase estimation accuracy for metric Jx ∈ {J1, J2, J3}, cluster c and measurement iM +c (j) is given as +AJx +c,iM +c (j) = 100 × +� +1 − +� 1 +� +P est,Jx +c,iM +c (j) − P true +c +̸= 0 +� +Nc +� +, +(28) +where P est,Jx +c,iM +c (j) denote the phase estimation vector for cluster input metric Jx, cluster c and measurement +iM +c (j). +The estimation accuracy for consensus algorithm Sy is given as +AJx,Sy +c += 100 × +� +1 − +�Nc +n=1 1 +� +P est,Jx,Sy +c +− P true +c +̸= 0 +� +Nc +� +, +(29) +Since there are three base metrics denoted as Jx ∈ {J1, J2, J3} and five consensus models (including +the naïve model) denoted as Sy ∈ {S0, S1, S2, S3, S4}, therefore, we evaluate in total 13 phase identification +models (the naïve model, S0, is performed for J1 only). In numerical results, we will compare the benefits and +shortcomings of these models. Estimation accuracy averaged over Q Monte Carlo simulations are denoted +as ¯AJx,Sy +c +. +2) Confidence factor for phase identification: Note that models S1, S3, and S4 provide coefficients that +add up to 1 (node-wise). Thus, GJx,S3 +c +and GJx,S4 +c +can be used to indicate probabilities of phase estimation. +For S1, the estimation probabilities can be calculated by dividing P est,Jx,S1 +c +with the number of measurements +in a cluster, Mc. +We define the confidence factor as the minimum distance between the factor associated with the correct +phase and the maximum of the two incorrect phases, over all nodes in the cluster c. For the model, S2 +we normalized the confidence factor with the range of variation of GJx,S2 +c +. The proposed confidence factor +will provide us with additional information about the robustness of our phase estimation output. Note, the +proposed confidence factor can lie in the range ∈ [−1, 1] for S1, S3, and S4. A confidence factor close +to 1 implies very high confidence in our phase estimation output. The confidence factor for measurement +k ∈ {1, .., Mc} and cluster c is given as F Jx,Sy +c,k +. The confidence factor for a cluster c for all Monte Carlo +scenarios is given as +¯F Jx,Sy +c += +1 +Q × Mc +Q +� +q=1 +Mc +� +k=1 +F Jx,Sy +c,k +, +(30) +where w ∈ {1, .., W} denotes the Monte Carlo scenarios. + +26 +3) Standard deviation with measurement error: Q Monte Carlo simulations are considered for minimizing +the measurement error biases on phase estimation. For each of Monte Carlo iteration q ∈ {1, ..., Q}, calculate +the standard deviation of the measure used for calculating P est,Jx,Sy +c +, denoted as DJx,Sy +c +. + +27 +VI. NUMERICAL RESULTS +The numerical case study considers a German DN with 646 nodes and 331 loads connected to it. Out +of 331 loads, 313 loads (94.6% of total consumers) are single-phase loads. The phase connections are +widely unknown to Mitnetz Strom. The objective of the case study is to assess the phase identification +algorithm proposed in this work, benchmarked over naïve phase identification model. The selected DN +is part of the demo network selected for evaluation in the EUniversal project. Mitnetz Strom placed 53 +3 − φ measurement devices in the DN. These measurement points will be considered as references used for +PCI. The flexibility participants will be provided with a Home Energy Management System (HEMS) which +will provide measurements of load and voltages at the point of common coupling. In the case studies, we +assume the time series of voltage measurements of all single-phase users are known. In the real world, only +measurements of consumers with HEMS will be available. +There are four case studies performed in this paper. The first case study compares the 12 phase identifica- +tion models on different phase mappings. The second case study quantifies the impact of reference location +in a DN on the phase identification metrics proposed in the work. The third case study assesses the impact +of measurement error at the reference and/or at the consumer location on phase identification metrics. The +last case study compares the phase identification metrics for DN with and without the neutral conductor +model. As most European DNs are four-wire, it is crucial to quantify the impact. +Prior to case studies, we detail the test DN clustering results and the performance of the naïve phase +identification algorithm. The benefits of the proposed phase identification algorithms are compared to the +naïve model. +A. Clustering of distribution network +Fig. 11 shows the location of single-phase consumers and measurement points in the DN. We apply the +clustering algorithm for identifying the clusters in the DN. +Since the number of clusters to be formed is not clear, we utilize the silhouette score plotted in Fig. 12 +for fixing the number of clusters. Observe that the silhouette score is maximized for 3 clusters with a value +of 0.872. However, we select the number of clusters to be 7 as maximization of silhouette score is not the +only goal. We also need to quantify how many clusters will make the problem tractable by explaining the +DN sufficient. Note there is a sharp decline in silhouette score if the number of clusters is increased beyond +7. +Previously, we defined βc as the voltage change threshold for cluster c. Note βc will vary with different +clusters, as voltage fluctuation in different zones will vary drastically. Fig. 13 shows the variation of nodal + +28 +Fig. 11: Consumers (blue) and measurement points (orange) in the DN +voltages in seven clusters of the Mitnetz Strom example LV distribution network. It can be observed that the +voltage variation in cluster 2 is very small, ranging from 0.995 to 1.002. This narrowband is due to cluster +2 including the substation and the slack bus, where the voltage is regulated at 1 per unit level. +Fig. 14 shows the DN clusters. We can also comment that the clusters identified are indeed stable, which is +validated by 100 Monte Carlo (MC) simulations. The stability of clusters, impacted due to initializations are +discussed in [64], [65]. Clustering based on k-means is sensitive to randomized initializations of centroids, +and if not stable would provide different clusters in different iterations. Observe that the numbering of +clusters in Fig. 14 is based on randomized initialization of the centroids, and indeed would vary in a +different iteration. +Tab. II details the phase mapping scenarios. In the first case study, we assess the impact of different +phase mappings. C1 denotes balanced, C2 denotes moderately balanced, and C3 denotes unbalanced phase +mapping based on the annual cumulative load on each phase. For the rest of the paper, if phase mapping is +not explicitly mentioned, then C2 phase mapping is used, see Tab. II. + +29 +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +Number of clusters +0.5 +0.55 +0.6 +0.65 +0.7 +0.75 +0.8 +0.85 +0.9 +Silhouette Score +2 +3 +4 +5 +6 +7 +8 +0.75 +0.8 +0.85 +0.9 +X 7 +Y 0.832668 +X 7 +Y 0.832668 +X 3 +Y 0.872196 +(b) +(a) +Fig. 12: Choosing the number of clusters of DN based on the silhouette coefficient. In (a) the variation of silhouette coefficient +is plotted with increasing number of cluster. In (b) we zoom into the plot (a). Note the silhouette coefficient is maximum for 3 +clusters, however, the best-suited number of clusters selected is 7 [53]. +Fig. 13: Distribution of voltage in phases in different clusters for Mitnetz Strom DN for cluster 1 to 7. + +60 +40 +40 +20 +50 +20 +0 +0.85 +0 +0.9 +0.95 +1 +1.05 +0.85 +0.9 +0.95 +1 +1.05 +0.85 +0.9 +0.95 +1 +1.05 +Cluster5 +Cluster 6 +100 +100 +Cluster7 +100 +(e) +80 +80 +(f) +(g) +60 +60 +60 +40 +40 +40 +20 +20 +20 +0 +0 +0.85 +0.9 +0.95 +1 +1.05 +0.85 +0.9 +0.95 +1 +1.05 +0.85 +0.9 +0.95 +1 +1 +Voltage in per unit.85 +0.9 +0.95 +1 +1.05 +Phase A +Phase B +Phase C +05Cluster 1 +80 +Cluster 2 +Cluster 3 +80 +100 +150 +(b) +60 +60 +80 +(c) +(a) +tan +100Cluster4 +(d)30 +Fig. 14: Clusters of DN. +TABLE II: Phase mapping scenarios +Annual cumulative load (MWh) +ID +Cases +Phase A +Phase B +Phase C +C1 +Highly balanced +274.6 +282.3 +275.7 +C2 +Fairly balanced +288.6 +292.4 +251.6 +C3 +Highly unbalanced +240.4 +257.7 +334.4 +The details of the number of consumers, measurement points, and voltage variation for phase mapping +C2 (see Table II) are provided in Table III. +B. Performance of naïve model +As the majority of prior works utilize a single voltage reference for PCI, we would show the performance +of this model, referred to as the naïve model, prior to evaluation of the proposed phase identification models. +We utilize 4 different measurement points close to the transformer for evaluating the naïve phase identification +model. The measurement points are shown in Fig. 11. It is also indicated that nodes 1, 72, 74, and 511 are + +5 +631 +TABLE III: Network attributes for C2 phase mapping +Cluster ID +Nc +Mc +βc +Max voltage deviation +1 +20 +4 +0.056 +0.136 +2 +73 +8 +0.008 +0.018 +3 +21 +9 +0.064 +0.164 +4 +56 +2 +0.051 +0.172 +5 +46 +9 +0.0493 +0.112 +6 +38 +8 +0.040 +0.091 +7 +59 +13 +0.031 +0.058 +connected to feeders going towards clusters 6, 4, 5, and 7 respectively, see Fig. 14. A zoomed-in plot of +measurement points and the location of the nodes of measurement is shown in Fig. 15. +The results of PCI using the naïve model is detailed in Table IV. It lists the cluster-wise PCI accuracy. +Observe that naïve model correctly estimates the phase connectivity for the cluster with which it is directly +connected, however, the estimation accuracy for other clusters can be as low as 0%. This is also shown in +Fig. 16. In Fig. 16, the measurement points are indicated by a black square, the correctly estimated consumer +phase is indicated by a green circle and red circles show as the incorrect identification. We can observe that +the selection of a reference highly affects the phase connectivity identification accuracy in a multi-feeder +distribution network. +TABLE IV: PCI accuracy with naïve model +Node 1 +Node 72 +Node 74 +Node 511 +Cluster 1 +33.33 +52.94 +100 +61.11 +Cluster 2 +44.44 +50 +56.76 +29.03 +Cluster 3 +82.35 +47.37 +100 +73.68 +Cluster 4 +12.5 +100 +35.14 +57.14 +Cluster 5 +69.77 +23.26 +100 +61.36 +Cluster 6 +100 +0 +63.64 +15.63 +Cluster 7 +47.06 +15.56 +19.57 +100 +overall accuracy +47.92 +44.09 +55.59 +52.72 +It is clear from the numerical evaluations that +• Naïve model for phase connectivity identification is very sensitive towards the selection of the reference +voltage node which is utilized for phase identification of a single phase consumer, and +• Naïve model does not consider multiple reference measurements in a DN, + +32 +Node 1: towards cluster 6 +Node 74: towards cluster 5 +Node 72: towards cluster 4 +Node 511: towards cluster 7 +Fig. 15: Measurement locations for naïve model evaluation +• The mean PCI accuracy with naïve model in a multi-feeder DN considered in this work is below 56%. +The numerical case studies are presented subsequently. +C. Case study 1: Comparing phase identification models +Previously, we proposed three metrics {J1, J2, J3} and four consensus algorithms {S1, S2, S3, S4}. In +this case study, we compare 12 phase connectivity identification algorithms are proposed and applied to the +German DN. This case study provides recommendations for the best-suited metric and consensus algorithm +to be used for phase identification. Note that these recommendations could vary for other DNs. All results +consider a measurement error of 1%. In order to eliminate the impact of measurement error on PCI, we +perform 1000 MC simulations with different measurement errors calculated using (2). +The majority rule consensus algorithm (S1) outperforms all other models proposed for all clusters except +for cluster 2. For all other clusters, the majority rule provides 100% accurate rules with a confidence factor +of 1. As detailed earlier, cluster 2 is the part of the DN around the substation. The voltage deviation in this +cluster is very small. Table V shows the performance of the majority rule consensus algorithm for cluster +2. Observe that all three metrics of phase identification are very poor. Thus, we drop the majority rule +consensus algorithm in subsequent evaluations. + +249 +49 +253 +249 +254 +246 +252 +514 +806 +207 +6 +51 +4 +74 +511 +1233 +(a) Node 1 as reference +(b) Node 72 as reference +(c) Node 74 as reference +(d) Node 511 as reference +Fig. 16: Phase connectivity identification accuracy with naïve model. The correct phase estimations are +indicated with green circles, and incorrect with red circles. The location of the reference is indicated with +a black square. The substation is marked with a yellow square. +From Fig. 17, the confidence factor deteriorates from model C1 which is balanced phase mapping to C3 +which is highly unbalanced phase mapping. No noticeable PCI accuracy change is observed for consensus +algorithms S2, S3, and S4. The mean confidence factor for cases C1, C2, and C3 are 0.201, 0.174, and +0.133 respectively. Thus, correlation-based phase connectivity identification tends to be more accurate for +more balanced phase mapping. +Fig. 18 shows the mean phase estimation accuracy for metrics {J1, J2, J3}. Observe that mean estimation + +34 +TABLE V: Majority rule (S1) model for cluster 2 +Case +Metric (Jx) +Accuracy +Confidence factor +Sensitivity +( ¯AJx,Sy +c +) in % +( ¯F +Jx,Sy +c +) +(D +Jx,Sy +c +) +C1 +J1 +50.99 +0 +0.3714 +J2 +50.99 +0 +0.3714 +J3 +59.99 +0 +0.4641 +C2 +J1 +60.76 +0 +0.3878 +J2 +60.75 +0 +0.3879 +J3 +61.14 +0 +0.4612 +C3 +J1 +58.91 +0 +0.3791 +J2 +59.83 +0 +0.3792 +J3 +59.41 +0 +0.4628 +Balanced (C1) +Fairly balanced (C2) +Unbalanced (C3) +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +Confidence factor +Mean confidence factor +Mean=0.174 +Mean=0.133 +Mean=0.201 +Fig. 17: Confidence factor for three-phase mappings for metrics {J1, J2, J3} and consensus algorithms {S2, S3, S4}. +accuracy is deteriorating for metrics J1 to J3. Further, the accuracy is higher for balanced phase mapping +compared to unbalanced phase mapping. Thus, Fig. 17 and 18 are used to evaluate the impact of phase +mappings {C1, C2, C3} on phase identification. Further, we observe that metric J1 outperforms others. J1 +is more robust as a measure of phase identification. +For each cluster {1,...,7} and phase mappings {C1, C2, C3} we rank the phase identification methods + +35 +0 +12 +24 +36 +48 +60 +72 +84 +40 +60 +80 +100 +% +Estimation accuracy for metric J1 +0 +12 +24 +36 +48 +60 +72 +84 +40 +60 +80 +100 +% +Estimation accuracy for metric J2 +0 +12 +24 +36 +48 +60 +72 +84 +40 +60 +80 +100 +% +Estimation accuracy for metric J3 +Mean = 96.5 % +C1: Balanced +C3: Unbalanced +C2: Fairly unbalanced +Mean = 94.01 % +Mean = 95.8 % +Fig. 18: Accuracy for three-phase mappings. +based on the three metrics (a) estimation accuracy in % denoted as ¯AJx,Sy +c +, (b) confidence factor denoted +as ¯F Jx,Sy +c +, and (c) sensitivity towards measurement errors denoted as DJx,Sy +c +. Out of these 21 models, 13 +times consensus algorithm S3 and metric J1 denoted as S3-J14, outperformed all the other combinations, +see Fig. 19. Therefore, for all subsequent studies, we will utilize S3-J1 as the best-suited model for phase +identification for the DN considered. +Table VI shows the consolidated results for metric J1. Observe that the sensitivity towards phase identi- +4Model Sy-Jx denotes that the phase identification algorithm utilizes Jx as the metric and Sy as the consensus algorithm + +36 +3 +5 +13 +0 +2 +4 +6 +8 +10 +12 +14 +S2-J1 +S4-J1 +S3-J1 +Fig. 19: For three different phase mappings and 7 clusters, S3-J1 phase identification model outperforms others for most of the +time. The plot shows the histogram best model for a cluster. +fication due to measurement error is more than 10 times higher for cluster 2 (close to the substation with +low βc) compared to other clusters. This effect can again be attributed to low voltage fluctuations in cluster +2, see Fig. 13. +D. Case study 2: Effect of the proximity of measurement on PCI +The goal of this case study is to assess the impact of measurement point proximity on phase identification +performance metrics. The simplified representation of the original network model in Fig. 14 is shown in +Fig. 20. +Zonal connections are listed in Table VII. L0 denotes the zone where the 1 − φ consumer is located. L1 +denotes the zones which are directly connected to L0. Similarly, L2 denotes zones that are not connected +directly to L0 but via L1; second-order neighbor, and so on. +Tables VIII, IX, and X list the numerical results with mean phase estimation metrics for each consumer +and each measurement point in the DN. The following observations are made: +• Phase estimation accuracy deteriorates as the electrical distance between the measurement point used +as the reference in correlation-based phase identification and the 1 − φ consumer with unknown phase +connection increases, +• The confidence of estimation deteriorates as the distance between measurement and consumer increases, + +37 +TABLE VI: PCI estimation for metric J1 +Balanced (C1) +Fairly balanced (C2) +Unbalanced (C3) +Cluster +Consensus +¯AJx,Sy +c +¯F +Jx,Sy +c +D +Jx,Sy +c +¯AJx,Sy +c +¯F +Jx,Sy +c +D +Jx,Sy +c +¯AJx,Sy +c +¯F +Jx,Sy +c +D +Jx,Sy +c +1 +S2 +100 +0.5592 +0.0295 +100 +0.2951 +0.0344 +40 +-0.0669 +0.0386 +1 +S3 +100 +0.4509 +0.016 +100 +0.376 +0.0137 +100 +0.0615 +0.0068 +1 +S4 +100 +0.4361 +0.0179 +100 +0.3723 +0.017 +100 +0.0654 +0.0076 +2 +S2 +86.7973 +-0.0022 +3.6282 +91.2137 +-0.0027 +6.94 +85.4356 +-0.0008 +4.0251 +2 +S3 +88.9466 +-0.0049 +0.1293 +87.2301 +-0.0038 +0.1338 +88.3151 +-0.0049 +0.1344 +2 +S4 +91.6849 +-0.0076 +0.1202 +90.3699 +-0.0065 +0.1249 +91.2589 +-0.007 +0.1217 +3 +S2 +100 +0.4791 +0.0264 +100 +0.1388 +0.0567 +100 +0.4958 +0.0261 +3 +S3 +100 +0.4115 +0.0144 +100 +0.2644 +0.0103 +100 +0.3907 +0.0138 +3 +S4 +100 +0.3856 +0.0159 +100 +0.2569 +0.0122 +100 +0.3575 +0.0157 +4 +S2 +91.5196 +-0.0035 +7.2585 +100 +0.1442 +0.069 +99.6857 +0.0337 +3.4869 +4 +S3 +100 +0.1738 +0.0166 +100 +0.2467 +0.0225 +100 +0.1704 +0.0168 +4 +S4 +100 +0.1722 +0.01471 +100 +0.261 +0.236 +100 +0.1607 +0.0172 +5 +S2 +100 +0.3729 +0.0399 +100 +0.2052 +0.0558 +100 +0.2329 +0.0422 +5 +S3 +100 +0.3827 +0.0235 +100 +0.3451 +0.02 +100 +0.2453 +0.0159 +5 +S4 +100 +0.3783 +0.0265 +100 +0.3287 +0.0242 +100 +0.2245 +0.0194 +6 +S2 +100 +0.2579 +0.0509 +100 +0.2887 +0.0466 +100 +0.1096 +0.1027 +6 +S3 +100 +0.3457 +0.0191 +100 +0.2927 +0.0201 +100 +0.2376 +0.0196 +6 +S4 +100 +0.3153 +0.0235 +100 +0.2923 +0.0226 +100 +0.1968 +0.0217 +7 +S2 +100 +0.0741 +0.0961 +99.9983 +0.0746 +0.5477 +100 +0.0406 +0.1595 +7 +S3 +100 +0.1803 +0.0461 +100 +0.1 +0.0347 +100 +0.2281 +0.0458 +7 +S4 +100 +0.1732 +0.0491 +100 +0.0955 +0.0479 +100 +0.1587 +0.0484 +Zone 1 +Zone 3 +Zone 7 +Zone 5 +Zone 6 +Zone 4 +Zone 2 +Fig. 20: Zones in a simplified network diagram. + +38 +TABLE VII: Zonal connections +L0 +1 +2 +3 +4 +5 +6 +7 +L1 +5 +[4,5,6,7] +5 +2 +[1,2,3] +2 +2 +L2 +[2,3] +[1,3] +[1,2] +[5,6,7] +[4,6,7] +[4,5,7] +[4,5,6] +L3 +[4,6,7] +- +[4,6,7] +[1,3] +- +[1,3] +[1,3] +TABLE VIII: Mean PCI accuracy with reference selection +level +1 +2 +3 +4 +5 +6 +7 +L0 +100 +90.44 +100 +100 +100 +100 +100 +L1 +100 +43.94 +100 +100 +100 +99.39 +78.78 +L2 +100 +35.71 +100 +29.57 +44.94 +75.18 +48.44 +L3 +34.76 +- +47.31 +27.37 +- +30.47 +19.55 +TABLE IX: Mean confidence factor ( ¯F Jx,Sy +c +) with reference selection +level +1 +2 +3 +4 +5 +6 +7 +L0 +0.372 +-0.007 +0.256 +0.261 +0.329 +0.293 +0.096 +L1 +0.357 +-0.005 +0.262 +0.141 +0.317 +0.039 +-0.003 +L2 +0.333 +-0.005 +0.240 +-0.004 +-0.014 +-0.025 +-0.005 +L3 +-0.049 +- +-0.051 +-0.004 +- +-0.016 +-0.003 +TABLE X: Mean PCI STD (DJx,Sy +c +) with reference selection +level +1 +2 +3 +4 +5 +6 +7 +L0 +0.017 +0.125 +0.012 +0.023 +0.024 +0.022 +0.048 +L1 +0.019 +0.223 +0.018 +0.074 +0.039 +0.078 +0.108 +L2 +0.044 +0.259 +0.042 +0.099 +0.088 +0.100 +0.124 +L3 +0.068 +- +0.066 +0.094 +- +0.101 +0.116 +• The mean estimation variance increases (implying estimation becomes more sensitive to measurement +errors) as the distance between the measurement point used as the reference in correlation based phase +identification and the single phase consumer with unknown phase connection increases. +The phase connectivity identification metrics proposed in this work not only quantitatively provide the +estimation accuracy in % but also qualitatively provide the confidence factor (the higher the better) and +the sensitivity to measurement errors (the lower the better). Using these metrics, we observe that selecting +of measurement points in proximity is better in terms of phase estimation quality. This also validates our + +39 +zonal phase identification approach we have established in this work. +E. Case study 3: Impact of measurement error +In case study 1, we identified the best-suited model for phase identification as the S3-J1. Previously, we +observed that an imbalance in DN leads to worse estimation compared to balanced phase mapping. In order +to not have pessimistic or optimistic phase identification results, we utilize C2 phase mapping. We apply +different levels of measurement errors at the point of reference and at the point of single phase consumer +point of connection. In order to not be biased by one sample of estimation error, we perform 1000 MC +simulations. The three-phase identification metrics are shown in Fig. 21. The three metrics are almost equally +influenced by measurement error at reference voltage measurement or 1-φ consumer end. Observe from Fig. +21 +• Mean phase estimation accuracy for 1% error in reference voltage and consumer voltage measurement +is 98.7%. Implying, our consensus-based phase estimation algorithm was able to estimate more than +308 out of 313 of single-phase consumer phases accurately on average. +• For 10% error in reference voltage and consumer voltage measurement is still 82%. Thus, the phase +estimation proposed in this work is largely immune to measurement errors. +• The confidence factor deteriorates with an increase in measurement errors, see Fig. 21(b). +• The sensitivity of phase estimation towards measurement error, shown as the variance in estimation +Fig. 21(c) increases with an increase in measurement error. +F. Case study 4: Effect of size of neutral conductor +European DNs are 4-wire systems with a neutral conductor. The impact of modeling the neutral conductor +is not assessed in any of the prior works. This is especially crucial if the digital twin is used for generating +synthetic data. Fig. 22 shows the phase identification metrics for model S3-J1 and phase mapping C2 with and +without neutral conductor considerations. It is clear that modeling the neutral conductor improves estimation +accuracy for all three metrics. + +40 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +Effect of measurement error on PCI accuracy +100 +99.9 +99.6 +99.1 +98.5 +97.8 +97.4 +96.8 +96 +95.3 +94.6 +99.7 +98.7 +97.6 +97 +96.5 +95.9 +95.2 +94.4 +93.8 +93.2 +92.5 +98 +97.1 +96.4 +95.5 +94.6 +93.9 +93.1 +92.5 +91.8 +91.4 +90.6 +97.2 +96.2 +95.2 +93.9 +93.1 +92.3 +91.7 +91.1 +90.3 +89.8 +89.3 +96.7 +95.5 +94.1 +92.9 +92.1 +91.2 +90.6 +90 +89.4 +88.8 +96.3 +94.8 +93.4 +92.2 +91.3 +90.6 +89.9 +89.3 +88.7 +96.1 +94.3 +92.7 +91.6 +90.8 +90.1 +89.2 +88.4 +95.4 +93.6 +92.2 +91 +90.2 +89.3 +95.3 +93.4 +91.5 +90.5 +89.5 +88.8 +94.9 +92.8 +91.3 +90.2 +89.1 +94.4 +92.2 +90.7 +89.6 +88.7 +88.1 +88 +87.1 +87.6 +87 +86.1 +88.3 +87.4 +86.9 +85.8 +85.2 +87.7 +86.8 +85.9 +84.9 +84.1 +88.3 +87.2 +86.1 +85.1 +83.9 +83.1 +87.6 +86.2 +85.2 +84.2 +83.1 +82 +85 +90 +95 +100 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +Error in reference 3-phase voltage measurement (%) +Confidence factor of PCI +0.279 +0.251 +0.232 +0.221 +0.21 +0.197 +0.184 +0.17 +0.16 +0.14 +0.134 +0.232 +0.229 +0.221 +0.211 +0.199 +0.187 +0.177 +0.162 +0.148 +0.138 +0.127 +0.211 +0.209 +0.202 +0.193 +0.183 +0.171 +0.16 +0.148 +0.136 +0.123 +0.113 +0.198 +0.195 +0.188 +0.179 +0.17 +0.158 +0.143 +0.13 +0.119 +0.186 +0.183 +0.175 +0.165 +0.152 +0.14 +0.128 +0.114 +0.176 +0.174 +0.164 +0.153 +0.137 +0.124 +0.167 +0.164 +0.154 +0.137 +0.125 +0.158 +0.154 +0.141 +0.127 +0.111 +0.15 +0.144 +0.131 +0.116 +0.144 +0.138 +0.12 +0.139 +0.132 +0.113 +0.104 +0.094 +0.098 +0.083 +0.074 +0.109 +0.097 +0.083 +0.071 +0.061 +0.11 +0.095 +0.079 +0.068 +0.055 +0.045 +0.093 +0.081 +0.066 +0.057 +0.045 +0.041 +0.099 +0.083 +0.067 +0.056 +0.044 +0.039 +0.03 +0.104 +0.086 +0.072 +0.056 +0.043 +0.037 +0.03 +0.026 +0.098 +0.078 +0.064 +0.047 +0.038 +0.03 +0.025 +0.02 +0.05 +0.1 +0.15 +0.2 +0.25 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +Error in 1-phase voltage measurement (%) +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +Variance of PCI +0.079 +0.093 +0.107 +0.119 +0.132 +0.143 +0.154 +0.071 +0.085 +0.098 +0.11 +0.12 +0.131 +0.142 +0.15 +0.079 +0.091 +0.103 +0.114 +0.124 +0.133 +0.141 +0.15 +0.074 +0.087 +0.098 +0.108 +0.119 +0.128 +0.137 +0.144 +0.153 +0.07 +0.082 +0.094 +0.104 +0.114 +0.124 +0.133 +0.142 +0.149 +0.156 +0.07 +0.079 +0.089 +0.1 +0.11 +0.12 +0.13 +0.138 +0.146 +0.153 +0.161 +0.077 +0.086 +0.097 +0.107 +0.117 +0.127 +0.136 +0.144 +0.152 +0.16 +0.166 +0.086 +0.093 +0.102 +0.113 +0.123 +0.132 +0.142 +0.15 +0.158 +0.164 +0.17 +0.095 +0.1 +0.11 +0.12 +0.129 +0.138 +0.148 +0.156 +0.163 +0.17 +0.175 +0.102 +0.105 +0.115 +0.124 +0.134 +0.144 +0.153 +0.161 +0.169 +0.175 +0.181 +0.105 +0.112 +0.121 +0.129 +0.14 +0.149 +0.158 +0.165 +0.172 +0.179 +0.186 +0 +0.026 +0.046 +0.063 +0.023 +0.039 +0.056 +0.039 +0.05 +0.065 +0.052 +0.061 +0.061 +0 +0.05 +0.1 +0.15 +(a) +(b) +(c) +% +Fig. 21: Impact of measurement accuracy on PCI metrics. (a) shows the accuracy, (b) shows the confidence factor and (c) shows +the variance of PCI respectively. + +41 +1 +2 +3 +4 +5 +6 +7 +60 +70 +80 +90 +100 +Estimation accuracy in % +Neutral size same as phase conductor +No neutral considered +1 +2 +3 +4 +5 +6 +7 +0 +0.2 +0.4 +0.6 +Confidence factor +1 +2 +3 +4 +5 +6 +7 +Cluster ID +0 +0.05 +0.1 +0.15 +Sensitivity towards measurement error +(a) +(c) +(b) +Fig. 22: Impact of neutral conductor modeling on phase estimation metrics. +VII. CONCLUSIONS +We propose a phase connectivity identification (PCI) algorithm that utilize voltage time series, distribution +network (DN) zones, and multiple measurements as references for improving the PCI accuracy. The proposed +phase identification algorithm builds a consensus among multiple estimations in a zone. This method is +extended to consider metrics derived from voltage time series, which filters larger voltage deviations. Due +to the consideration of multiple measurements, the PCI is drastically exceeding the performance compared +to the widely used naïve model. Further, our approach is immune to network topology and measurement +errors, as PCI accuracy is not dependent on only one reference measurement point. We also propose phase +connectivity identification metrics that not only quantitatively describe the estimation accuracy, but also +qualitatively describe how good the PCI is. We utilize a real German DN with limited observability for phase + +42 +identification. This network has 602 nodes and 313 single-phase consumers. It is observed that the original +voltage time series without salient feature extraction and absolute weighted consensus algorithm outperforms +all the other phase estimation algorithms. The proposed algorithm identifies the phase connections on average +of over 308 consumers accurately for 1% measurement error at the consumer end and reference measurements +for 1000 Monte Carlo simulations. Thus, the proposed algorithm is robust towards uncertainty with high +precision. +In future work, we will consider synchronization errors in measurement while limiting DN observability +even further. Further assessment is needed for selecting the best-suited algorithm for phase identification with +varying network layouts, load profiles, and PV penetration. Finally, we will extend this work for executing +the algorithms on real measurement data and verify algorithm efficacy by intrusive phase measurements. +ACKNOWLEDGEMENT +This work is supported by the H2020 EUniversal project, grant agreement ID: 864334 (https://euniversal. +eu/). We would like to thank Clara Gouveia and Gil Silva Sampaio at INESC TEC, Porto for their comments +on problem formulation. 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Vetrov, Evaluation of stability of k-means cluster ensembles with respect to random initialization, IEEE transactions +on pattern analysis and machine intelligence 28 (11) (2006) 1798–1808. + diff --git a/v9E2T4oBgHgl3EQfggeM/content/tmp_files/load_file.txt b/v9E2T4oBgHgl3EQfggeM/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..aae0da72e7f8bec8e6857a6dc43286886d91223a --- /dev/null +++ b/v9E2T4oBgHgl3EQfggeM/content/tmp_files/load_file.txt @@ -0,0 +1,2928 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf,len=2927 +page_content='1 Consensus based phase connectivity identification for distribution network with limited observability Md Umar Hashmi1 , David Brummund2, Rickard Lundholm1, Arpan Koirala1 , and Dirk Van Hertem1 Abstract The mitigation of distribution network (DN) unbalance and the use of single-phase flexibility for congestion mitigation requires accurate phase connection information, which is often not available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' For a large DN, the naïve phase identification proposed in the majority of the prior works using a single voltage reference does not scale well for a multi-feeder DN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' We present a consensus algorithm-based phase identification mechanism which uses multiple three-phase reference points to improve the prediction of phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Due to the absence of real measurements for a real- suburban German DN, the algorithms are developed and evaluated over synthetic data using a digital twin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' To utilize strongly correlated measurements, the DN is clustered into zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' We observe those reference measurements located in the same zone as the single-phase consumer leads to accurate prediction of DN phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Four consensus algorithms are developed and compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Using numerical results, we recommend the most robust phase identification mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' In our evaluation, measurement error, and the impact of the neutral conductor are also assessed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' We assume limited DN observability and apply our findings to a German DN without smart meters, but only less than 8% of nodes have measurement boxes along with single-phase consumers with a home energy management system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Voltage time series for 1 month (hourly sampled) is utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The numerical results indicate that for 1% accuracy class measurement, the phase connectivity of 308 out of 313 single-phase consumers in a German DN can be identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Further, we also propose metrics quantifying the goodness of the phase identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The phase identification framework based on consensus algorithms for DN zones is scalable for large DN and robust towards measurement errors as the estimation is not dependent on a single measurement point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Index Terms Data-driven, distribution network, machine learning, phase identification, voltage time series Corresponding author email: mdumar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='hashmi@kuleuven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='be 1Md Umar Hashmi, Rickard Lundholm, Arpan Koirala and Dirk Van Hertem are with KU Leuven, division Electa & EnergyVille, Genk, Belgium 2David Brummund is with MITNETZ STROM, Germany arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='03938v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='SY] 10 Jan 2023 2 CONTENTS I Introduction 4 I-A Observations of this paper .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 6 II Low observability in DN: the German case 8 II-A Metering of German DN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 8 II-B Roadmap of meter rollout .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 19 IV-B Correlation based metrics .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 20 IV-B1 J1: correlation of voltage time series .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 21 IV-B2 J2: Salient features with voltage difference time series .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 21 IV-B3 J3: Salient features with voltage magnitude time series .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 22 V Consensus algorithm 23 V-A Naïve phase identification .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 24 V-C3 Maximum value of correlation as a measure .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 24 V-D Metrics for phase identification models .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 25 V-D1 Modelling accuracy .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 25 V-D2 Confidence factor for phase identification .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 25 V-D3 Standard deviation with measurement error .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 26 VI Numerical results 27 VI-A Clustering of distribution network .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 27 VI-B Performance of naïve model .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 30 VI-C Case study 1: Comparing phase identification models .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 39 VI-F Case study 4: Effect of size of neutral conductor .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 39 VII Conclusions 41 References 43 4 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' INTRODUCTION The low voltage distribution network (DN) in Europe consists predominantly of single phase (1-φ) loads, inverter interfaced PV, storage, and electric vehicle charging infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Often the phase connectivity of such resources is not accurately known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' This lack of DN observability will restrict the monitoring and control of DN imbalances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Existing DN consists of multiple measurements at the feeder level and end of the feeder, which provides utilities with some degree of observability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The goal of the paper is to develop a scalable and robust phase connectivity identification (PCI) framework that considers multiple measurement points for improving the PCI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Parameter used 64% 29% 7% Voltage Power Voltage and Power Tool used 42% 27% 19% 12% Statistical or ML Correlationship Clustering Optimization (b) (a) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 1: Classifying of phase connectivity identification literature based on parameter(s) and tool(s) used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Phase identification methodologies can be broadly classified as intrusive and non-intrusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' As the name suggests, intrusive methods require manual identification of phases and are often labor-intensive and/or hardware-based [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' On the other hand, non-intrusive methods are often data-driven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' A brief summary of non-intrusive phase identification methods is detailed in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' These data-driven methods can be classified based on the parameter used and tool used for phase identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' For the literature summarized in Table I, the classification is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 1, we observe that 64% of existing works utilize voltage magnitude time series and approximately 27% utilize correlation as a tool for PCI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' In this work, we also utilize these widely used techniques for developing a consensus-based phase identification framework using voltage time series and correlation as the tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Voltage time series-based phase identification is more 5 TABLE I: Literature review on non-intrusive phase identification Ref Measurement dependency Proposed solution Remarks Methodology Input [1] AMI voltage time series, partially incorrect phase label information Spectral clustering with a sliding window;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' does not require substation measurement 91% accuracy, Google street view analysed for phase identification Clustering / Unsupervised ML voltage [2] Voltage magnitude (denoted as |V |) Spectral clustering is utilized and MILP model is used for unbalance mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 156 user DN in China is used for validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Majority rule is applied to over predictions over new data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Clustering / Unsupervised ML voltage [3] Voltage magnitude k-means clustering with Gaussian Mixture Model algorithm for phase id 91% accurate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' with salient features accuracy is 100% Clustering / Unsupervised ML voltage [4] Voltage magnitude k-means clustering is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Use multiple references with Majority rule based estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='90% accuracy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='Clustering / Unsupervised ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='ML ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='voltage ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='[5] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='Voltage magnitude ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='k-medoids clustering is with denoised data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='Singular value decomposition is used for denois- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='ing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='Clustering / Unsupervised ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='ML ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='voltage ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='[6] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='Voltage magnitude ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='principal components are used to extract feature vectors over ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='which constrained k-means clustering is applied ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='90% accuracy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='Clustering / Unsupervised ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='ML ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='voltage ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='[7] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='Voltage magnitude ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='k-means clustering with principal component analysis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='Phase identification is applied for multiple days ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='separately and a majority rule is applied ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='Clustering / Unsupervised ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='ML ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='voltage ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='[8] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='Active power measurements,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' sub- station as reference extract distinct features from load profiles and correlate with phase load;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' limitation: high granularity data needed 93% accuracy with 10% SMs in DN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' results compared with [9] Correlation power [10] Voltage magnitude Correlation based;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' the salient features of the time series are extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Large enough data sheet leads to 100% accuracy for 75 consumer DN Correlation voltage [11] Voltage magnitude Relies on graph theory and the notion of maximum spanning tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Correlation based PCI for a four wire DN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Closer the measurement points are geographi- cally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' the stronger the correlation between the voltages Correlation voltage [12] Voltage magnitude Difference matrix is created 82% accuracy Correlation voltage [13] Voltage magnitude time series Correlation between voltage measurements of SMs with constrained k-means substation voltage is used as reference for phase identification Correlation with unsuper- vised ML voltage [14] Active power and voltage time se- ries data Correlationship with clustering is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Ensemble learning combines voltage and power-based estimation re- sults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Impact of different SM accuracy class is evalu- ated Correlation with unsuper- vised ML voltage and power [9] Active power time series at the transformer and consumers integer programming along with branch and bound search algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Access sensitivity of the ratio of measurement points and total number of consumers MILP based solution depends on the principle of conservation of energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Optimization power [15] P, Q and |V | measurements MILP with Bender’s decomposition is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Accuracy of phase id is governed by number of data points, SM class, data resolution For large EU feeder, the runtime with 5% SM error is 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='2 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Difficult to scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Optimization power (P & Q) and voltage [16] P, Q, |V | measurements Utilize state-estimation with MILP, also considers errors in layout information Data needs are smaller compared to statistical and ML-based techniques Optimization power and voltage [17] Active power time series LASSO based data driven approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Also considers SM accuracy class 97% accuracy with 60% SMs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' LASSO immune ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='to noise unlike [9] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='Statistical or ML ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='power ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='[18]– ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='[20] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='Energy measurement time series ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='data-driven approach with Principal component analysis & ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='graph theory interpretations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='Also considers noisy data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='Statistical or ML ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='power ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='[21] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='Voltage magnitude time series ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='Develops multi-dimensional calibration in phase id based on ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='voltage characteristics in LVDN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='Observe that voltage characteristics are more ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='robust under incomplete data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='Statistical or ML ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='voltage ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='[22] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='Voltage magnitude ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='Linear regression and voltage drop relationship for phase ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='identification ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='Observe close measurement are strongly cor- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='related ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='Statistical or ML ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='voltage ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='[19] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='Voltage phasor time series mea- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='sured using microPMUs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='Phase id analyses cross correlations over voltage magnitudes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='and detects the phase angle difference between reference and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='test nodes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='multi-phase connections are also considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Fine resolution of 120 samples per sec is used Statistical or ML voltage phasor [23] P, |V | time series Using statistical analysis of AMI data over a day for DN with PV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Regression model is used to model substation voltage using nodal P, V and substation power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Explore data needs, granularity and impact of PV penetration levels Statistical or ML voltage and power [24] Voltage magnitude and phase info of a small representative set Train an ML model of constrained function of voltage time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Since manual measurements are needed thus may not be scalable 5% selected representative set leads to accuracy of 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='9% Statistical or ML voltage [25] Voltage phasor time series Use sequence component for phase identification Also utilize SM data for learning the topology of the DN Statistical or ML voltage phasor [26] Voltage magnitude Supervised machine learning with theory of information loss Accuracy up to 97% Supervised ML [27] Voltage phasor time series Topology and phase identification using linearized model of three-phase unbalanced DN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 120 Hz PMU measurements are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Data col- lected for 1 second to 1 minute is used for estimation Statistical or ML voltage phasors 6 robust to limited observability in a DN, also observed in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The power-based methods rely on the law of conservation of energy and require a high degree of observability in a DN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Motivation: With the increasing generation from renewable energy sources and the growing addition of flexible loads like electric vehicles, heat pumps, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' congestions, voltage violations and phase imbalances in the grid will likely become more frequent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Therefore, suitable corrective measures must be found and implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' An important basis for mitigation measures is the detection of congestions and thus, firstly, improving the network model of the low-voltage grid, which is still largely unmonitored in many parts of the world today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Accurate network topology is assumed to be known in many works [30], [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' However, this assumption is not accurate in the case of EUniversal’s demo networks for the German DN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Observations of this paper The contributions and observations of the paper are as follows: A tailor-made solution for the German DSO, Mitnetz Strom, is proposed for phase identification which considers multiple measurements in a DN zone for improving the PCI accuracy, this is detailed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The proposed framework can also be applied for other DNs with limited DN observability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Twelve phase identification models are benchmarked over the naïve model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The naïve model considers only one 3−φ measurement reference in a DN for phase estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The proposed phase identification models build a consensus among multiple measurements for robust phase estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Metrics are proposed for evaluating phase identification models using (a1) accuracy of estimation, (a2) confidence factor, and (a3) sensitivity towards measurement errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Metrics (a2) and (a3) provide a qualitative metric for evaluating estimation accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' A detailed description is provided for synthetic data generation, which utilizes an example suburban LV DN grid model of Mitnetz Strom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' This is also crucial for DNs with limited or no historical measurement data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Four case studies are performed in the numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The performance of the naïve model is used for benchmarking and comparing the proposed phase identification algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' – Firstly, the proposed phase identification models are compared for the read German DN – Secondly, we quantify the impact of measurement proximity on phase identification metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' We observe that for a DN partitioned into zones, the estimation quality deteriorates as the measurement reference selected is farther away from the zone where the consumer is located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' – Thirdly, the impact of measurement errors on PCI is assessed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' For 1% accuracy class measurements, an estimation accuracy exceeding 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='6% is achieved for a DN with 646 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 7 – Most European DNs are four-wire system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' In our last case study, we quantify the impact of the neutral conductor model on phase estimation accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' We observe that if the neutral conductor is not modeled, then a pessimistic estimation is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Based on the knowledge of the authors, the neutral conductor impact assessment is done for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Phase mapping (true phase connectivity) Network Layout Multi-period power flow Consumer load profile selector Meta data for DN Synthetic data Inject measurement errors Assess estimation accuracy Consensus based phase identification 3-ϕ Reference measurements Estimation accuracy Confidence of est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Sensitivity to measurement errors Evaluation metrics Digital twin for synthetic data generation (Section 3) Consensus based Phase identification (Section 4 and 5) A A Zonal Clustering Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 2: Consensus-based phase identification, synthetic data generation and metrics used The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Section II presents the German DN case of low observability and the need for enhanced phase information for future DN operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Section III outlines the different modeling steps used for generating synthetic data used for phase connectivity identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Section IV presents the methodology, and Section V presents the consensus algorithms used for phase identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Section VI presents the four numerical case studies, and section VII concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 8 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' LOW OBSERVABILITY IN DN: THE GERMAN CASE The low voltage grid serves households and small consumers connecting at 230 V or 400 V [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' German authorities have decided to implement an optional smart meter (SM) roll-out as the information security standards need to be adjusted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Presently, less than 5% of residential customers are equipped with SMs [33], [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Thus, the smart meter infrastructure is not widespread at a low-voltage level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The German Energy Industry Act requires that customers with a yearly consumption of over 6000 kWh are provided with smart measurement systems (when technically possible) [35], [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The requirement also applies to generators with an installed capacity above 7 kW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' However, these requirements leave the majority of German households unaffected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The lack of sufficiently granular metering equipment at the household level is currently a barrier for implementing imbalance sensitive flexibility activation for solving DN issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Metering of German DN Mitnetz Strom is one of the largest regional distribution system operators in Eastern Germany and is responsible for supplying electricity to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='2 million electricity consumers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The grid area of Mitnetz Strom covers an area of 30,804 km2 and is characterized by rural conditions with a high share of renewables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The installed capacity of renewable energy reached an all-time high of more than 10,000 MW (more than 64,0000 plants) in 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' This development was spurred primarily by rapid growth in solar energy, as the number of photovoltaic installations increased by more than 17 percent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' In Germany, the metering and DSO roles are decoupled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The Meter Point Operator (MPO) is responsible for the installation, operation, data gathering, and maintenance of energy meters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Note, in many locations, system operators also perform as MPO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' However, the electricity consumer could opt for an independent MPO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' This is in accordance with the § 43 German MsbG (measuring point operation law).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' In principle, also a third party can be commissioned as a meter operator with the operation of the metering point on the free market [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Furthermore, there is presently no general legal obligation to share grid-relevant information obtained from smart meters with the DSO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Due to these constraints, DSO needs to request for receiving historical data thus cannot be utilized for short-term grid operation, congestion mitigation, etc (due to delays in DSO making a request and receiving the measurement data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Thus, observability in DNs are limited not only by SM penetration level but also by data-sharing policies targeting system operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Roadmap of meter rollout Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 3 shows the meter rollout phases in German.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' It was expected that by 2032, all German consumers are to be equipped with modern metering devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' (§ 29 para.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 3 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='1 MsbG) compared to other countries, 9 thus it will still take several years before the DSO has sufficient data from SMs at its disposal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Complicating this schedule are also the legal issues concerning safety and privacy of smart meter operation and usage1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The continued operation and installation of smart metering systems as defined by the Act by the MPOs is thus still possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' However, there is no longer an obligation to install them [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' This makes the need for DN parameter estimation and mechanisms to enhance observability even more crucial for ensuring the operational integrity of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 3: Rollout Plan for smart meters in Germany by 2032 [40] Mitnetz Strom has invested a total of 19 million euros in 2022 in the conversion to digital local network stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The plan is to install a total of 226 digiONS in the year 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' By 2026, up to 30 percent of the transformer stations and cable distributors in the network area are to be digitally equipped or retrofitted with the corresponding metering technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Measured secondary substations are an important component in the digital transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' They ensure better controllability and transparency of medium and low-voltage grids, which directly benefits the implementation of the energy transition and security of supply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Increasing feed-in from renewable energies, rising demand for charging power for electro-mobility, extreme weather conditions that endanger the energy supply, especially in areas with overhead lines - the reasons for the digital monitoring and control of electricity grids are many and have one goal: security of supply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 1On 20 May 2022, the Federal Office for Information Security (BSI) withdrew the general ruling of 7 February 2020 on the determination of technical feasibility pursuant § 30 MsbG (so-called market declaration on the rollout of smart metering systems) with effect for the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' In addition, the BSI issued a general ruling pursuant to §19 (6) MsbG in which it determined that the use and installation of smart metering systems available on the market do not pose any significant risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 2016 2017 2018 2019 2020 2021 2022 2023 2024 2025 20262027 2028 2029 2030 2031 2032 >100,000kWhp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='and interruptiblesystems(914aEnWG) >10,000kWhp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' PilotPhase >6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='000kwhp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' RESandCHP>100kW RESandCHP>7kW Optional>=6,000kWhp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Optionalgeneration>1-7KW(fornewinstallations) <=6,o00kWhp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='andRES+CHP>=7kW (fornewbuildings/renovationimmediately)10 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Demo network for EUniversal In EUniversal, Mitnetz Strom is testing the use of flexibility services and markets and is leading the German demonstration together with the parent company E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='ON SE [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The German demonstration tries to combine principles of the German mandatory process Redispatch 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='0 [42] and a market-based approach to mitigate grid constraints in a cascaded operation across multiple voltage levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The goal is to provide DSOs with access to flexibility from grid customers across the LV/MV level for their active system management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' To this end, the EUniversal consortium is testing various optimization algorithms with the aim of minimizing activation costs while ensuring the secure operation of the grid and is developing concepts for grid state estimation of smart grids, of which the first interim results and experiences will be presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' In the German demo of EUniversal, Mitnetz Strom and its partners are investigating the use of flexibility markets in low voltage grids for congestion management and voltage maintenance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' An attempt is being made to develop an iterative procedure that will prevent new congestion from occurring when flexibility is activated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 1) Network features and meter placement: The network considered for numerical evaluation is a typical low-voltage network in Mitnetz’s network area in a small town in Eastern Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' There are already some flexible plants, but the penetration with them is not yet significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Due to application cases, certain locations in the LV grid are particularly interesting when it comes to equipping with measurement technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' In particular, the end of the feeder is often an important indicator for the evaluation of potential voltage band violations, while the current at the beginning of the feeder is important for the thermal constraints in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Unfortunately, these points are often not available in practice due to ownership issues and the partially pronounced building development in the localities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Therefore, cable distribution cabinets were selected and equipped with measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' For EUniversal devices are a bundle of voltage and current sensors (Rogowksi coils), gateway, and power supply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Need for phase information LV DN topology identification is essential for efficient network operation, monitoring, and control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' This also assists in planning the phases for new resources connected to the DN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Next, there is a real issue Mitnetz Strom faced due to the connection of 8 out of 9 electric vehicle chargers was made using a single-phase (1 − φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' This led to thermal limit violations in that particular phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' With the topology identification, the phase connections were optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' According to DIN ISO 50160, [43], limits of voltage deviations are defined up to ±10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' LV networks are asymmetrically biased by 1-φ loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Unbalanced new loads, such as electric fans, heat pumps, and unbalanced charging could amplify this effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Unbalanced loaded DNs lose a substantial amount of their power transmission capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' In a research project for an automatic phase 11 switch in EV charging showed the effects in charging for 1 − φ or two-phase connected EVs or hybrids in the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' These findings presented here are also applicable to 1 − φ heaters, inverter interfaced PV, storage, and other unbalanced loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 4: Line loading for unplanned EV charger placement Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 5: Line loading after phase redistribution In the further evaluation of this field test, it is shown that with the average existing charging capacity, the penetration rate with EVs until a line limit is violated increased from 16% to 47% with symmetrical CurrentMeasurements EVs withoutphase selection 40,00 35,00 30,00 25,00 /[A] 20,00 15,00 10,00 5,00 0,00 Time of day [h] L1 [A] , 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='2[A1 L3[ACurrent Measurements EVs with phase selection 15,00 10,00 5,00 0,00 Time of day [h] L1 [A] —L2[A] —L3 [A]12 utilization of DN with approximately balanced phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' In the case of flexibility markets, this means the possibility of using flexibility decreases for ensuring grid integrity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' This underlines the importance of having accurate knowledge of the phase connections, as flexibility activation should not further increase imbalance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' In theory, flexibility activation could also limit DN imbalance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Mitnetz Strom and most DSO’s follow a passive way of identifying the phase connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The accurate topology identification is performed manually in case of a follow-up on customer complaints on power quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' This motivates us to propose a scalable and robust phase identification mechanism using historical measurement data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 13 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' SYNTHETIC DATA GENERATION In this section, we detail the steps we took for generating synthetic data for the DSO grid layout provided in DigSilent format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' A digital twin is used for the process of phase and load placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Further, the neutral conductor modeling, noise injection, and zonal clustering model used in this work are elaborated in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' DGS parser for network JSON A parser has been created to convert the DGS (DIgSilent) [44] file format into a JSON file that is readable to the PowerModels script.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The parser is derived from the GridCal python package [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The main difference between the two formats is that the DGS data format contains different classes with different information in a hierarchical structure, whereas the JSON file just requires information on the buses, branches, and devices in the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 1) Bus information: The DGS format relies on cubicle information, which can be seen as a connection point for the different elements in the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The cubicle information are the unique IDs given to each connection point, which are converted to simple grid ids starting from 1 and ending in the number of buses in the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The original ID is saved to allow for cross-checking data and for linking devices to the correct bus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 2) Branch information: The DGS branch format relies on cable type information, but the JSON file requires the values directly in per-unit (pu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Therefore, the r and x parameters (amongst other) needs to be converted from their ohmic values to pu of the total cable length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' This requires a multiplication of the base impedance and the total length of the cable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 3) Device information: The DGS file format can contain detailed information on different loads and static/synchronous generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The JSON file format only has information on the devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Therefore, the parser extracts the relevant P and Q data from each device in the DGS file and compiles it as a separate entity in the device file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Additional information included is bus ID, PV size, and connected phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 4) Switches: A crucial element of the parser is removing the switches that connect branches to substations and cabins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The switches are removed as they add numerical complexities when calculating the admittance matrix in PowerModels [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Setting the r and x values to zero can lead to infinity values during the admittance calculation, but setting it to a very low value leads to inefficiencies when running the power flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Removal of the switches not only removes numerical complexities but, since switches make up 7% of the branches in the system, by removing the extra nodes the computation time of the simulation significantly increases (in the order of a 10%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 14 Once the switches have been removed, the IDs should be renumbered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' This also serves the purpose of removing empty nodes in the grid, which has a positive effect on the computation time of the simulation as the admittance matrix only contains non-zero components, which helps reduce its size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' This also tidies up the network data, making it easier to view from a simulation perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Mitnetz Strom DN and metadata Along with the grid data, metadata on the loads in the grids are also provided in a separate file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' This metadata contains details associated with different consumer devices in the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The information includes a node number to connect it to the grid data, load type (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' household, PV, CHP, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' ), the annual energy consumption, single or three-phase connection, and the available active and reactive power output (if applicable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The information is compiled and appended to the device’s JSON file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 6 shows the spread of the annual cumulative energy consumption of loads connected to the test DN used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Note that the exact phase connection of single-phase loads is not known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The goal of this paper is to present a framework for identifying the phase connection of such single-phase consumers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 6: Metadata for annual kWh consumption of 331 consumers in the test LV DN considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Note that 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='5% of consumers have an annual consumption of lower than 6000 kWh for the test DN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Randomized phase mapping DSOs actively try to balance the phases so that the load distribution is fairly balanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Madeira island case study in [47] and DSO questionnaire in [48] details the phase assignment procedure of a DSO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Number of consumers in LV feeo 10 8 6 4 2 0 0 1000 2000 3000 4000 5000 6000 7000 8000 AnnualConsumptioninkWh9000 1000014 1215 In order to generate synthetic data, randomized load mapping is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' For a single phase load with different levels of annual kWh consumption, a phase is randomly selected from phases A, B, and C with equal likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 7: Phase load distribution of three-phase distribution network for 100000 phase mapping scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The randomized phase mapping is evaluated based on the sum of the absolute error in phase load (AEPL), which is given as AEPL = 1 3 1 D D � i=1 � φ∈{A,B,C} |LD φ − ¯LD|, where D denotes the number of Monte Carlo phase mapping scenarios, and ¯LD denotes the mean load in all the phases and is given as ¯LD = � φ∈{A,B,C} LD φ /3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Using 100000 Monte Carlo simulations for phase mapping, we observe that randomized phase mapping performs fairly well, with a maximum and mean per phase load deviation of 30% and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='5% with respect to the mean load met by the three phases in the worst case (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 8 shows the distribution of the phase load errors while performing randomized phase mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Thus, in this work, we utilize randomized phase mapping for data generation for evaluating our proposed probabilistic phase identification mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Histogram for phase loads 5000 phase A 2500 5000 phase B 2500 of instances 5000 phase C Number 2500 150000 200000 250000 300000 350000 400000 45000016 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 8: Sum of the absolute value of the difference of phase load and mean load of all the phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Neutral modelling for four-wire DN European LV DN is usually different from the North-American one with a larger size distribution trans- former with multiple low voltage feeders supplying a large number of consumers per transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The German low voltage feeders normally follow a four-wire three-phase configuration with single-grounded neutral [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Such systems are usually reduced to three-wire equivalent using Kron’s reduction [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' In Kron’s reduction, it is assumed that the neutral is grounded multiple times and for a perfectly grounded neutral2, the neutral voltage equals zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' However, for the DN considered in this paper, this assumption is not true as the neutral is isolated from consumer grounding and is grounded only at the sub-station (See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The inclusion of sparsely grounded neutral in modeling can be done by taking an exact four-wire model with four-wire power flow solvers or reducing it to a three-wire equivalent and solving by using the three-wire solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' In [51] a new reduction method is proposed for sparsely grounded European LV feeders so that the impact of neutral is represented as equivalent as a four-wire model without the necessity of carrying around extra variables and measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' In such reduction, the 4×4 impedance matrix is transformed to 3×3 matrix is given in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 2A perfectly grounded neutral refers to grounding resistance of zero ohms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Typically the grounding resistance is ≈ 5 ohms which leads to a small voltage drop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' In this work, we assume perfectly grounded neutral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 6000 5000 4000 umber 3000 N 2000 1000 0 10 152025 35 40 Absolute error in %17 b zl,aa zl,bb Ilij,a Ilij,b Ilij,c Ilij,n Ui,a zl,cc zl,nn zl,ab zl,bc zl,cn Ui,b Ui,c Ui,n Uj,a Uj,b Uj,c Uk,c Uj,n Uk,n Uk,b Uk,a isolated neutral c a k j i n g substation grounding Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 9: Isolated neutral model of distribution network Impedance matrix = � ���� zs l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='aa − zs l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='na − zs l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='an + zs l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='nn zs l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='ab − zs l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='nb − zs l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='an + zs l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='nn zs l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='ac − zs l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='nc − zs l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='an + zs l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='nn zs l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='ba − zs l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='na − zs l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='bn + zs l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='nn zs l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='bb − zs l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='nb − zs l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='bn + zs l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='nn zs l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='bc − zs l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='nc − zs l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='bn + zs l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='nn zs l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='ca − zs l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='na − zs l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='cn + zs l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='nn zs l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='cb − zs l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='nb − zs l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='cn + zs l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='nn zs l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='cc − zs l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='nc − zs l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='cn + zs l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='nn � ���� (1) In (1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' zs l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='aa is self-impedance of the phase a of the branch l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' zs l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='nn is self-impedance of the neutral of the branch l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Similarly, zs l,ab is the mutual-impedance between phase a and b of branch l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' This transformation is exact and eliminates the error introduced by Kron’s reduction in three-phase DN modeling for sparsely grounded system [51], [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Furthermore, a minor boost in computation time is achieved compared to the exact four-wire model as the necessity of carrying extra variables for neutral voltage is removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' This reduction is more relevant as the measured voltages in German demo-grid are also phase-to-neutral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Interested readers are guided to [51] for details about the transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Metering noise injection model Prior works [14], [15], [17], [20] consider smart meter measurement error based on the accuracy class of metering infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Frequently, the measurement error is considered using Gaussian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Further, the measurement accuracy is considered to hold true for three sigma of the times, which corresponds to 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='7% of total instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The standard deviation of the Gaussian noise is related to the tolerance τ of the measuring device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The noisy measurement is given as ˆZ = Z × Norm(1, τ/3), (2) 18 where Z denotes the true measurement, Norm(µ, σ) denotes a sample of a normal distribution with mean µ and standard deviation σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' In order to evaluate the impact of measurement noise, 1000 Monte Carlo simulations are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Zonal clustering of Mitnetz Strom DN Identifying the zones of an LV DN can be helpful to the DSO in planning the flexibility needs of a network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Due to the large numbers of DN feeders, a standardized approach to divide zones based on electrical and/or geographical distances is deemed essential [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' In this section, the summary of the clustering framework to identify the best-suited LV DN zonal partition using electrical distance as a measure is presented, which is explained in detail in [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' This zonal partition method uses an incidence matrix-based measure, which can be obtained with the help of spectral decomposition of the admittance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The adequate number of zones is obtained based on the maximization of silhouette score while considering the desired number of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The zonal partition divides nodes N into c ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=', C} clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The spectral clustering proposed in [54], [55] for the creation of zones or network reduction is used for zonal partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' A double stochastic matrix is formed, which is a special type of Markov matrix where not only each row but also each column add to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' For this transformed matrix, all eigenvalues are real and smaller than or equal to 1, with one eigenvalue exactly equal to 1 [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' For identifying C partitions in a graph, the C highest eigenvalues and corresponding orthonormal eigenvectors are identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The eigenvector matrix of the order N × C is used for DN partitioning, in effect reduces the dimensionality of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' k-means clustering is used to partition the spectral data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The goodness of a cluster is measured using the mean silhouette index of the network cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The silhouette coefficient of a node is a confidence indicator of its association in a group [54], [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Power flows for synthetic data Power flow equations translate the load information of consumers to the nodal voltage and nodal currents when the network topology and impedances are known using the first equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Three-phase unbalanced power flow equations were used to create the pseudo-measurement point based on the given load data and network topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Open-source power flow solver of PowerModelsDistribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='jl was used for creating such pseudo measurement points [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 19 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' PHASE IDENTIFICATION METHODOLOGY Using the synthetic data generated in the previous section, we develop correlation based voltage matrices used for consensus algorithm base PCI algorithms in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Notation A three-phase distribution network (DN) consists of phases denoted as φ ∈ {A, B, C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The DN consists of branches, nodes, loads, and generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' A DN is represented as a directed graph by < N, E >, where N denotes the set of nodes in all the phases and E denotes the set of branches connecting a pair of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' For each node i in the phase φ at any time t, have two variables: (i) voltage magnitude denoted as Vφ,i,t and phase angle denoted as θφ,i,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The voltage phasor at a node and phase is governed by power injections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The branch denoted as (i, j) ∈ E is characterized by line admittance denoted as Yφ,ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The line admittance governs power flow and line losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Nd ⊂ N denotes the nodes with loads connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' For these nodes, the active and reactive power is given as P d φ,i,t and Qd φ,i,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Ng ⊂ N denotes the nodes with generators connected, have active and reactive power generation denoted as P g φ,i,t and Qg φ,i,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The time t is sampled hourly, and its range is given as t ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='., T}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Distribution network zone for cluster C # of 1-phase loads Lc # of measurements Mc Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 10: A distribution network cluster with measurement points and single-phase loads with unknown phase connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The DN is clustered into c ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=', C} clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' A cluster c consists of Mc number of three-phase reference measurement points, Lc number of single-phase consumers and Nc are the number of nodes present in that cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' We assume that all the reference measurements are aligned and there are no synchronization delays considered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' A stylized representation of a zone with 1 − φ consumers and 3 − φ 000020 reference measurement points are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' DN clustering is performed such that Ni ⊂ N and Ni ∩ Nj = ∅, ∀i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The set of nodes where Mc measurements are placed is denoted as iM c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The set of nodes where Lc single phase consumers are located is denoted as iL c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' For a vector parameter K, ¯K is the mean value, and |K| denotes its absolute value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' For a vector K, C(K) denotes its cardinality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 1(condition) returns 1 if the condition is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Correlation based metrics In this work, we utilized voltage magnitude time series as a parameter for estimating phases of the single- phase consumers in each of the clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Each of the measurement points is utilized for estimating the phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Thus, a unique reference matrix is created using the phase voltage magnitudes at the measurement node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' For cluster c, measurement iM c (j) where j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='., Mc}, the reference voltage matrix is given as V ref iM c (j) = � ���������� VA,iM c (j),1 VB,iM c (j),1 VC,iM c (j),1 VA,iM c (j),2 VB,iM c (j),2 VC,iM c (j),2 : : : : : : VA,iM c (j),T VB,iM c (j),T VC,iM c (j),T � ���������� (3) The dimension of V ref iM c (j) is T × 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Note time t is hourly, therefore, C(t ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='., T}) = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The single phase consumer nodal voltage time series in cluster c forms a column of the matrix denoted as V L c , and given as V L c = � ���������� Vφ,iL c (1),1 Vφ,iL c (2),1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='. Vφ,iL c (Nc),1 Vφ,iL c (1),2 Vφ,iL c (2),2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='. Vφ,iL c (Nc),2 : : .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='. : : : .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='. : Vφ,iL c (1),T Vφ,iL c (2),T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='. Vφ,iL c (Nc),T � ���������� (4) The dimension of V L c is T ×Nc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The connection phase of 1−φ consumers are assumed to be not known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' For the calculation of correlation between reference 3−φ voltage and 1−φ consumer voltage time series, Pearson correlation3 is utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Three models are presented for generating metrics for phase identification using a consensus algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' These three metrics are described next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 3The Pearson correlation between two vectors X and Y is given as ρ(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Y ) = �(X − ¯ X)(Y − ¯Y ) ��(X − ¯ X)2 �(Y − ¯Y )2 (5) 21 1) J1: correlation of voltage time series: The correlation matrix for measurement iM c (j) is denoted as ρc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='iM c (j) J1 = � ������� ρ(ViL c (1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' VA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='iM c (j)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' ρ(ViL c (1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' VB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='iM c (j)) ρ(ViL c (1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' VC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='iM c (j)) : : : : : : ρ(ViL c (Nc),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' VA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='iM c (j)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' ρ(ViL c (Nc),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' VB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='iM c (j)) ρ(ViL c (Nc),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' VC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='iM c (j)) � ������� (6) Note that the voltage time series is denoted by dropping t in the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' For instance, ViL c (w) denotes the voltage time series for t ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='., T} for single phase consumer located at node id iL c (w) : w ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='., Nc}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' For single-phase consumers with unknown phases, the phase notation is also dropped to avoid confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 2) J2: Salient features with voltage difference time series: Salient features in voltage time series could help in improving phase identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The use of salient features has been explored in [3], [8], [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' In this work, we utilize the difference matrix and a zonal voltage fluctuation threshold for identifying the salient features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The difference matrix for the reference voltage matrix for cluster c and measurement iM c (j) is given as ∆V ref iM c (j) = � Vφ,iM c (j),t+1 − Vφ,iM c (j),t ∀ t, ∀ φ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' (7) The dimension of ∆V ref iM c (j) is (T − 1) × 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' βc denotes the voltage change threshold for cluster c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The salient features are extracted using the ∆V ref iM c (j) matrix as isalient c,iM c (j) = arg 1 � |∆V ref iM c (j)(t)| > βc ∀t ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='., T − 1} � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' (8) The voltage difference matrix for the connected load matrix is denoted as ∆V L c = diff(V L c ), (9) where diff operator finds the difference of adjacent rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The dimension of ∆V L c matrix is (T − 1) × Nc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The new reference and load matrix extracts the rows with salient features in the reference matrix and are given as ∆V ref, J2 iM c (j) = � ∆V ref iM c (j)(isalient c,iM c (j), :) � , (10) ∆V L, J2 c = � ∆V L c (isalient c,iM c (j), :) � , (11) The correlation matrix with salient features using the voltage difference as a metric is given as ρc,iM c (j) J2 = � ������� ρ(∆V L, J2 c (1), ∆V ref, J2 A,iM c (j)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='. ρ(∆V L, J2 c (1), ∆V ref, J2 C,iM c (j)) : : : : : : ρ(∆V L, J2 c (Nc), ∆V ref, J2 A,iM c (j)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='. ρ(∆V L, J2 c (Nc), ∆V ref, J2 C,iM c (j)) � ������� (12) 22 3) J3: Salient features with voltage magnitude time series: Previously, we used the voltage difference as a metric for identifying the salient features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The salient features when projected onto the voltage magnitude would require the previous time stamp to capture the voltage change trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' This trajectory captured will improve the correlation-based metric we are utilizing for phase identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The new salient feature matrix is given as isal, plus c,iM c (j) = unique( � isalient c,iM c (j), isalient c,iM c (j) + 1 � ), (13) with unique operator finding unique time stamps, considering there could be repetitions that will be eliminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The new reference and load voltage matrix are given as V ref, J3 iM c (j) = � V ref iM c (j)(isal, plus c,iM c (j), :) � , (14) V L, J3 c = � ∆V L c (isal, plus c,iM c (j), :) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' (15) The correlation matrix for model J3 is given as ρc,iM c (j) J3 = � ������ ρ(V L, J3 c (1), V ref, J3 A,iM c (j)), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='. ρ(ViL c (1), V ref, J3 C,iM c (j)) : : : : : : ρ(V L, J3 c (Nc), V ref, J3 A,iM c (j)), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='. ρ(ViL c (Nc), V ref, J3 C,iM c (j)) � ������ (16) The dimension of ρc,iM c (j) J1 , ρc,iM c (j) J2 and ρc,iM c (j) J3 equals Nc × 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 23 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' CONSENSUS ALGORITHM In Section IV, we calculated correlation matrices using the voltage time series for measurement reference located at node iM c (j), j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='., Mc}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Thus, with multiple measurement points in a cluster, independent phase identifications can be performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' These estimations can be taken into consideration using consensus algorithms to be presented in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' A consensus algorithm is a strategy that a group of agents use to agree with each other on what’s true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' In a multi-sensor PCI scenario, there is just one true phase placement (ground truth), which is given as P true c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Each of the measurements used as a reference for models J1, J2, and J3 as metrics are used for estimating the true phases of single-phase consumers in cluster c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The advantage of the consensus algorithm is that no one measurement point limits the PCI accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' One of the most widely used consensus algorithms is in blockchain technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Consensus algorithms are widely used in state estimation [59]–[61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' In this work, we use consensus for phase identification in a distribution network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Naïve phase identification The naïve phase identification, denoted as S0, uses only one of the 3 − φ reference measurement points, typically the substation measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' P est,Jx,S0 c = arg max ρc,isel Jx , (17) where isel denotes the node id for the reference measurement point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Most literature on phase identification uses this naïve model with substation time series measurement as the reference, see Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Majority rule The majority rule, denoted as S1, is one of the most commonly used consensus algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' It identifies the most agreed-upon estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Earlier works such as [62], [63] detail the applications of the majority rule in building consensus among agents (sensors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' We note that for a cluster c, measurement point located at node iM c (j), and metric Jx, we can calculate ρc,iM c (j) Jx .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' This correlation matrix is used for calculating the estimated phases, as P est,Jx c,iM c (j) = arg max ρc,iM c (j) Jx , P est,Jx,S1 c = fS1 � P est,Jx c,iM c (j) ∀j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='., Mc} � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' (18) The function fS1 calculates the majority among the estimated phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Consider there are 7 measurement points in a cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' For a node, consider 3 of the estimations that predict phase B, and 2 for phases A and C respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' In this case, the majority rule predicts the phase to be estimated as phase B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 24 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Weighted measure Previously, for a majority rule-based consensus algorithm, we assumed all agents to be of equal importance (or weights).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' However, if we use the physical laws governing the system, we can calculate the weights for different agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' In phase identification, earlier works point out that measurement points in geographical proximity will have a greater voltage correlation among similar phases [11], [22], see Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' In this work, we use the correlation value as a weighing factor for calculating the estimated phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' A correlation value of 1 implies 100% correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 1) Correlation as a measure: ¯ρc Jx = Mc � k=1 3 � φ=1 ρc,k Jx, (19) where ¯ρc Jx is Nc × 1 vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The normalized correlation coefficients are given as GJx,S2 c = �Mc k=1 ρc,k Jx ¯ρc Jx , (20) where GJx,S2 c is Nc × 3 matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The estimated phases are given as P est,Jx,S2 c = arg max GJx,S2 c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' (21) 2) Absolute value of correlation as a measure: ¯ρc Jx,abs = Mc � k=1 3 � φ=1 |ρc,k Jx|, (22) where ¯ρc Jx,abs is Nc × 1 vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The normalized correlation coefficients are given as GJx,S3 c = �Mc k=1 |ρc,k Jx| ¯ρc Jx,abs , (23) where GJx,S3 c is Nc × 3 matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The estimated phases are given as P est,Jx,S3 c = arg max GJx,S3 c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' (24) 3) Maximum value of correlation as a measure: ¯ρc Jx,max = max k=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='.,Mc max φ=1,2,3 |ρc,k Jx|, (25) where ¯ρc Jx,max is Nc × 1 vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The normalized correlation coefficients are given as GJx,S4 c = maxk=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='.,Mc |ρc,k Jx| ¯ρc Jx,max , (26) where GJx,S4 c is Nc × 3 matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The estimated phases are given as P est,Jx,S4 c = arg max GJx,S4 c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' (27) Note that (20), (23) and (26) denotes element wise division of vector of length Nc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 25 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Metrics for phase identification models 1) Modelling accuracy: Consider, the true phase information in a cluster is given as P true c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The estimation accuracy is denoted as Estimation accuracy = number of correct phase estimation total number of single phase consumers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The phase estimation accuracy for metric Jx ∈ {J1, J2, J3}, cluster c and measurement iM c (j) is given as AJx c,iM c (j) = 100 × � 1 − � 1 � P est,Jx c,iM c (j) − P true c ̸= 0 � Nc � , (28) where P est,Jx c,iM c (j) denote the phase estimation vector for cluster input metric Jx, cluster c and measurement iM c (j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The estimation accuracy for consensus algorithm Sy is given as AJx,Sy c = 100 × � 1 − �Nc n=1 1 � P est,Jx,Sy c − P true c ̸= 0 � Nc � , (29) Since there are three base metrics denoted as Jx ∈ {J1, J2, J3} and five consensus models (including the naïve model) denoted as Sy ∈ {S0, S1, S2, S3, S4}, therefore, we evaluate in total 13 phase identification models (the naïve model, S0, is performed for J1 only).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' In numerical results, we will compare the benefits and shortcomings of these models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Estimation accuracy averaged over Q Monte Carlo simulations are denoted as ¯AJx,Sy c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 2) Confidence factor for phase identification: Note that models S1, S3, and S4 provide coefficients that add up to 1 (node-wise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Thus, GJx,S3 c and GJx,S4 c can be used to indicate probabilities of phase estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' For S1, the estimation probabilities can be calculated by dividing P est,Jx,S1 c with the number of measurements in a cluster, Mc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' We define the confidence factor as the minimum distance between the factor associated with the correct phase and the maximum of the two incorrect phases, over all nodes in the cluster c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' For the model, S2 we normalized the confidence factor with the range of variation of GJx,S2 c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The proposed confidence factor will provide us with additional information about the robustness of our phase estimation output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Note, the proposed confidence factor can lie in the range ∈ [−1, 1] for S1, S3, and S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' A confidence factor close to 1 implies very high confidence in our phase estimation output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The confidence factor for measurement k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='., Mc} and cluster c is given as F Jx,Sy c,k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The confidence factor for a cluster c for all Monte Carlo scenarios is given as ¯F Jx,Sy c = 1 Q × Mc Q � q=1 Mc � k=1 F Jx,Sy c,k , (30) where w ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='., W} denotes the Monte Carlo scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 26 3) Standard deviation with measurement error: Q Monte Carlo simulations are considered for minimizing the measurement error biases on phase estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' For each of Monte Carlo iteration q ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=', Q}, calculate the standard deviation of the measure used for calculating P est,Jx,Sy c , denoted as DJx,Sy c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 27 VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' NUMERICAL RESULTS The numerical case study considers a German DN with 646 nodes and 331 loads connected to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Out of 331 loads, 313 loads (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='6% of total consumers) are single-phase loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The phase connections are widely unknown to Mitnetz Strom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The objective of the case study is to assess the phase identification algorithm proposed in this work, benchmarked over naïve phase identification model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The selected DN is part of the demo network selected for evaluation in the EUniversal project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Mitnetz Strom placed 53 3 − φ measurement devices in the DN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' These measurement points will be considered as references used for PCI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The flexibility participants will be provided with a Home Energy Management System (HEMS) which will provide measurements of load and voltages at the point of common coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' In the case studies, we assume the time series of voltage measurements of all single-phase users are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' In the real world, only measurements of consumers with HEMS will be available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' There are four case studies performed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The first case study compares the 12 phase identifica- tion models on different phase mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The second case study quantifies the impact of reference location in a DN on the phase identification metrics proposed in the work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The third case study assesses the impact of measurement error at the reference and/or at the consumer location on phase identification metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The last case study compares the phase identification metrics for DN with and without the neutral conductor model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' As most European DNs are four-wire, it is crucial to quantify the impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Prior to case studies, we detail the test DN clustering results and the performance of the naïve phase identification algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The benefits of the proposed phase identification algorithms are compared to the naïve model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Clustering of distribution network Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 11 shows the location of single-phase consumers and measurement points in the DN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' We apply the clustering algorithm for identifying the clusters in the DN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Since the number of clusters to be formed is not clear, we utilize the silhouette score plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 12 for fixing the number of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Observe that the silhouette score is maximized for 3 clusters with a value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='872.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' However, we select the number of clusters to be 7 as maximization of silhouette score is not the only goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' We also need to quantify how many clusters will make the problem tractable by explaining the DN sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Note there is a sharp decline in silhouette score if the number of clusters is increased beyond 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Previously, we defined βc as the voltage change threshold for cluster c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Note βc will vary with different clusters, as voltage fluctuation in different zones will vary drastically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 13 shows the variation of nodal 28 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 11: Consumers (blue) and measurement points (orange) in the DN voltages in seven clusters of the Mitnetz Strom example LV distribution network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' It can be observed that the voltage variation in cluster 2 is very small, ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='995 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' This narrowband is due to cluster 2 including the substation and the slack bus, where the voltage is regulated at 1 per unit level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 14 shows the DN clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' We can also comment that the clusters identified are indeed stable, which is validated by 100 Monte Carlo (MC) simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The stability of clusters, impacted due to initializations are discussed in [64], [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Clustering based on k-means is sensitive to randomized initializations of centroids, and if not stable would provide different clusters in different iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Observe that the numbering of clusters in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 14 is based on randomized initialization of the centroids, and indeed would vary in a different iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' II details the phase mapping scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' In the first case study, we assess the impact of different phase mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' C1 denotes balanced, C2 denotes moderately balanced, and C3 denotes unbalanced phase mapping based on the annual cumulative load on each phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' For the rest of the paper, if phase mapping is not explicitly mentioned, then C2 phase mapping is used, see Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 29 0 10 20 30 40 50 60 70 80 90 100 Number of clusters 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='9 Silhouette Score 2 3 4 5 6 7 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='9 X 7 Y 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='832668 X 7 Y 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='832668 X 3 Y 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='872196 (b) (a) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 12: Choosing the number of clusters of DN based on the silhouette coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' In (a) the variation of silhouette coefficient is plotted with increasing number of cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' In (b) we zoom into the plot (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Note the silhouette coefficient is maximum for 3 clusters, however, the best-suited number of clusters selected is 7 [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 13: Distribution of voltage in phases in different clusters for Mitnetz Strom DN for cluster 1 to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 60 40 40 20 50 20 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='85 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='95 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='95 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='95 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='05 Cluster5 Cluster 6 100 100 Cluster7 100 (e) 80 80 (f) (g) 60 60 60 40 40 40 20 20 20 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='95 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='95 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='95 1 1 Voltage in per unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='95 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='05 Phase A Phase B Phase C 05Cluster 1 80 Cluster 2 Cluster 3 80 100 150 (b) 60 60 80 (c) (a) tan 100Cluster4 (d)30 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 14: Clusters of DN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' TABLE II: Phase mapping scenarios Annual cumulative load (MWh) ID Cases Phase A Phase B Phase C C1 Highly balanced 274.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='6 282.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='3 275.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='7 C2 Fairly balanced 288.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='6 292.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='4 251.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='6 C3 Highly unbalanced 240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='4 257.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='7 334.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='4 The details of the number of consumers, measurement points, and voltage variation for phase mapping C2 (see Table II) are provided in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Performance of naïve model As the majority of prior works utilize a single voltage reference for PCI, we would show the performance of this model, referred to as the naïve model, prior to evaluation of the proposed phase identification models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' We utilize 4 different measurement points close to the transformer for evaluating the naïve phase identification model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The measurement points are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' It is also indicated that nodes 1, 72, 74, and 511 are 5 631 TABLE III: Network attributes for C2 phase mapping Cluster ID Nc Mc βc Max voltage deviation 1 20 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='056 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='136 2 73 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='018 3 21 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='064 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='164 4 56 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='051 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='172 5 46 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='0493 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='112 6 38 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='091 7 59 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='031 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='058 connected to feeders going towards clusters 6, 4, 5, and 7 respectively, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' A zoomed-in plot of measurement points and the location of the nodes of measurement is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The results of PCI using the naïve model is detailed in Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' It lists the cluster-wise PCI accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Observe that naïve model correctly estimates the phase connectivity for the cluster with which it is directly connected, however, the estimation accuracy for other clusters can be as low as 0%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' This is also shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 16, the measurement points are indicated by a black square, the correctly estimated consumer phase is indicated by a green circle and red circles show as the incorrect identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' We can observe that the selection of a reference highly affects the phase connectivity identification accuracy in a multi-feeder distribution network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' TABLE IV: PCI accuracy with naïve model Node 1 Node 72 Node 74 Node 511 Cluster 1 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='33 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='94 100 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='11 Cluster 2 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='44 50 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='76 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='03 Cluster 3 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='35 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='37 100 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='68 Cluster 4 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='5 100 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='14 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='14 Cluster 5 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='77 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='26 100 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='36 Cluster 6 100 0 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='64 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='63 Cluster 7 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='06 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='56 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='57 100 overall accuracy 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='92 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='09 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='59 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='72 It is clear from the numerical evaluations that Naïve model for phase connectivity identification is very sensitive towards the selection of the reference voltage node which is utilized for phase identification of a single phase consumer, and Naïve model does not consider multiple reference measurements in a DN, 32 Node 1: towards cluster 6 Node 74: towards cluster 5 Node 72: towards cluster 4 Node 511: towards cluster 7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 15: Measurement locations for naïve model evaluation The mean PCI accuracy with naïve model in a multi-feeder DN considered in this work is below 56%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The numerical case studies are presented subsequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Case study 1: Comparing phase identification models Previously, we proposed three metrics {J1, J2, J3} and four consensus algorithms {S1, S2, S3, S4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' In this case study, we compare 12 phase connectivity identification algorithms are proposed and applied to the German DN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' This case study provides recommendations for the best-suited metric and consensus algorithm to be used for phase identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Note that these recommendations could vary for other DNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' All results consider a measurement error of 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' In order to eliminate the impact of measurement error on PCI, we perform 1000 MC simulations with different measurement errors calculated using (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The majority rule consensus algorithm (S1) outperforms all other models proposed for all clusters except for cluster 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' For all other clusters, the majority rule provides 100% accurate rules with a confidence factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' As detailed earlier, cluster 2 is the part of the DN around the substation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The voltage deviation in this cluster is very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Table V shows the performance of the majority rule consensus algorithm for cluster 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Observe that all three metrics of phase identification are very poor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Thus, we drop the majority rule consensus algorithm in subsequent evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 249 49 253 249 254 246 252 514 806 207 6 51 4 74 511 1233 (a) Node 1 as reference (b) Node 72 as reference (c) Node 74 as reference (d) Node 511 as reference Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 16: Phase connectivity identification accuracy with naïve model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The correct phase estimations are indicated with green circles, and incorrect with red circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The location of the reference is indicated with a black square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The substation is marked with a yellow square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 17, the confidence factor deteriorates from model C1 which is balanced phase mapping to C3 which is highly unbalanced phase mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' No noticeable PCI accuracy change is observed for consensus algorithms S2, S3, and S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The mean confidence factor for cases C1, C2, and C3 are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='201, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='174, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='133 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Thus, correlation-based phase connectivity identification tends to be more accurate for more balanced phase mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 18 shows the mean phase estimation accuracy for metrics {J1, J2, J3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Observe that mean estimation 34 TABLE V: Majority rule (S1) model for cluster 2 Case Metric (Jx) Accuracy Confidence factor Sensitivity ( ¯AJx,Sy c ) in % ( ¯F Jx,Sy c ) (D Jx,Sy c ) C1 J1 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='99 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='3714 J2 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='99 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='3714 J3 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='99 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='4641 C2 J1 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='76 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='3878 J2 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='75 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='3879 J3 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='14 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='4612 C3 J1 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='91 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='3791 J2 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='83 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='3792 J3 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='41 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='4628 Balanced (C1) Fairly balanced (C2) Unbalanced (C3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='6 Confidence factor Mean confidence factor Mean=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='174 Mean=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='133 Mean=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='201 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 17: Confidence factor for three-phase mappings for metrics {J1, J2, J3} and consensus algorithms {S2, S3, S4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' accuracy is deteriorating for metrics J1 to J3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Further, the accuracy is higher for balanced phase mapping compared to unbalanced phase mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Thus, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 17 and 18 are used to evaluate the impact of phase mappings {C1, C2, C3} on phase identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Further, we observe that metric J1 outperforms others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' J1 is more robust as a measure of phase identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' For each cluster {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=',7} and phase mappings {C1, C2, C3} we rank the phase identification methods 35 0 12 24 36 48 60 72 84 40 60 80 100 % Estimation accuracy for metric J1 0 12 24 36 48 60 72 84 40 60 80 100 % Estimation accuracy for metric J2 0 12 24 36 48 60 72 84 40 60 80 100 % Estimation accuracy for metric J3 Mean = 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='5 % C1: Balanced C3: Unbalanced C2: Fairly unbalanced Mean = 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='01 % Mean = 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='8 % Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 18: Accuracy for three-phase mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' based on the three metrics (a) estimation accuracy in % denoted as ¯AJx,Sy c , (b) confidence factor denoted as ¯F Jx,Sy c , and (c) sensitivity towards measurement errors denoted as DJx,Sy c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Out of these 21 models, 13 times consensus algorithm S3 and metric J1 denoted as S3-J14, outperformed all the other combinations, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Therefore, for all subsequent studies, we will utilize S3-J1 as the best-suited model for phase identification for the DN considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Table VI shows the consolidated results for metric J1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Observe that the sensitivity towards phase identi- 4Model Sy-Jx denotes that the phase identification algorithm utilizes Jx as the metric and Sy as the consensus algorithm 36 3 5 13 0 2 4 6 8 10 12 14 S2-J1 S4-J1 S3-J1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 19: For three different phase mappings and 7 clusters, S3-J1 phase identification model outperforms others for most of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The plot shows the histogram best model for a cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' fication due to measurement error is more than 10 times higher for cluster 2 (close to the substation with low βc) compared to other clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' This effect can again be attributed to low voltage fluctuations in cluster 2, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Case study 2: Effect of the proximity of measurement on PCI The goal of this case study is to assess the impact of measurement point proximity on phase identification performance metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The simplified representation of the original network model in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 14 is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Zonal connections are listed in Table VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' L0 denotes the zone where the 1 − φ consumer is located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' L1 denotes the zones which are directly connected to L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Similarly, L2 denotes zones that are not connected directly to L0 but via L1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' second-order neighbor, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Tables VIII, IX, and X list the numerical results with mean phase estimation metrics for each consumer and each measurement point in the DN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The following observations are made: Phase estimation accuracy deteriorates as the electrical distance between the measurement point used as the reference in correlation-based phase identification and the 1 − φ consumer with unknown phase connection increases,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The confidence of estimation deteriorates as the distance between measurement and consumer increases,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 37 TABLE VI: PCI estimation for metric J1 Balanced (C1) Fairly balanced (C2) Unbalanced (C3) Cluster Consensus ¯AJx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='Sy c ¯F Jx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='Sy c D Jx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='Sy c ¯AJx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='Sy c ¯F Jx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='Sy c D Jx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='Sy c ¯AJx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='Sy c ¯F Jx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='Sy c D Jx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='Sy c 1 S2 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='5592 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='0295 100 0.' metadata={'source': 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+page_content='1732 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='0491 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='0955 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='0479 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='1587 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='0484 Zone 1 Zone 3 Zone 7 Zone 5 Zone 6 Zone 4 Zone 2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 20: Zones in a simplified network diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 38 TABLE VII: Zonal connections L0 1 2 3 4 5 6 7 L1 5 [4,5,6,7] 5 2 [1,2,3] 2 2 L2 [2,3] [1,3] [1,2] [5,6,7] [4,6,7] [4,5,7] [4,5,6] L3 [4,6,7] [4,6,7] [1,3] [1,3] [1,3] TABLE VIII: Mean PCI accuracy with reference selection level 1 2 3 4 5 6 7 L0 100 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='44 100 100 100 100 100 L1 100 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='94 100 100 100 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='39 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='78 L2 100 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='71 100 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='57 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='94 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='18 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='44 L3 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='76 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='31 27.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='116 The mean estimation variance increases (implying estimation becomes more sensitive to measurement errors) as the distance between the measurement point used as the reference in correlation based phase identification and the single phase consumer with unknown phase connection increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The phase connectivity identification metrics proposed in this work not only quantitatively provide the estimation accuracy in % but also qualitatively provide the confidence factor (the higher the better) and the sensitivity to measurement errors (the lower the better).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Using these metrics, we observe that selecting of measurement points in proximity is better in terms of phase estimation quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' This also validates our 39 zonal phase identification approach we have established in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Case study 3: Impact of measurement error In case study 1, we identified the best-suited model for phase identification as the S3-J1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Previously, we observed that an imbalance in DN leads to worse estimation compared to balanced phase mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' In order to not have pessimistic or optimistic phase identification results, we utilize C2 phase mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' We apply different levels of measurement errors at the point of reference and at the point of single phase consumer point of connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' In order to not be biased by one sample of estimation error, we perform 1000 MC simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The three-phase identification metrics are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The three metrics are almost equally influenced by measurement error at reference voltage measurement or 1-φ consumer end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Observe from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 21 Mean phase estimation accuracy for 1% error in reference voltage and consumer voltage measurement is 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Implying, our consensus-based phase estimation algorithm was able to estimate more than 308 out of 313 of single-phase consumer phases accurately on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' For 10% error in reference voltage and consumer voltage measurement is still 82%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Thus, the phase estimation proposed in this work is largely immune to measurement errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The confidence factor deteriorates with an increase in measurement errors, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 21(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The sensitivity of phase estimation towards measurement error, shown as the variance in estimation Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 21(c) increases with an increase in measurement error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Case study 4: Effect of size of neutral conductor European DNs are 4-wire systems with a neutral conductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The impact of modeling the neutral conductor is not assessed in any of the prior works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' This is especially crucial if the digital twin is used for generating synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 22 shows the phase identification metrics for model S3-J1 and phase mapping C2 with and without neutral conductor considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' It is clear that modeling the neutral 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='15 (a) (b) (c) % Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 21: Impact of measurement accuracy on PCI metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' (a) shows the accuracy, (b) shows the confidence factor and (c) shows the variance of PCI respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 41 1 2 3 4 5 6 7 60 70 80 90 100 Estimation accuracy in % Neutral size same as phase conductor No neutral considered 1 2 3 4 5 6 7 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='6 Confidence factor 1 2 3 4 5 6 7 Cluster ID 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content='15 Sensitivity towards measurement error (a) (c) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' 22: Impact of neutral conductor modeling on phase estimation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' CONCLUSIONS We propose a phase connectivity identification (PCI) algorithm that utilize voltage time series, distribution network (DN) zones, and multiple measurements as references for improving the PCI accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The proposed phase identification algorithm builds a consensus among multiple estimations in a zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' This method is extended to consider metrics derived from voltage time series, which filters larger voltage deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Due to the consideration of multiple measurements, the PCI is drastically exceeding the performance compared to the widely used naïve model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Further, our approach is immune to network topology and measurement errors, as PCI accuracy is not dependent on only one reference measurement point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' We also propose phase connectivity identification metrics that not only quantitatively describe the estimation accuracy, but also qualitatively describe how good the PCI is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' We utilize a real German DN with limited observability for phase 42 identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' This network has 602 nodes and 313 single-phase consumers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' It is observed that the original voltage time series without salient feature extraction and absolute weighted consensus algorithm outperforms all the other phase estimation algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' The proposed algorithm identifies the phase connections on average of over 308 consumers accurately for 1% measurement error at the consumer end and reference measurements for 1000 Monte Carlo simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Thus, the proposed algorithm is robust towards uncertainty with high precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' In future work, we will consider synchronization errors in measurement while limiting DN observability even further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Further assessment is needed for selecting the best-suited algorithm for phase identification with varying network layouts, load profiles, and PV penetration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' Finally, we will extend this work for executing the algorithms on real measurement data and verify algorithm efficacy by intrusive phase measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' ACKNOWLEDGEMENT This work is supported by the H2020 EUniversal project, grant agreement ID: 864334 (https://euniversal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' eu/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' We would like to thank Clara Gouveia and Gil Silva Sampaio at INESC TEC, Porto for their comments on problem formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQfggeM/content/2301.03938v1.pdf'} +page_content=' We would like to thank Marta Vanin (KU Leuven), Deepjyoti Deka (Los Alamos National Lab), Lucas Pereira (Técnico Lisboa) for their insightful comments on the paper.' metadata={'source': 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b/vdFKT4oBgHgl3EQf4C7n/content/tmp_files/2301.11932v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..659897cbbb4e679f575072fb7f0dc010d8c5a1c7 --- /dev/null +++ b/vdFKT4oBgHgl3EQf4C7n/content/tmp_files/2301.11932v1.pdf.txt @@ -0,0 +1,341 @@ +RGB Arabic Alphabets Sign Language Dataset +Authors +• Muhammad Al-Barham a,* +• Adham Alsharkawi b +• Musa Al-Yaman b +• Mohammad Al-Fetyani c +• Ashraf Elnagar d +• Ahmad Abu Sa’aleek e +• Mohammad Al-Odat f +Affiliations +• a MLALP Research Group, University of Sharjah, United Arab Emirates +• b Mechatronics Engineering Department, The University of Jordan +• c AppsWave for Information Technology, Jordan +• d Department of Computer Science, University of Sharjah, United Arab Emirates +• e Al-Wefaq Control Systems, Doha, Qatar +• f Student Guidance Department, The University of Jordan, Jordan +Corresponding author’s email address and Twitter handle +• muhammadal-barham@ieee.org +• twitter: @MuhammadBarham +Keywords +• Sign-Language +• Dataset +• Deaf +• Arabic +• Alphabet +1 +arXiv:2301.11932v1 [cs.CV] 30 Jan 2023 + +Abstract +This paper introduces the RGB Arabic Alphabet Sign Language (AASL) dataset. AASL comprises 7,857 raw +and fully labelled RGB images of the Arabic sign language alphabets, which to our best knowledge is the first +publicly available RGB dataset. The dataset is aimed to help those interested in developing real-life Arabic +sign language classification models. AASL was collected from more than 200 participants and with different +settings such as lighting, background, image orientation, image size, and image resolution. Experts in the field +supervised, validated and filtered the collected images to ensure a high-quality dataset. AASL is made available +to the public on Kaggle.1 +Specifications table +Subject +Computer Science, Computer Vision, Pattern Recognition +Specific subject area +RGB-Image Based Arabic Sign Language Classification +Type of data +Images +How the data were +acquired +Images in this dataset were acquired using different types of cameras (we- +bcam, digital camera, and camera phone). +Data format +Labelled filtered RGB images with different extensions (’.jpg’: 6545, ’.jpeg’: +1211, ’.JPG’: 80, ,’.JPEG’: 21) +Description of data +collection +Participants were asked to submit their captured images through a form. +Arabic sign language alphabets are grouped into five main categories and +each category consists of a number of Arabic sign language alphabets. Ges- +tures of the Arabic sign language alphabets are shown to the participants +to follow. +The quality and suitability of submitted images are checked +manually. +Data source location +Jordan. +Data accessibility +The Data is available on Kaggle under CC BY-NC-SA 4.0, publicly +available via the link https://kaggle.com/datasets/59761a7132888de252 +ded8443ced1c7fb21ad28be5598f1f6ca43c663c32b40b +Data identification number: It will be provided once the paper is accepted +and the dataset become publicly available. +1https://kaggle.com/datasets/59761a7132888de252ded8443ced1c7fb21ad28be5598f1f6ca43c663c32b40b +2 + +Value of the Data +• The data is versatile as it is collected with different settings such as lighting, background, image orientation, +image size, and image resolution. +• The dataset is suitable for developing machine learning algorithms for Arabic sign language classification. +• The dataset is verified and validated by experts in the field. +• This dataset is - to our best knowledge - the first RGB high-resolution and publicly available dataset for +Arabic sign language. +Data Description +The RGB Arabic Alphabet Sign Language (AASL) dataset is the result of a collaborative effort among more +than 200 participants who shared one or more alphabets. Most of the images were taken by different types of +cameras including webcams, digital cameras, and phone cameras. The AASL dataset introduces 7,857 labeled +images for the Arabic sign language. A group of Arabic sign language experts supervised, validated and filtered +the images to ensure a high-quality dataset. +The dataset is organized into 31 folders, each folder represents a single alphabet. Table 2 highlights the +number of images in each folder, while Fig 1 presents a sample of images for different alphabets. +Table 2: Dataset distribution. +# +Letter name in +English Script +Letter name in +Arabic Script +# of Images +# +Letter name in +English Script +Letter name in +Arabic Script +# of Images +1 +ALEF +� ������ �� +287 +17 +ZAH +��� ��� +�� +232 +2 +BEH +������ �� +307 +18 +AIN +� ������ � +244 +3 +TEH +������ �� +226 +19 +GHAIN +� ������� �� +231 +4 +THEH +������ �� +305 +20 +FEH +������ +�� +255 +5 +JEEM +������� �� +210 +21 +QAF +� ������ �� +219 +6 +HAH +����� � +246 +22 +KAF +� ����� � +264 +7 +KHAH +��� ��� � +250 +23 +LAM +���� � +260 +8 +DAL +����� � +235 +24 +MEEM +������ � +253 +9 +THAL +������ �� +202 +25 +NOON +� ������ �� +237 +10 +REH +����� � +227 +26 +HEH +����� � +253 +11 +ZAIN +��� ���� �� +201 +27 +WAW +����� � +249 +12 +SEEN +� ������ � +266 +28 +YEH +������ �� +272 +13 +SHEEN +� ���� ��� +�� +278 +29 +TEH MARBUTA ��������� ����� �� +257 +14 +SAD +����� � +270 +30 +AL +�� +276 +15 +DAD +��� ��� +�� +266 +31 +LAA +� +268 +16 +TAH +����� � +227 +3 + +Figure 1: Sample from the dataset. +Experimental design, materials and methods: +With the aim of contributing to the Arabic sign language classification, we asked experts in the field of ArSL +interpretation to provide and verify ground-truth images that represent static ArSL alphabets. The experts +also helped in providing tips on how to perform each of the alphabets. +An online form with a set of instructions was prepared for data collection. The alphabets were distributed +into five different categories for the participants, the first 4 categories have 6 alphabets and the fifth and last +category has the remaining 7 alphabets. Participants had the option to submit images of the alphabets that +they felt comfortable performing them. Hence, there was not any restriction on the number of images that a +participant should submit. +The link to the online form was posted on different social media platforms. We had participants from schools +and universities with different ages and genders. Images were captured by the participants using different types +of cameras, backgrounds, light conditions, and image sizes. The identity of the participants was kept anonymous. +4 + +ALEF +BEH +TEH +THEH +DAL +Jeem +KHAH +HAH +(c) +(caji) T +(s) +() +2 () +(s) +(es) +(Jls) s +SHEEN +THAL +REH +TAH +ZAIN +SEEN +SAD +ZAH +(sb) ) +(st) co +(cw) cw +(slo) o +(J13) 3 +(i) s +(sb) j +(b) b +KAF +AIN +FEH +QAF +LAM +GHAIN +ZAH +() +(irs) E +(ic) E +(b) +(s) +y +(cb) b +NOON +WAW +AL +MEEM +HEH +YEH +TEH MARBUTA +LAA +(9l9) 9 +(cl) s +(ogi) ‘ +(Po) +(rlo) 0 +(abgy sb) 8(1) Ground-Truth image. +(2) Correct image. +(3) Wrong image. +Figure 2: Geem ArSL alphabet. +The data collection started in March 2022 and lasted for five months. Two of our research team were given +the task of evaluating each and every submitted image manually. They were mainly responsible for checking +the label of an image and the match between a submitted image and the ground-truth image of a particular +alphabet. Fig 2 shows an example of a ground-truth image of an alphabet (left), a correctly performed alphabet +(center), and a wrongly performed alphabet (right). +The whole dataset then went through one final round of evaluation where one of our research team double- +checked all submitted images for correctness. The evaluation process resulted in a dataset size reduction going +from 8,042 images to 7,857 correct images. +Finally the whole dataset was labelled automatically by running a simple script. Each of the images is +labeled as ”AphabetName ID”. The ID started from 0 till reaching the total number of images of a certain +alphabet in a specific folder. +On a final note, images of our dataset are raw in nature, and thus interested researchers are left to perform +any necessary processing they may need. Also, this work has been inspired by ArASL (Arabic Alphabets Sign +Language) Dataset [1]. +CRediT author statement +Muhammad Al-Barham: Conceptualization, Validation, Methodology, Writing- Original draft prepara- +tion, Software, Data Curation +Adham Alsharkawi: Writing- Reviewing and Editing +Musa Al-Yaman: Conceptualization, Writing- Original draft preparation, Resources +Mohammad Al-Fetyani: Writing- Reviewing and Editing, Software, Data Curation +Ashraf Elnagar: Writing- Reviewing and Editing +Ahmad Abu Sa’Aleek: Conceptualization, Methodology, Validation +Mohammad Al-Odat: Validation, Methodology +Acknowledgments +We would like to thank the Student Counseling Department at the University of Jordan for their guidance +on how to get the right and correct images based on their experiences. We would like also to thank Jana M. +AlNatour and Raneem F. Abdelraheem for their help in the data collection process. +5 + +References +[1] +Ghazanfar Latif et al. “ArASL: Arabic Alphabets Sign Language Dataset”. In: Data in Brief 23 (2019), +p. 103777. issn: 2352-3409. doi: https://doi.org/10.1016/j.dib.2019.103777. url: https://www.sciencedir +ect.com/science/article/pii/S2352340919301283. +6 + diff --git a/vdFKT4oBgHgl3EQf4C7n/content/tmp_files/load_file.txt b/vdFKT4oBgHgl3EQf4C7n/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..52ca262d395cc3f840f9d292aff0779defd05f0a --- /dev/null +++ b/vdFKT4oBgHgl3EQf4C7n/content/tmp_files/load_file.txt @@ -0,0 +1,244 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf,len=243 +page_content='RGB Arabic Alphabets Sign Language Dataset Authors Muhammad Al-Barham a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content='* Adham Alsharkawi b Musa Al-Yaman b Mohammad Al-Fetyani c Ashraf Elnagar d Ahmad Abu Sa’aleek e Mohammad Al-Odat f Affiliations a MLALP Research Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' University of Sharjah,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' United Arab Emirates b Mechatronics Engineering Department,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' The University of Jordan c AppsWave for Information Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' Jordan d Department of Computer Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' University of Sharjah,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' United Arab Emirates e Al-Wefaq Control Systems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' Doha,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' Qatar f Student Guidance Department,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' The University of Jordan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' Jordan Corresponding author’s email address and Twitter handle muhammadal-barham@ieee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content='org twitter: @MuhammadBarham Keywords Sign-Language Dataset Deaf Arabic Alphabet 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content='11932v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content='CV] 30 Jan 2023 Abstract This paper introduces the RGB Arabic Alphabet Sign Language (AASL) dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' AASL comprises 7,857 raw and fully labelled RGB images of the Arabic sign language alphabets, which to our best knowledge is the first publicly available RGB dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' The dataset is aimed to help those interested in developing real-life Arabic sign language classification models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' AASL was collected from more than 200 participants and with different settings such as lighting, background, image orientation, image size, and image resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' Experts in the field supervised, validated and filtered the collected images to ensure a high-quality dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' AASL is made available to the public on Kaggle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content='1 Specifications table Subject Computer Science, Computer Vision, Pattern Recognition Specific subject area RGB-Image Based Arabic Sign Language Classification Type of data Images How the data were acquired Images in this dataset were acquired using different types of cameras (we- bcam, digital camera, and camera phone).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' Data format Labelled filtered RGB images with different extensions (’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content='jpg’: 6545, ’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content='jpeg’: 1211, ’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content='JPG’: 80, ,’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content='JPEG’: 21) Description of data collection Participants were asked to submit their captured images through a form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' Arabic sign language alphabets are grouped into five main categories and each category consists of a number of Arabic sign language alphabets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' Ges- tures of the Arabic sign language alphabets are shown to the participants to follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' The quality and suitability of submitted images are checked manually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' Data source location Jordan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' Data accessibility The Data is available on Kaggle under CC BY-NC-SA 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content='0, publicly available via the link https://kaggle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content='com/datasets/59761a7132888de252 ded8443ced1c7fb21ad28be5598f1f6ca43c663c32b40b Data identification number: It will be provided once the paper is accepted and the dataset become publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' 1https://kaggle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content='com/datasets/59761a7132888de252ded8443ced1c7fb21ad28be5598f1f6ca43c663c32b40b 2 Value of the Data The data is versatile as it is collected with different settings such as lighting, background, image orientation, image size, and image resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' The dataset is suitable for developing machine learning algorithms for Arabic sign language classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' The dataset is verified and validated by experts in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' This dataset is - to our best knowledge - the first RGB high-resolution and publicly available dataset for Arabic sign language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' Data Description The RGB Arabic Alphabet Sign Language (AASL) dataset is the result of a collaborative effort among more than 200 participants who shared one or more alphabets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' Most of the images were taken by different types of cameras including webcams, digital cameras, and phone cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' The AASL dataset introduces 7,857 labeled images for the Arabic sign language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' A group of Arabic sign language experts supervised, validated and filtered the images to ensure a high-quality dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' The dataset is organized into 31 folders, each folder represents a single alphabet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' Table 2 highlights the number of images in each folder, while Fig 1 presents a sample of images for different alphabets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' Table 2: Dataset distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content='# ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content='268 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content='TAH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content='����� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content='227 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content='Figure 1: Sample from the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' Experimental design, materials and methods: With the aim of contributing to the Arabic sign language classification, we asked experts in the field of ArSL interpretation to provide and verify ground-truth images that represent static ArSL alphabets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' The experts also helped in providing tips on how to perform each of the alphabets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' An online form with a set of instructions was prepared for data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' The alphabets were distributed into five different categories for the participants, the first 4 categories have 6 alphabets and the fifth and last category has the remaining 7 alphabets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' Participants had the option to submit images of the alphabets that they felt comfortable performing them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' Hence, there was not any restriction on the number of images that a participant should submit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' The link to the online form was posted on different social media platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' We had participants from schools and universities with different ages and genders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' Images were captured by the participants using different types of cameras, backgrounds, light conditions, and image sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' The identity of the participants was kept anonymous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' 4 ALEF BEH TEH THEH DAL Jeem KHAH HAH (c) (caji) T (s) () 2 () (s) (es) (Jls) s SHEEN THAL REH TAH ZAIN SEEN SAD ZAH (sb) ) (st) co (cw) cw (slo) o (J13) 3 (i) s (sb) j (b) b KAF AIN FEH QAF LAM GHAIN ZAH () (irs) E (ic) E (b) (s) y (cb) b NOON WAW AL MEEM HEH YEH TEH MARBUTA LAA (9l9) 9 (cl) s (ogi) ‘ (Po) (rlo) 0 (abgy sb) 8(1) Ground-Truth image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' (2) Correct image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' (3) Wrong image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' Figure 2: Geem ArSL alphabet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' The data collection started in March 2022 and lasted for five months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' Two of our research team were given the task of evaluating each and every submitted image manually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' They were mainly responsible for checking the label of an image and the match between a submitted image and the ground-truth image of a particular alphabet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' Fig 2 shows an example of a ground-truth image of an alphabet (left), a correctly performed alphabet (center), and a wrongly performed alphabet (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' The whole dataset then went through one final round of evaluation where one of our research team double- checked all submitted images for correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' The evaluation process resulted in a dataset size reduction going from 8,042 images to 7,857 correct images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' Finally the whole dataset was labelled automatically by running a simple script.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' Each of the images is labeled as ”AphabetName ID”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' The ID started from 0 till reaching the total number of images of a certain alphabet in a specific folder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' On a final note, images of our dataset are raw in nature, and thus interested researchers are left to perform any necessary processing they may need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' Also, this work has been inspired by ArASL (Arabic Alphabets Sign Language) Dataset [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' CRediT author statement Muhammad Al-Barham: Conceptualization,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' Validation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' Methodology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' Writing- Original draft prepara- tion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' Software,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' Data Curation Adham Alsharkawi: Writing- Reviewing and Editing Musa Al-Yaman: Conceptualization,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' Writing- Original draft preparation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' Resources Mohammad Al-Fetyani: Writing- Reviewing and Editing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' Software,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' Data Curation Ashraf Elnagar: Writing- Reviewing and Editing Ahmad Abu Sa’Aleek: Conceptualization,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' Methodology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' Validation Mohammad Al-Odat: Validation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' Methodology Acknowledgments We would like to thank the Student Counseling Department at the University of Jordan for their guidance on how to get the right and correct images based on their experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' We would like also to thank Jana M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' AlNatour and Raneem F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' Abdelraheem for their help in the data collection process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' 5 References [1] Ghazanfar Latif et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' “ArASL: Arabic Alphabets Sign Language Dataset”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' In: Data in Brief 23 (2019), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' 103777.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' issn: 2352-3409.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' doi: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content='dib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content='103777.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' url: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content='sciencedir ect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content='com/science/article/pii/S2352340919301283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} +page_content=' 6' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFKT4oBgHgl3EQf4C7n/content/2301.11932v1.pdf'} diff --git a/wNE0T4oBgHgl3EQf-AKd/content/tmp_files/2301.02809v1.pdf.txt b/wNE0T4oBgHgl3EQf-AKd/content/tmp_files/2301.02809v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..841b63d72c7e1d4c0bae1f00f3f429ae1accae33 --- /dev/null +++ b/wNE0T4oBgHgl3EQf-AKd/content/tmp_files/2301.02809v1.pdf.txt @@ -0,0 +1,931 @@ +A Brain-inspired Memory Transformation based +Differentiable Neural Computer for Reasoning-based +Question Answering +Yao Liang1,3,‡, +Hongjian Fang1,3,‡, +Yi Zeng1,2,3,4∗, +Feifei Zhao1 +1Brain-Inspired Cognitive Intelligence Lab, Institute of Automation, +Chinese Academy of Sciences, Beijing 100190, China +2Center for Excellence in Brain Science and Intelligence Technology, +Chinese Academy of Sciences, Shanghai 200031, China +3University of Chinese Academy of Sciences, Beijing 100190, China +4National Laboratory of Pattern Recognition, Institute of Automation, +Chinese Academy of Sciences, Beijing 100190, China +‡Co-first authors with equal contribution +{liangyao2020,fanghongjian2017,yi.zeng}@ia.ac.cn +Abstract +Reasoning and question answering as a basic cognitive function for humans, is +nevertheless a great challenge for current artificial intelligence. Although the +Differentiable Neural Computer (DNC) model could solve such problems to a +certain extent, the development is still limited by its high algorithm complexity, +slow convergence speed, and poor test robustness. Inspired by the learning and +memory mechanism of the brain, this paper proposed a Memory Transformation +based Differentiable Neural Computer (MT-DNC) model. MT-DNC incorporates +working memory and long-term memory into DNC, and realizes the autonomous +transformation of acquired experience between working memory and long-term +memory, thereby helping to effectively extract acquired knowledge to improve rea- +soning ability. Experimental results on bAbI question answering task demonstrated +that our proposed method achieves superior performance and faster convergence +speed compared to other existing DNN and DNC models. Ablation studies also +indicated that the memory transformation from working memory to long-term +memory plays essential role in improving the robustness and stability of reasoning. +This work explores how brain-inspired memory transformation can be integrated +and applied to complex intelligent dialogue and reasoning systems. +1 +Introduction +Reasoning and Question Answering (QA) are advanced cognitive functions of human beings, and +they are also one of the important challenges in the field of deep learning. Differentiable Neural +Computers (DNC) model proposed by Graves et al. (2016) provides a feasible solution to study +reasoning and QA. DNC consists of a DNN-based computational controller and an external memory +in which the neural network could learn and communicate (read and write) with memory module, +and the memory module could represent and store learned structure. +The DNC model has achieved good performance on various image reasoning and QA tasks Graves +et al. (2016); Rasekh and Safi-Esfahani (2020). However, it faces several main challenges such as +∗Corresponding authors +Preprint. Under review. +arXiv:2301.02809v1 [cs.AI] 7 Jan 2023 + +high algorithm complexity, slow convergence speed, and high average test error rate, which limit +its further development and wider application. The BrsDNC model Franke et al. (2018) improves +DNC model with the introduction of Normalization and Dropout, which shows to be more robust and +salable. Actually, the main issues of current DNC models lie in the restricted memory may lead to +missing critical knowledge. As the working time becomes longer, reading and writing pressure on +memory module increases rapidly, thus limiting the training speed and performance of the model. +Besides, existing methods lack references from brain learning and memory mechanisms. Thus, there +is still much room for improvement. +Memory in the brain includes short-term and long-term memory, etc Baddeley (2007); Lee and +Wilson (2002); Winocur et al. (2010); Marshall and Born (2007); Ji and Wilson (2007). They play +an important role in various cognitive functions such as learning, decision making, and reasoning, +respectively. Short-term memory has limited storage space and therefore cannot retain information +indefinitely Diamond (2013). Then some memories are forgotten, others that are repeatedly reused +will be retained and transferred to long-term memory. Information can be stored in long-term memory +for a longer period of time, continuously aiding learning and reasoning Atkinson and Shiffrin (1968). +The collaboration and division between working memory and long-term memory help the human +brain to consolidate and use the acquired knowledge more efficiently, and therefore perform multiple +cognitive tasks more robustly Kitamura et al. (2017). +Inspired by the learning and memory mechanism of the brain, we proposed a brain-inspired Memory +Transformation based Differentiable Neural Computer (MT-DNC) model. MT-DNC expands only +one memory area in the original DNC model into two different memory areas, namely the working +memory area and the long-term memory area. Working memory stores information that is more +closely related to the current task, while long-term memory stores more meaningful long-term +knowledge. These two memory areas are connected through a Memory Transformation algorithm. +The core principles of the memory transformation algorithm are: repeatedly visited knowledge will be +transferred to long-term memory area, useless information will be discarded from working memory +area Zhao et al. (2017); LeCun et al. (2015). +The innovations of our method are mainly reflected in the following aspects: +• MT-DNC combines short-term working memory with long-term memory to promote a more +comprehensive memory of the acquired knowledge. +• We proposed a brain-inspired memory transformation algorithm to dynamically store and +extract useful information and filter out the useless ones. +• Reasoning-based question answering experiments demonstrated that MT-DNC could im- +prove the accuracy and convergence speed on bAbI task compared to current DNC-based +methods. The experimental results also illustrated the great significance of introducing the +neural mechanism of memory transformation to improve the stability and robustness of +model reasoning. +2 +Related work +Neural Turing Machine (NTM) : The core idea is to combine neural networks with external memory +to expand the capabilities of neural networks and interact through an attention mechanism Graves +et al. (2014). To a certain extent, NTM can be compared to Turing machine Xiong et al. (2016); +Zaremba and Sutskever (2015), and the experiment verifies the Turing completeness of NTM Tao +et al. (2021); Zaremba and Sutskever (2015). The biggest advantage of NTM in terms of function is +to deal with complex tasks that require memory participation. +Differentiable Neural Computer (DNC): DNC, known as the second version of NTM. Its core idea +is consistent with NTM, based on external memory to improve the ability of neural networks Graves +et al. (2016); Santoro et al. (2016); Lake et al. (2017). Compared with the original NTM, DNC has +made important improvements in the addressing mechanism Hassabis et al. (2017); Chan et al. (2018), +removing the index shift operation, and better supporting the functions of allocate and de-allocate for +memory. Compared with the original NTM, there is also a certain improvement in performance. +In recent years, several works have made some improvements on the basis of the DNC structure. +Franke et al. (2018) improved the performance of the model by optimizing the memory module of +2 + +DNC, increasing the bidirectional connection between memory modules, and introducing the layer +normalization Ba et al. (2016b) training method. By improving the details of addressing and memory +allocation in DNC, Csordás and Schmidhuber (2019) made the model achieve better accuracy on +the bAbI dataset. Rasekh and Safi-Esfahani (2020) integrated the NeuroEvolution algorithm into +the DNC structure, and showed faster encoding speed in various cognitive tasks, and the model +performance was improved. +To sum up, none of these approaches fundamentally address the issues of low accuracy and slow +convergence speed of DNC due to the constrained external memory. This paper takes inspiration +from the learning and memory mechanism of the brain, and proposes the multiple memory modules +coordinated MT-DNC model that integrates working memory and long-term memory Seo et al. +(2016); Ba et al. (2016a); Le et al. (2019, 2020). The proposed model could improve the accuracy +and convergence speed, and bring about superior performance compared to other DNC-based models. +3 +Method +In this section, we will give a comprehensive introduction to our MT-DNC method. MT-DNC extends +the memory module of the DNC to a working memory module and a long-term memory module. The +overall framework of MT-DNC includes controller layer, memory layer and linear layer, as shown in +Figure 1. +Figure 1: Overall architecture of MT-DNC. +1) The controller layer is responsible for encoding and processing the input data and the output of the +previous Memory Layer, learning the time series information from the training data, and transmitting +the output results to the memory layer and the linear layer. +2) Memory Layer is responsible for storing the output of the controller layer and extracting the +useful information through a series of storage and transformation mechanisms. The memory layer +incorporates memory transformation between working memory and long-term memory modules, +enabling the MT-DNC model to have strong memory and reasoning capabilities. +3) The linear layer combines the output of the memory Layer and control layer as input, and outputs +the final prediction result after linear transformation. +A more detailed description of our MT-DNC method is presented as follows: +3.1 +Controller Layer +Control layer combines the original input data xt and the output of the previous time step memory +layer OutputM +t−1 as input. After linear operation with weight WC +t and Normalization Klambauer +3 + +Linear Layer +Long-term +Working +Memory Module +Memory Module +Memory Layer +Controller LayerFigure 2: MT-DNC incorporates working memory module and long-term memory module. +et al. (2017); Franke et al. (2018), the output OutputC +t is transmitted to working memory module of +memory layer. +OutputC +t = Normalization(WC +t (xt + OutputM +t−1) + bC +t ) +(1) +Where +xt ∈ RX, OutputC +t +∈ RC, bC +t +∈ RC, WC +t +∈ R(2×W ×R+C+X)×C, OutputM +t−1 ∈ +R2×W ×R+C. +Among them: X represents the dimension of the input data, C represents the output dimension of the +control layer, M represents the output of the memory layer, R represents the head number of the read +memory area, and W represents the width of the memory area. +3.2 +Memory Layer +The memory layer contains the working memory module, the long-term memory module and +the memory transformation algorithm. The working memory module stores the latest interaction +information from the controller layer, while the long-term memory holds the information that is +frequently used with high importance but will be deleted by the working memory. In both working +memory and long-term memory, a dynamic update and extract rule is required to continuously replace +the store information. The memory transformation algorithm selectively transfers the data from +working memory to long-term memory for processing, and finally the memory layer combines the +output of the controller layer, the output of working memory and the output of long-term memory +to make a decision. The information processing procedure of our MT-DNC method is depicted in +Figure 2. +3.2.1 +Working Memory Module. +The working memory module is functionally designed to store interactive information from the output +of the controller layer in real time, updating and extracting related information according to the output +of the controller layer. Due to the limitation of storage space, we take inspiration from the update and +decay mechanism of memory in the human brain and replace the information that is similar to the +current interaction information (OutputC +t ). In addition, information that has already been extracted +or used is more likely to be replaced to retain as much innovative information as possible. +The memory area read, write and gating related signals are first generated from OutputC +t by a +linear transformation, detailed as Kw +t ∈ RW , Kl +t ∈ RW , Ew +t ∈ RW , El +t ∈ RW , Vw +t ∈ RW , Kw,i +t +∈ +RR×W , Kl,i +t +∈ RR×W , βw +t ∈ R, fw,i +t +∈ RR. +4 + +Controller Layer +Linear Layer +gindino +Output? +Read and Write +Update Memory +Extract Information +Memory Layer +Working Memory +Long-Term Memory +Transform +Uy +UWorking Memory Updating Algorithm. The updating of working memory is based on the follow- +ing principles: +1) Delete the items that has not been used for a long time. +2) Delete the ones that has just been extracted. +3) Delete similar items. +4) Retain the novel ones that have been updated recently. +Based on the above principles, we update the working memory in real time according to the dynamic +addressing algorithm in Graves et al. (2016). +ψw +t = +R +� +i=1 +(1 − fw,i +t +Ww,i +t−1) +Uw +t = (Uw +t−1 + Ww +t−1 − (Uw +t−1 ◦ Ww +t−1)) ◦ ψw +t +φw +t = SortIndiceAscending(Uw +t ) +aw +t [φw +t [j]] = (1 − Uw +t [φw +t [j]]) +j−1 +� +i=1 +Uw +t [φw +t [j]]] +(2) +Where ψw +t ∈ RN is the result of scaling and accumulating the read weight matrix Ww,i +t−1 of the +previous time step through the fw,i +t +gated tensor, The φw +t ∈ RN tensor is the index tensor sorted in +ascending order by the memory area management tensor Uw +t ∈ RN, N represents the length of the +memory area, aw +t ∈ RN is the dynamic addressing based write weight of the working memory area. +Ww,c +t += +exp(d(Kw +t , Mw +t )βw +t ) +� exp(d(Kw +t , Mw +t )βw +t ) +(3) +Where +Ww,c +t +∈ RN, βw +t ∈ R, Kw +t ∈ RW . N represents the length of the memory area, and W +represents the width of the memory area. +Ww +t = γw +t [gw +t aw +t + (1 − gw +t )Ww,c +t +] +Mw +t = Mw +t−1 − Mw +t−1 ◦ Ww +t (Ew +t )T + Ww +t (Vw +t )T +(4) +Where +Mw +t ∈ RN×W . gw +t ∈ [0, 1] represents the write weight allocation gate tensor in the working +memory area, which is used to control the allocation proportion of the two addressing modes in the +final write. γw +t ∈ [0, 1] is used to avoid data from being flushed +Working Memory Extracting Algorithm. The working memory extracts the information that most +relevant to the current interactive information (OutputC +t ), then this returned information is the used +item at the current moment. The extracting algorithm is Graves et al. (2016). +Ww,i +t += +exp(d(Kw,i +t +, Mw +t )βw,i +t +) +� exp(d(Kw,i +t +, Mw +t )βw,i +t +) +(5) +Where +Ww,i +t +∈ RR×N, Kw,i +t +∈ RR×W , βw,i +t +∈ RR, read R times in total, The label is i. N +represents the length of the memory area, and W represents the width of the memory area. +Rw,i +t += (Mw +t )T Ww,i +t +(6) +Where +Rw,i +t +∈ RR×W . +3.2.2 +Memory Transformation from Working Memory to Long-term Memory. +The DNC-based model Graves et al. (2016); Franke et al. (2018) directly maps the output of the +working memory (Rw,i +t +) to the linear layer, and since the used items will be deleted in the working +memory, this will lead to the loss of some important information, which in turn affects the performance +5 + +and robustness. In this paper, we design a memory transformation algorithm to transfer the extracted +information from working memory to long-term memory, thus compensating for the information loss +caused by the update in working memory. +The algorithm for updating and extracting information in long-term memory are similar to that +in working memory, the only difference is that the input in working memory originates from the +controller layer and the input in long-term memory originates from the working memory layer. The +update formula for the long-term memory layer is as follows: +Wl +t = γl +t[gl +tal +t + (1 − gl +t)Wl,c +t ] +Bw +t = +R +� +i=1 +Rw,i +t +Ml +t = Ml +t−1 − Ml +t−1 ◦ Wl +t(El +t)T + Wl +t(Bw +t )T +(7) +Where +Ml +t ∈ RN×W , Bw +t ∈ RW . gl +t ∈ [0, 1] represents the long-term memory write weight +allocation gate tensor, which is used to control the allocation proportion of the two addressing modes +in the final write. +The information extraction from the long-term memory layer integrates the information from the +long-term memory layer Rl,i +t as well as the information from the working memory layer Rw,i +t +and is +calculated as follows: +OutputM +t += Rw,i +t ++ Rl,i +t +(8) +Where +Ri +t ∈ R2×R×W , OutputC +t ∈ RC, OutputM +t +∈ R2×W ×R+C. +Algorithm 1 Execution algorithm for MT-DNC +Input: Training set xt,yt. +Output: The MT-DNC model. +1: randomly initialize weight W. +2: for e = 0; e < Epoch;e + + do +3: +%Forward propagation +4: +xt = xt + inverse(xt). +5: +As shown in Eq 1, xt and OutputM +t−1 are used as input data for the control layer. +6: +After processing by Eq 2, 3, 4, 5 and 6, the output Rw,i +t +of the working memory area is obtained. +7: +Rw,i +t +will be input to the long-term memory area and processed by Eq 2, 3, 7, 5 and 6 to +generate Rl,i +t . +8: +After the processing of Eq 8, the memory layer output tensor OutputM +t +is obtained. +9: +After the processing of Eq 9, the model output ˆyt is obtained. +10: +%Back propagation updating W +11: +The difference between yt and ˆyt is optimized by Cross Entropy. +12: end for +13: return MT-DNC model +3.3 +Linear Layer +The output of the linear layer ˆyt is determined by the output of the controller layer OutputC +t after +Dropout processing Franke et al. (2018); Gal and Ghahramani (2016); Srivastava et al. (2014) and +the output of the memory layer OutputM +t , given by: +ˆyt = Softmax(WO +t (OutputM +t ++ Dropout(OutputC +t )) + bO +t ) +(9) +Where +ˆyt ∈ RY , WO +t ∈ R(2×W ×R+C)×Y is the output parameter matrix, bO +t ∈ RY is the bias +matrix. +The detailed procedure of our MT-DNC is shown in Algorithm 1. +6 + +4 +Experiments +4.1 +bAbI task description +The bAbI1 is a reasoning-based text question-and-answer task Weston et al. (2015); Kumar et al. +(2016). We choose en-10k as the experimental dataset, which contains 20 sub-tasks, each consisting +of a training dataset text file containing 10k questions and a test dataset text file containing 1k +questions. A joint training approach is used to verify the text comprehension and reasoning ability +of the MT-DNC model. Unlike other previous related work, our method uses end-to-end training +without any pre-processing on the bAbI dataset itself. +4.2 +Training details +bAbI question and answer task composed of 20 sub-tasks is combined in one training session. A +training sample is generated for each subtask in the dataset in terms of different stories. The detailed +generation process is as follows: +1. The text sequence training samples are processed by removing digits, converting words from +upper case to lower case, removing line breaks, etc. +2. Cut the text sequence training sample into a list of sequences with words (including 3 +punctuation marks). +3. Replace ’answer words’ in the list with ’-’, and get a list of training input samples after +encoded into word vectors by one-hot word vector processor. The length of the list is the +length of the largest text sequence of the current batch, and the text with insufficient length +is filled with ’0’. A word in the list is represented as xt ∈ RZ, where Z is the length of the +word vector with a value of 159. +4. All training input samples and target samples form the training sample list. +5. 10% of the data in the training sample list is used as the validation dataset. +6. MT-DNC model performs 300 epochs of training, validation and testing. +The total number of parameters in the model is 1267337 and the batch size of the data is 32. The +number of control level nodes is 172. Both memory areas have a length of 128 and a width of 64, 4 +read heads, 1 write head and a dropout of 0.9. The learning rate is 0.0003, the Momentum value of +the optimizer Rmsprop is 0.9 Kingma and Ba (2014), and the gradient clipping value is 10. It takes +about 1 hour to run an epoch using an RTX3060 graphics card, an Intel i7 processor, and 16G of +RAM for training. +4.3 +Experimental results +To verify the effectiveness of the proposed MT-DNC model, we conducted comparison experiments +with DNC, EntNet Henaff et al. (2016), LSTM Hochreiter and Schmidhuber (1997), SDNC Rae +et al. (2016), BrsDNC Franke et al. (2018) and other models on the bAbI question and answer task +dataset. In addition, we also compared the MT-DNC-DI model (our MT-DNC without memory +transformation) to verify the effectiveness of memory transformation. The MT-DNC-DI model refers +to our model without memory transformation, which directly uses independent memory modules +with dual regions of working memory and long-term memory (both receive input from the controller +layer). Table 1 shows the average word error rate (WER) of different methods under seven different +initialized parameters. +According to the experimental results, it can be seen that the MT-DNC model achieves a lower average +error rate (only 2.5% of mean WER) than other models, especially the current BrsDNC model which +reaches State Of The Art (3.2% of mean WER) on the 20 bAbI tasks with joint training. In particular, +on the tasks of 8th, 14th, and 15th sub-tasks, all other methods have errors, while our method +achieves an error rate of 0. For 16th and 17th sub-tasks, compared to the best performance achieved +by BrsDNC, our method could significantly reduces the error rate by 6.7% and 4.7%, respectively. +We also counted the number of failed tasks (more than 5% errors) among the 20 tasks, as shown +1https://research.facebook.com/downloads/babi/ +7 + +Task +DNC +EntNet +LSTM +SDNC +BrsDNC +MT-DNC-DI +MT-DNC +1: 1 supporting fact +9.0 ± 12.6 +0.0 ± 0.1 +28.4 ± 1.5 +0.0 ± 0.0 +0.1 ± 0.1 +0.0 ± 0.1 +0.0 ± 0.1 +2: 2 supporting facts +39.2 ± 20.5 +15.3 ± 15.7 +56.0 ± 1.5 +7.1 ± 14.6 +0.8 ± 0.2 +0.7 ± 0.2 +0.1 ± 0.1 +3: 3 supporting facts +39.6 ± 16.4 +29.3 ± 26.3 +51.3 ± 1.4 +9.4 ± 16.7 +2.4 ± 0.6 +3.6 ± 0.8 +2.8 ± 0.4 +4: 2 argument relations +0.4 ± 0.7 +0.1 ± 0.1 +0.8 ± 0.5 +0.1 ± 0.1 +0.0 ± 0.0 +0.0 ± 0.0 +0.0 ± 0.0 +5: 3 argument relations +1.5 ± 1.0 +0.4 ± 0.3 +3.2 ± 0.5 +0.9 ± 0.3 +0.7 ± 0.1 +0.8 ± 0.2 +0.7 ± 0.1 +6: yes/no questions +6.9 ± 7.5 +0.6 ± 0.8 +15.2 ± 1.5 +0.1 ± 0.2 +0.0 ± 0.0 +0.0 ± 0.0 +0.0 ± 0.0 +7: counting +9.8 ± 7.0 +1.8 ± 1.1 +16.4 ± 1.4 +1.6 ± 0.9 +1.0 ± 0.5 +1.5 ± 0.1 +1.2 ± 0.2 +8: lists/sets +5.5 ± 5.9 +1.5 ± 1.2 +17.7 ± 1.2 +0.5 ± 0.4 +0.5 ± 0.3 +0.1 ± 0.2 +0.0 ± 0.0 +9: simple negation +7.7 ± 8.3 +0.0 ± 0.1 +15.4 ± 1.5 +0.0 ± 0.1 +0.1 ± 0.2 +0.0 ± 0.0 +0.0 ± 0.0 +10: indefinite knowledge +9.6 ± 11.4 +0.1 ± 0.2 +28.7 ± 1.7 +0.3 ± 0.2 +0.0 ± 0.0 +0.1 ± 0.0 +0.0 ± 0.1 +11: basic coreference +3.3 ± 5.7 +0.2 ± 0.2 +12.2 ± 3.5 +0.0 ± 0.0 +0.0 ± 0.0 +0.0 ± 0.0 +0.0 ± 0.0 +12: conjunction +5 ± 6.3 +0.0 ± 0.0 +5.4 ± 0.6 +0.2 ± 0.3 +0.0 ± 0.1 +0.0 ± 0.0 +0.0 ± 0.0 +13: compound coreference +3.1 ± 3.6 +0.0 ± 0.1 +7.2 ± 2.3 +0.1 ± 0.1 +0.0 ± 0.0 +0.0 ± 0.0 +0.0 ± 0.0 +14: time reasoning +11 ± 7.5 +7.3 ± 4.5 +55.9 ± 1.2 +5.6 ± 2.9 +0.8 ± 0.7 +1.1 ± 0.1 +0.0 ± 0.0 +15: basic deduction +27.2 ± 20.1 +3.6 ± 8.1 +47.0 ± 1.7 +3.6 ± 10.3 +0.1 ± 0.1 +0.0 ± 0.0 +0.0 ± 0.0 +16: basic induction +53.6 ± 1.9 +53.3 ± 1.2 +53.3 ± 1.3 +53.0 ± 1.3 +52.6 ± 1.6 +48.5 ± 0.9 +45.9 ± 1.0 +17: positional reasoning +32.4 ± 8 +8.8 ± 3.8 +34.8 ± 4.1 +12.4 ± 5.9 +4.8 ± 4.8 +5.1 ± 2.8 +0.1 ± 0.2 +18: size reasoning +4.2 ± 1.8 +1.3 ± 0.9 +5.0 ± 1.4 +1.6 ± 1.1 +0.4 ± 0.4 +1.0 ± 0.8 +0.1 ± 0.1 +19: path finding +64.6 ± 37.4 +70.4 ± 6.1 +90.9 ± 1.1 +30.8 ± 24.2 +0.0 ± 0.0 +0.0 ± 0.0 +0.0 ± 0.0 +20: agents motivation +0.0 ± 0.1 +0.0 ± 0.0 +1.3 ± 0.4 +0.0 ± 0.0 +0.1 ± 0.1 +0.0 ± 0.0 +0.0 ± 0.0 +Mean WER: +16.7 ± 7.6 +9.7 ± 2.6 +27.3 ± 0.8 +6.4 ± 2.5 +3.2 ± 0.5 +3.1 ± 0.1 +2.5 ± 0.1 +Failed Tasks (>5%): +11.2 ± 5.4 +5.0 ± 1.2 +17.1 ± 1.0 +4.1 ± 1.6 +1.4 ± 0.5 +1.4 ± 0.4 +1.0 ± 0.0 +Table 1: The average word error rate(WER) of different models on bAbi task +in the last row of Table 1. Our method has only 1 failed task and achieves superior performance +compared to other methods, significantly outperforming DNC (with 11 failed tasks) and LSTM (with +17 failed tasks) methods. +Figure 3: Validation loss (A) and training loss (B) of DNC, BrsDNC, MT-DNC-DI and MT-DNC. +The horizontal coordinate represents the number of Epochs and the vertical coordinate represents the +changing of loss. +Figure 3 depicts the changing of loss for different methods during validation (Figure 3A) and training +(Figure 3B) processes. It can be seen that our MT-DNC model shows lower loss, higher performance +and faster convergence speed compared with DNC and BrsDNC models. In addition, the variances of +the learning curves shown in Figure 3A and Figure 3B illustrate that our method is more stable, with +minimal fluctuations in variance, while BrsDNC model exhibits extremely fluctuating and unstable +learning process. Overall, our MT-DNC model can improve the convergence speed and performance, +and has better stability. +Ablation Study. To further analyze the validity of our proposed model, we designed a series of +experiments for the ablation analysis. The main innovation of our model is the introduction of +long-term memory and the memory transformation algorithm. In MT-DNC, the long-term memory +module receives information input from the working memory module by the memory transformation +algorithm. Therefore, we verified the effectiveness of the memory transformation mechanism by +comparing MT-DNC-DI and MT-DNC. In MT-DNC-DI, the long-term memory module receives +inputs from the controller layer (with different parameters of working memory module). From +Table 1 and Figure 3, We discover that MT-DNC achieves superior performance compared with the +MT-DNC-DI, both in terms of WER on each sub-task and in terms of average WER. In addition, +the MT-DNC-DI model achieves higher performance and lower loss compared with DNC, BrsDNC, +8 + +A +B +DNC Model +3.5 +DNC Model +3.0 +BrsDNC Model +BrsDNC Model +MT-DNC Model +3.0 +MT-DNC Model +2.5 - +MT-DNC-DI Model +MT-DNC-DI Model +2.5 +Loss +2.0 +Loss +2.0 +Train I +1.5 +1.5 +1.0 +1.0 +0.5 +0.5 +0.0 +0 +5 +10 +15 +20 +25 +30 +35 +40 +0 +5 +10 +15 +20 +25 +30 +35 +40 +Epochs +Epochsand other models, indicating that the long-time memory itself has a certain effect, and the memory +transformation can further improve the performance of the model. +Figure 4: Mean Word Error Rate of MT-DNC-32, MT-DNC-64, MT-DNC-128, MT-DNC-256, +MT-DNC-512, BrsDNC. The horizontal coordinate represents the number of Epochs and the vertical +coordinate represents the changing of Mean Word Error Rate. +In addition, We analyzed the effect of storage space of long-term memory and working memory on +the experimental results. Figure 4 illustrates the changing of mean word error rate during learning +process at different storage spaces sizes. We also compared the change in mean WER of BrsDNC +model (black line in Figure 4) during the learning process. The experimental results reveal that when +the memory space is too small (only 32 and 64), the performance of the model will be affected. Our +model is only inferior to (with comparable performance) the BrsDNC model under very small 32 and +64 lengths of memory space, while the length of memory space in BrsDNC is 128 (larger than us). +And our MT-DNC model significantly outperforms the BrsDNC model at 128 and 256 memory space +lengths. In addition, we found that too much memory space (with 512 length) does not continue to +improve the performance, but leads to performance degradation. Overall, our model is robust and +adaptable to different memory space lengths, and too large or too small memory spaces do not work +as well as the most appropriate length. +5 +Conclusion +In this paper, inspired by the memory transformation mechanism of the human brain, we propose a +multiple memory modules coordinated MT-DNC model. Through the linkage between the working +memory area and the long-term memory area, this model solves the dilemma faced by DNC to a +certain extent, that is, as the amount of information in the memory area increases, its addressing and +training speed gradually increase, which greatly affects the convergence rate and final performance of +the model. Of course, there are still many points worth improving in this work in the future, such +as the ratio of the two memory areas and the optimization of the memory transfer mechanism. In +conclusion, this work is inspired from the cognitive mechanism of human learning new knowledge, +extends the memory module of the DNC model, improves the training speed of the original model +and the final performance, and surpasses the DNC-based reasoning models in 20 types of reasoning +tasks on the bAbI dataset. +6 +Acknowledgments +This work is supported by the National Key Research and Development Program (Grant No. +2020AAA0107800), the Strategic Priority Research Program of the Chinese Academy of Sciences +(Grant No. XDB32070100). +9 + +6.0 +5.5 + Error Rate +5.0 +4.5 +Mean Word +4.0 +3.5 +-□—MT-DNC-32 +-□— MT-DNC-64 +3.0 +—MT-DNC-128 +□—MT-DNC-256 +2.5 - +MT-DNC-512 +OBrsDNC +2.0 +15.0 +17.5 +20.0 +22.5 +25.0 +27.5 +30.0 +32.5 +35.0 +EpochsReferences +Atkinson, R. C. and Shiffrin, R. M. (1968). Human memory: A proposed system and its control +processes. In Psychology of learning and motivation (Elsevier), vol. 2. 89–195 +Ba, J., Hinton, G. E., Mnih, V., Leibo, J. 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of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Shanghai 200031,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' China 3University of Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Beijing 100190,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' China 4National Laboratory of Pattern Recognition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Institute of Automation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Beijing 100190,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' China ‡Co-first authors with equal contribution {liangyao2020,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='fanghongjian2017,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='zeng}@ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='cn Abstract Reasoning and question answering as a basic cognitive function for humans, is nevertheless a great challenge for current artificial intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Although the Differentiable Neural Computer (DNC) model could solve such problems to a certain extent, the development is still limited by its high algorithm complexity, slow convergence speed, and poor test robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Inspired by the learning and memory mechanism of the brain, this paper proposed a Memory Transformation based Differentiable Neural Computer (MT-DNC) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' MT-DNC incorporates working memory and long-term memory into DNC, and realizes the autonomous transformation of acquired experience between working memory and long-term memory, thereby helping to effectively extract acquired knowledge to improve rea- soning ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Experimental results on bAbI question answering task demonstrated that our proposed method achieves superior performance and faster convergence speed compared to other existing DNN and DNC models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Ablation studies also indicated that the memory transformation from working memory to long-term memory plays essential role in improving the robustness and stability of reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' This work explores how brain-inspired memory transformation can be integrated and applied to complex intelligent dialogue and reasoning systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' 1 Introduction Reasoning and Question Answering (QA) are advanced cognitive functions of human beings, and they are also one of the important challenges in the field of deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Differentiable Neural Computers (DNC) model proposed by Graves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' (2016) provides a feasible solution to study reasoning and QA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' DNC consists of a DNN-based computational controller and an external memory in which the neural network could learn and communicate (read and write) with memory module, and the memory module could represent and store learned structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' The DNC model has achieved good performance on various image reasoning and QA tasks Graves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Rasekh and Safi-Esfahani (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' However, it faces several main challenges such as ∗Corresponding authors Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Under review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='02809v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='AI] 7 Jan 2023 high algorithm complexity, slow convergence speed, and high average test error rate, which limit its further development and wider application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' The BrsDNC model Franke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' (2018) improves DNC model with the introduction of Normalization and Dropout, which shows to be more robust and salable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Actually, the main issues of current DNC models lie in the restricted memory may lead to missing critical knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' As the working time becomes longer, reading and writing pressure on memory module increases rapidly, thus limiting the training speed and performance of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Besides, existing methods lack references from brain learning and memory mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Thus, there is still much room for improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Memory in the brain includes short-term and long-term memory, etc Baddeley (2007);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Lee and Wilson (2002);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Winocur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' (2010);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Marshall and Born (2007);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Ji and Wilson (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' They play an important role in various cognitive functions such as learning, decision making, and reasoning, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Short-term memory has limited storage space and therefore cannot retain information indefinitely Diamond (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Then some memories are forgotten, others that are repeatedly reused will be retained and transferred to long-term memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Information can be stored in long-term memory for a longer period of time, continuously aiding learning and reasoning Atkinson and Shiffrin (1968).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' The collaboration and division between working memory and long-term memory help the human brain to consolidate and use the acquired knowledge more efficiently, and therefore perform multiple cognitive tasks more robustly Kitamura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Inspired by the learning and memory mechanism of the brain, we proposed a brain-inspired Memory Transformation based Differentiable Neural Computer (MT-DNC) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' MT-DNC expands only one memory area in the original DNC model into two different memory areas, namely the working memory area and the long-term memory area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Working memory stores information that is more closely related to the current task, while long-term memory stores more meaningful long-term knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' These two memory areas are connected through a Memory Transformation algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' The core principles of the memory transformation algorithm are: repeatedly visited knowledge will be transferred to long-term memory area, useless information will be discarded from working memory area Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' LeCun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' The innovations of our method are mainly reflected in the following aspects: MT-DNC combines short-term working memory with long-term memory to promote a more comprehensive memory of the acquired knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' We proposed a brain-inspired memory transformation algorithm to dynamically store and extract useful information and filter out the useless ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Reasoning-based question answering experiments demonstrated that MT-DNC could im- prove the accuracy and convergence speed on bAbI task compared to current DNC-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' The experimental results also illustrated the great significance of introducing the neural mechanism of memory transformation to improve the stability and robustness of model reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' 2 Related work Neural Turing Machine (NTM) : The core idea is to combine neural networks with external memory to expand the capabilities of neural networks and interact through an attention mechanism Graves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' To a certain extent, NTM can be compared to Turing machine Xiong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Zaremba and Sutskever (2015), and the experiment verifies the Turing completeness of NTM Tao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Zaremba and Sutskever (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' The biggest advantage of NTM in terms of function is to deal with complex tasks that require memory participation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Differentiable Neural Computer (DNC): DNC, known as the second version of NTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Its core idea is consistent with NTM, based on external memory to improve the ability of neural networks Graves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Santoro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Lake et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Compared with the original NTM, DNC has made important improvements in the addressing mechanism Hassabis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Chan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' (2018), removing the index shift operation, and better supporting the functions of allocate and de-allocate for memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Compared with the original NTM, there is also a certain improvement in performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' In recent years, several works have made some improvements on the basis of the DNC structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Franke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' (2018) improved the performance of the model by optimizing the memory module of 2 DNC, increasing the bidirectional connection between memory modules, and introducing the layer normalization Ba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' (2016b) training method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' By improving the details of addressing and memory allocation in DNC, Csordás and Schmidhuber (2019) made the model achieve better accuracy on the bAbI dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Rasekh and Safi-Esfahani (2020) integrated the NeuroEvolution algorithm into the DNC structure, and showed faster encoding speed in various cognitive tasks, and the model performance was improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' To sum up, none of these approaches fundamentally address the issues of low accuracy and slow convergence speed of DNC due to the constrained external memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' This paper takes inspiration from the learning and memory mechanism of the brain, and proposes the multiple memory modules coordinated MT-DNC model that integrates working memory and long-term memory Seo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Ba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' (2016a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Le et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' (2019, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' The proposed model could improve the accuracy and convergence speed, and bring about superior performance compared to other DNC-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' 3 Method In this section, we will give a comprehensive introduction to our MT-DNC method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' MT-DNC extends the memory module of the DNC to a working memory module and a long-term memory module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' The overall framework of MT-DNC includes controller layer, memory layer and linear layer, as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Figure 1: Overall architecture of MT-DNC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' 1) The controller layer is responsible for encoding and processing the input data and the output of the previous Memory Layer, learning the time series information from the training data, and transmitting the output results to the memory layer and the linear layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' 2) Memory Layer is responsible for storing the output of the controller layer and extracting the useful information through a series of storage and transformation mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' The memory layer incorporates memory transformation between working memory and long-term memory modules, enabling the MT-DNC model to have strong memory and reasoning capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' 3) The linear layer combines the output of the memory Layer and control layer as input, and outputs the final prediction result after linear transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' A more detailed description of our MT-DNC method is presented as follows: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 Controller Layer Control layer combines the original input data xt and the output of the previous time step memory layer OutputM t−1 as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' After linear operation with weight WC t and Normalization Klambauer 3 Linear Layer Long-term Working Memory Module Memory Module Memory Layer Controller LayerFigure 2: MT-DNC incorporates working memory module and long-term memory module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Franke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' (2018), the output OutputC t is transmitted to working memory module of memory layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' OutputC t = Normalization(WC t (xt + OutputM t−1) + bC t ) (1) Where xt ∈ RX, OutputC t ∈ RC, bC t ∈ RC, WC t ∈ R(2×W ×R+C+X)×C, OutputM t−1 ∈ R2×W ×R+C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Among them: X represents the dimension of the input data, C represents the output dimension of the control layer, M represents the output of the memory layer, R represents the head number of the read memory area, and W represents the width of the memory area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='2 Memory Layer The memory layer contains the working memory module, the long-term memory module and the memory transformation algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' The working memory module stores the latest interaction information from the controller layer, while the long-term memory holds the information that is frequently used with high importance but will be deleted by the working memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' In both working memory and long-term memory, a dynamic update and extract rule is required to continuously replace the store information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' The memory transformation algorithm selectively transfers the data from working memory to long-term memory for processing, and finally the memory layer combines the output of the controller layer, the output of working memory and the output of long-term memory to make a decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' The information processing procedure of our MT-DNC method is depicted in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 Working Memory Module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' The working memory module is functionally designed to store interactive information from the output of the controller layer in real time, updating and extracting related information according to the output of the controller layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Due to the limitation of storage space, we take inspiration from the update and decay mechanism of memory in the human brain and replace the information that is similar to the current interaction information (OutputC t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' In addition, information that has already been extracted or used is more likely to be replaced to retain as much innovative information as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' The memory area read, write and gating related signals are first generated from OutputC t by a linear transformation, detailed as Kw t ∈ RW , Kl t ∈ RW , Ew t ∈ RW , El t ∈ RW , Vw t ∈ RW , Kw,i t ∈ RR×W , Kl,i t ∈ RR×W , βw t ∈ R, fw,i t ∈ RR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' 4 Controller Layer Linear Layer gindino Output?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Read and Write Update Memory Extract Information Memory Layer Working Memory Long-Term Memory Transform Uy UWorking Memory Updating Algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' The updating of working memory is based on the follow- ing principles: 1) Delete the items that has not been used for a long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' 2) Delete the ones that has just been extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' 3) Delete similar items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' 4) Retain the novel ones that have been updated recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Based on the above principles, we update the working memory in real time according to the dynamic addressing algorithm in Graves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' ψw t = R � i=1 (1 − fw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='i t Ww,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='i t−1) Uw t = (Uw t−1 + Ww t−1 − (Uw t−1 ◦ Ww t−1)) ◦ ψw t φw t = SortIndiceAscending(Uw t ) aw t [φw t [j]] = (1 − Uw t [φw t [j]]) j−1 � i=1 Uw t [φw t [j]]] (2) Where ψw t ∈ RN is the result of scaling and accumulating the read weight matrix Ww,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='i t−1 of the previous time step through the fw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='i t gated tensor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' The φw t ∈ RN tensor is the index tensor sorted in ascending order by the memory area management tensor Uw t ∈ RN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' N represents the length of the memory area,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' aw t ∈ RN is the dynamic addressing based write weight of the working memory area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Ww,c t = exp(d(Kw t , Mw t )βw t ) � exp(d(Kw t , Mw t )βw t ) (3) Where Ww,c t ∈ RN, βw t ∈ R, Kw t ∈ RW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' N represents the length of the memory area, and W represents the width of the memory area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Ww t = γw t [gw t aw t + (1 − gw t )Ww,c t ] Mw t = Mw t−1 − Mw t−1 ◦ Ww t (Ew t )T + Ww t (Vw t )T (4) Where Mw t ∈ RN×W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' gw t ∈ [0, 1] represents the write weight allocation gate tensor in the working memory area, which is used to control the allocation proportion of the two addressing modes in the final write.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' γw t ∈ [0, 1] is used to avoid data from being flushed Working Memory Extracting Algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' The working memory extracts the information that most relevant to the current interactive information (OutputC t ), then this returned information is the used item at the current moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' The extracting algorithm is Graves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Ww,i t = exp(d(Kw,i t , Mw t )βw,i t ) � exp(d(Kw,i t , Mw t )βw,i t ) (5) Where Ww,i t ∈ RR×N, Kw,i t ∈ RR×W , βw,i t ∈ RR, read R times in total, The label is i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' N represents the length of the memory area, and W represents the width of the memory area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Rw,i t = (Mw t )T Ww,i t (6) Where Rw,i t ∈ RR×W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='2 Memory Transformation from Working Memory to Long-term Memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' The DNC-based model Graves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Franke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' (2018) directly maps the output of the working memory (Rw,i t ) to the linear layer, and since the used items will be deleted in the working memory, this will lead to the loss of some important information, which in turn affects the performance 5 and robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' In this paper, we design a memory transformation algorithm to transfer the extracted information from working memory to long-term memory, thus compensating for the information loss caused by the update in working memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' The algorithm for updating and extracting information in long-term memory are similar to that in working memory, the only difference is that the input in working memory originates from the controller layer and the input in long-term memory originates from the working memory layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' The update formula for the long-term memory layer is as follows: Wl t = γl t[gl tal t + (1 − gl t)Wl,c t ] Bw t = R � i=1 Rw,i t Ml t = Ml t−1 − Ml t−1 ◦ Wl t(El t)T + Wl t(Bw t )T (7) Where Ml t ∈ RN×W , Bw t ∈ RW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' gl t ∈ [0, 1] represents the long-term memory write weight allocation gate tensor, which is used to control the allocation proportion of the two addressing modes in the final write.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' The information extraction from the long-term memory layer integrates the information from the long-term memory layer Rl,i t as well as the information from the working memory layer Rw,i t and is calculated as follows: OutputM t = Rw,i t + Rl,i t (8) Where Ri t ∈ R2×R×W , OutputC t ∈ RC, OutputM t ∈ R2×W ×R+C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Algorithm 1 Execution algorithm for MT-DNC Input: Training set xt,yt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Output: The MT-DNC model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' 1: randomly initialize weight W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' 2: for e = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' e < Epoch;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='e + + do 3: %Forward propagation 4: xt = xt + inverse(xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' 5: As shown in Eq 1, xt and OutputM t−1 are used as input data for the control layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' 6: After processing by Eq 2, 3, 4, 5 and 6, the output Rw,i t of the working memory area is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' 7: Rw,i t will be input to the long-term memory area and processed by Eq 2, 3, 7, 5 and 6 to generate Rl,i t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' 8: After the processing of Eq 8, the memory layer output tensor OutputM t is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' 9: After the processing of Eq 9, the model output ˆyt is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' 10: %Back propagation updating W 11: The difference between yt and ˆyt is optimized by Cross Entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' 12: end for 13: return MT-DNC model 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='3 Linear Layer The output of the linear layer ˆyt is determined by the output of the controller layer OutputC t after Dropout processing Franke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Gal and Ghahramani (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Srivastava et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' (2014) and the output of the memory layer OutputM t , given by: ˆyt = Softmax(WO t (OutputM t + Dropout(OutputC t )) + bO t ) (9) Where ˆyt ∈ RY , WO t ∈ R(2×W ×R+C)×Y is the output parameter matrix, bO t ∈ RY is the bias matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' The detailed procedure of our MT-DNC is shown in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' 6 4 Experiments 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 bAbI task description The bAbI1 is a reasoning-based text question-and-answer task Weston et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' We choose en-10k as the experimental dataset, which contains 20 sub-tasks, each consisting of a training dataset text file containing 10k questions and a test dataset text file containing 1k questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' A joint training approach is used to verify the text comprehension and reasoning ability of the MT-DNC model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Unlike other previous related work, our method uses end-to-end training without any pre-processing on the bAbI dataset itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='2 Training details bAbI question and answer task composed of 20 sub-tasks is combined in one training session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' A training sample is generated for each subtask in the dataset in terms of different stories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' The detailed generation process is as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' The text sequence training samples are processed by removing digits, converting words from upper case to lower case, removing line breaks, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Cut the text sequence training sample into a list of sequences with words (including 3 punctuation marks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Replace ’answer words’ in the list with ’-’, and get a list of training input samples after encoded into word vectors by one-hot word vector processor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' The length of the list is the length of the largest text sequence of the current batch, and the text with insufficient length is filled with ’0’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' A word in the list is represented as xt ∈ RZ, where Z is the length of the word vector with a value of 159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' All training input samples and target samples form the training sample list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' 10% of the data in the training sample list is used as the validation dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' MT-DNC model performs 300 epochs of training, validation and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' The total number of parameters in the model is 1267337 and the batch size of the data is 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' The number of control level nodes is 172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Both memory areas have a length of 128 and a width of 64, 4 read heads, 1 write head and a dropout of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' The learning rate is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0003, the Momentum value of the optimizer Rmsprop is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='9 Kingma and Ba (2014), and the gradient clipping value is 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' It takes about 1 hour to run an epoch using an RTX3060 graphics card, an Intel i7 processor, and 16G of RAM for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='3 Experimental results To verify the effectiveness of the proposed MT-DNC model, we conducted comparison experiments with DNC, EntNet Henaff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' (2016), LSTM Hochreiter and Schmidhuber (1997), SDNC Rae et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' (2016), BrsDNC Franke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' (2018) and other models on the bAbI question and answer task dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' In addition, we also compared the MT-DNC-DI model (our MT-DNC without memory transformation) to verify the effectiveness of memory transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' The MT-DNC-DI model refers to our model without memory transformation, which directly uses independent memory modules with dual regions of working memory and long-term memory (both receive input from the controller layer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Table 1 shows the average word error rate (WER) of different methods under seven different initialized parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' According to the experimental results, it can be seen that the MT-DNC model achieves a lower average error rate (only 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='5% of mean WER) than other models, especially the current BrsDNC model which reaches State Of The Art (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='2% of mean WER) on the 20 bAbI tasks with joint training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' In particular, on the tasks of 8th, 14th, and 15th sub-tasks, all other methods have errors, while our method achieves an error rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' For 16th and 17th sub-tasks, compared to the best performance achieved by BrsDNC, our method could significantly reduces the error rate by 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='7% and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='7%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' We also counted the number of failed tasks (more than 5% errors) among the 20 tasks, as shown 1https://research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='facebook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='com/downloads/babi/ 7 Task DNC EntNet LSTM SDNC BrsDNC MT-DNC-DI MT-DNC 1: 1 supporting fact 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 ± 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 2: 2 supporting facts 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='2 ± 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='3 ± 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='7 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 ± 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 3: 3 supporting facts 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='6 ± 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='4 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='3 ± 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='3 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='4 ± 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='4 4: 2 argument relations 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 5: 3 argument relations 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 6: yes/no questions 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='9 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='8 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 7: counting 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='8 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='2 8: lists/sets 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='5 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='2 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 9: simple negation 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='7 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 10: indefinite knowledge 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='6 ± 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='2 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 11: basic coreference 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='3 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='2 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='2 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 12: conjunction 5 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 13: compound coreference 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='2 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 14: time reasoning 11 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='3 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='5 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='6 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 15: basic deduction 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='2 ± 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='6 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='6 ± 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 16: basic induction 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='9 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='2 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='3 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='3 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='6 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='9 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 17: positional reasoning 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='4 ± 8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='8 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='8 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='8 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='4 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='8 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='2 18: size reasoning 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='9 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 19: path finding 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='6 ± 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='4 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='4 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='8 ± 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 20: agents motivation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 Mean WER: 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='7 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='7 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='6 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='4 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 Failed Tasks (>5%): 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='2 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='2 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 Table 1: The average word error rate(WER) of different models on bAbi task in the last row of Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Our method has only 1 failed task and achieves superior performance compared to other methods, significantly outperforming DNC (with 11 failed tasks) and LSTM (with 17 failed tasks) methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Figure 3: Validation loss (A) and training loss (B) of DNC, BrsDNC, MT-DNC-DI and MT-DNC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' The horizontal coordinate represents the number of Epochs and the vertical coordinate represents the changing of loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Figure 3 depicts the changing of loss for different methods during validation (Figure 3A) and training (Figure 3B) processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' It can be seen that our MT-DNC model shows lower loss, higher performance and faster convergence speed compared with DNC and BrsDNC models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' In addition, the variances of the learning curves shown in Figure 3A and Figure 3B illustrate that our method is more stable, with minimal fluctuations in variance, while BrsDNC model exhibits extremely fluctuating and unstable learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Overall, our MT-DNC model can improve the convergence speed and performance, and has better stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Ablation Study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' To further analyze the validity of our proposed model, we designed a series of experiments for the ablation analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' The main innovation of our model is the introduction of long-term memory and the memory transformation algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' In MT-DNC, the long-term memory module receives information input from the working memory module by the memory transformation algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Therefore, we verified the effectiveness of the memory transformation mechanism by comparing MT-DNC-DI and MT-DNC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' In MT-DNC-DI, the long-term memory module receives inputs from the controller layer (with different parameters of working memory module).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' From Table 1 and Figure 3, We discover that MT-DNC achieves superior performance compared with the MT-DNC-DI, both in terms of WER on each sub-task and in terms of average WER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' In addition, the MT-DNC-DI model achieves higher performance and lower loss compared with DNC, BrsDNC, 8 A B DNC Model 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='5 DNC Model 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 BrsDNC Model BrsDNC Model MT-DNC Model 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 MT-DNC Model 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='5 - MT-DNC-DI Model MT-DNC-DI Model 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='5 Loss 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 Loss 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 Train I 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 Epochs Epochsand other models, indicating that the long-time memory itself has a certain effect, and the memory transformation can further improve the performance of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Figure 4: Mean Word Error Rate of MT-DNC-32, MT-DNC-64, MT-DNC-128, MT-DNC-256, MT-DNC-512, BrsDNC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' The horizontal coordinate represents the number of Epochs and the vertical coordinate represents the changing of Mean Word Error Rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' In addition, We analyzed the effect of storage space of long-term memory and working memory on the experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Figure 4 illustrates the changing of mean word error rate during learning process at different storage spaces sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' We also compared the change in mean WER of BrsDNC model (black line in Figure 4) during the learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' The experimental results reveal that when the memory space is too small (only 32 and 64), the performance of the model will be affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Our model is only inferior to (with comparable performance) the BrsDNC model under very small 32 and 64 lengths of memory space, while the length of memory space in BrsDNC is 128 (larger than us).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' And our MT-DNC model significantly outperforms the BrsDNC model at 128 and 256 memory space lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' In addition, we found that too much memory space (with 512 length) does not continue to improve the performance, but leads to performance degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Overall, our model is robust and adaptable to different memory space lengths, and too large or too small memory spaces do not work as well as the most appropriate length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' 5 Conclusion In this paper, inspired by the memory transformation mechanism of the human brain, we propose a multiple memory modules coordinated MT-DNC model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Through the linkage between the working memory area and the long-term memory area, this model solves the dilemma faced by DNC to a certain extent, that is, as the amount of information in the memory area increases, its addressing and training speed gradually increase, which greatly affects the convergence rate and final performance of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' Of course, there are still many points worth improving in this work in the future, such as the ratio of the two memory areas and the optimization of the memory transfer mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' In conclusion, this work is inspired from the cognitive mechanism of human learning new knowledge, extends the memory module of the DNC model, improves the training speed of the original model and the final performance, and surpasses the DNC-based reasoning models in 20 types of reasoning tasks on the bAbI dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' 6 Acknowledgments This work is supported by the National Key Research and Development Program (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' 2020AAA0107800), the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' XDB32070100).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' 9 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='5 Error Rate 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='5 Mean Word 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='5 □—MT-DNC-32 □— MT-DNC-64 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 —MT-DNC-128 □—MT-DNC-256 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='5 - MT-DNC-512 OBrsDNC 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content='5 25.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} +page_content=' IET Intelligent Transport Systems 11, 68–75 11' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQf-AKd/content/2301.02809v1.pdf'} diff --git a/xdAzT4oBgHgl3EQf7v58/content/tmp_files/2301.01894v1.pdf.txt b/xdAzT4oBgHgl3EQf7v58/content/tmp_files/2301.01894v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e0927123a7bbbac8c37fa7ab88948f98b9c30f30 --- /dev/null +++ b/xdAzT4oBgHgl3EQf7v58/content/tmp_files/2301.01894v1.pdf.txt @@ -0,0 +1,1019 @@ + +Preprint submitted to +1 +Urban-Semantic Computer Vision: +A Framework for Contextual Understanding of People in Urban Spaces + + +Anthony Vanky, Ph.D., Corresponding author +Graduate School of Architecture, Planning and Preservation +Columbia University, New York, NY 10025 +Email: a.p.vanky@columbia.edu + +Ri Le +Graduate School of Architecture, Planning and Preservation +Columbia University, New York, NY 10025 +Email: r.le@columbia.edu + + + + +Preprint submitted to +2 +KEYWORDS + +artificial intelligence, computer vision, urban space, urban context, urbanism, semantic meaning, thick description, +evaluation + + +ABSTRACT + +Increasing computational power and improving deep learning methods have made computer vision technologies +pervasively common in urban environments. Their applications in policing, traffic management, and documenting +public spaces are increasingly common. Despite the often-discussed biases in the algorithms' training and unequally +borne benefits, almost all applications similarly reduce urban experiences to simplistic, reductive, and mechanistic +measures. There is a lack of context, depth, and specificity in these practices that enables semantic knowledge or +analysis within urban contexts, especially within the context of using and occupying urban space. This paper will +critique existing uses of artificial intelligence and computer vision in urban practices to propose a new framework +for understanding people, action, and public space. + +This paper revisits Geertz's use of thick descriptions in generating interpretive theories of culture and activity and +uses this lens to establish a framework to evaluate the varied uses of computer vision technologies that weigh +meaning. We discuss how the framework's positioning may differ (and conflict) between different users of the +technology. This paper also discusses the current use and training of deep learning algorithms and how this process +limits semantic learning and proposes three potential methodologies for gaining a more contextually specific, urban- +semantic, description of urban space relevant to urbanists. + +This paper contributes to the critical conversations regarding the proliferation of artificial intelligence by +challenging the current applications of these technologies in the urban environment by highlighting their failures +within this context while also proposing an evolution of these algorithms that may ultimately make them sensitive +and useful within this spatial and cultural milieu. + + + + + +Preprint submitted to +3 +1. INTRODUCTION + +Ever-powerful computational ability, the reduced cost of communications infrastructure, and the increase- +diminishing size of sensors have enabled the pervasive placement of technologies into the fabric of urban spaces, +birthing a movement of the “smart city.” For many in the so-called smart cities movement, the trend has been +towards the instrumentalization of cities, finding greater efficiencies, and the problematization of many facets of +urban living (Hollands, 2008). For technologists working in this field, the enumeration game is being applied to all +domains ranging from mobility and infrastructure to public safety and democratic participation (Eagle & Pentland, +2009; Goldsmith & Crawford, 2014; Jiang et al., 2013). Nevertheless, the defining social characteristics of urban +space defy a reduction to a mere optimization problem. + +These digital technologies and their resultant models and data outcomes have the ability to shape our perspective of +the built environment. No different than the well-publicized challenges of bias prevalently found in other +algorithmic processes (Crawford, 2018; Kirchner et al., 2016), the black box methodologies and opaque outcomes so +too can unduly influence our reading of the places we inhabit (Schwarzer, 2017). Despite this conflict between +reductivism and complexity, there is an urgent need to understand through new models and tools may open new +avenues for research into public space and urban form in light of rapid urbanization and the increased privatization +of urban space (E. Talen & Ellis, 2002). + +As if it were an iteration from the modernist use of photography in urban planning, the growing ubiquity of deep +learning and computer vision applications have created new opportunities to understand cities through imagery +(Lecun et al., 2015). While there is optimism in how these emerging technologies can allow for a more precise (and +perhaps, broad) method for understanding cities, questions remain. + +Within this computational milieu, this paper focuses specifically on the nascent, but growing role of these +algorithmic tools are being applied to urban planning and management. In one sense, they quantify human behavior +in urban space that offers the ability for decision-makers to base policy in more informed ways. In another, this +numerical reductivism applied to urban space is blind to the specificity and essential character that makes cities +unique places of inhabitation. As such, this paper argues that in addition to situated technologies’ reductivist +orientation, there is a need for a distinct approach to their use in the understanding of how people inhabit and use +these public urban spaces. Further, with the increased proliferation of computer-vision and image-based approaches +toward the instrumentalization of cities, an urban-specific lexicon to the training, implementation, and adoption of +urban technologies. + +While the use of computer vision technologies has spanned many facets of urban management, such as infrastructure +utilization and public safety, this paper considers explicitly using these technologies to understand how people +inhabit and use public space. This paper argues that image-based artificial intelligence is a natural progression in the +modernist use of photo imagery to capture data on the city. This paper also reviews, in brief, how artificial +intelligence is being applied to image-based data to illustrate the potential weaknesses in creating thicker, domain- +specific ontologies about the occupants and the spaces in which they inhabit. Further, this paper discusses how thin +data is being marketed by both public and private actors to the potential detriment of planning human-centric spaces. +However, this paper also discusses potential approaches to reconceptualize the methodologies currently used to get +toward an urban-semantic description of public space using these algorithmically-based methodologies despite these +limitations. + +2. “URBANISM”, AND THE ISSUE OF CONTEXT + +A fundamental challenge is defining what characteristics make a space or practice urban, especially when +contemporary city building is neo-liberal and capital driven. In other words, what makes Washington Square Park + + +Preprint submitted to +4 +urban while the Hudson Yards feel not, despite being just a few kilometers apart in Manhattan? Or what makes +Seoul’s Namdaemun Night Markets an urban experience versus the nearby, impressively large Lotte Shopping +Center? + +By their nature, urban space is where individual experience comes together with strangers, even though they seldom +share our values, history, and perspective. It is in these spaces where the mingling and contact with individuals and +groups differ in their social presentation, appearance, and experience. As urbanist Michael Brill (1989) frames these +experiences, it is where inhabitants “can seek and find excitement and extraordinariness in the productions and +presentations created by strangers, and in those they create themselves.” The spaces around them influence these +social dynamics; the form and configuration of urban space frame the interactions of those in them. Ultimately, +public life is uniquely able to be experienced in these spatial commons—the streets and public, urban spaces— +because of how they make inhabitants' acute awareness of their dependence on one another, as well as the societal +obligations that dependence spawned (Chidster, 1989). Cities are, in a sense, where individualism intermixes to +create collective experience. + +Defining these socio-spatial interrelationships is a difficult challenge, and there exist differing perspectives on the +nature of a good public realm. There exist divergent but parallel theories about the character of these spaces, which +take up a significant portion of the physical city: streets, sidewalks, squares, arcades, non-motorized transport, +greenways, and the like. Jane Jacobs (1961) describes the importance of streets as public spaces: + +“Streets are almost always public: owned by the public, and when we speak of the public realm, we are +speaking in large measure of streets. If we can develop and design streets so that they are wonderful, +fulfilling places to be, community-building places, attractive public places for all people of cities and +neighborhoods, then we will have successfully designed about one-third of the city directly and will have +had an immense impact on the rest.” + +The sidewalk is, as she called them, “the main public places of the city” and “its most vital organ.” In a similar vein, +Jan Gehl (1987) calls attention to the life between buildings. It is in the spaces between architecture where the social +connections of inhabitants are created and reaffirmed and where the space of movement coexists with the city's +social life. These aspects form inter-related patterns of movement and activity to which buildings, in turn, respond. +To Kevin Lynch (1960), this life also imbues these spaces with meaning, whereby they become “imageable,” +containing spatial qualities that strengthen the attention and meaning of urban space to its inhabitants. Such spaces +establish memorable relationships among buildings, paths, features, and amenities due to their proximity to one +another or the geometric configuration of the space itself. + +Ultimately, these concepts are intended to influence the practice of place-making or the shaping of urban spaces that +support the public activities mentioned. Designers are often unaware of their own biases and roots as influencing +their decision-making (A. Jacobs & Appleyard, 1987). The profession of urban planning would be helped by a +renewed quest for the elements of “good city form,” where the planning of the physical city is concentrated on +people and place (Emily Talen & Ellis, 2015). For designers and planners, normative limits such as time and +budgets are often drivers of the opportunities available to improve urban space. However, they are often exacerbated +by a lack of understanding of how both users and non-users perceive them and the characteristics of users (Chidster, +1989). Such an endeavor would have to deal with the complexities of aesthetic, ethical, and political theory to secure +its foundations and, as such, require domain-specific ontologies unique to the social and physical milieu of +urbanism. + + + + + + +Preprint submitted to +5 +3. IMAGERY AS URBAN DATA + +Imagery had enormous agency in being evidence of human activity in public space. Our contemporary era has seen +the ubiquity of cameras as tools used to documentation of social ills and injustice. Videos of vegetable seller +Mohamed Bouazizi setting himself on fire in protest in Tunisia (Noueihed, 2011) or of George Floyd gasping “I +can’t breathe” as police offers aggressively restrained him (J. Collins, 2020) have served as evidence for protests +demanding justice and change. Yet, the use of photographic imagery as a data source for learning and exploring +urban space dates as far back as the technology itself. + +Louis Daguerre’s View of the Boulevard du Temple, created in 1938, was the earliest photograph to include people +in a busy street. Although it depicts two men on a street corner, they appear only because one has his shoes polished +by the other and thus required them to be stationary. The long exposure required left no trace of moving traffic and +other people. In the 1860s, the nascent art of urban photography became a political tool for social and urban reform. +In Paris, Charles Meville’s commissions to photo-document the modernization efforts of Baron Georges-Eugène +Haussmann recorded both dirty and cramped “before” and grand, triumphalist “after” vignettes of the city. +Melville’s photographs have been the fodder for debates on whether they served more than an archive of a foregone +Paris on behalf of the city by also acting as propagandist justifications for the razing of many neighborhoods +(Dahlberg, 2015). Historian Shelley Rice’s (1997) critiques of Meville’s repetitious and “clinical” streetscapes point +to the photographer’s “passivity” toward the grand transformations of the city. On the other side of the Atlantic, +Jacob Riis, while working as a police reporter for the New York Tribune, documented the slum conditions on the +Lower East Side of Manhattan. Using lantern slide shows, illustrated articles, and printed books, Riis harnessed the +power of photographic imagery as evidence of the need for housing and social reform, which galvanized the +politically connected gentile and policymakers to action and reform. + +Where once rare, the twentieth century saw the increased ubiquity of cameras which served in the modernist era as +tools that could provide an unblinking empiricism in the documentation of human behavior. Powered by the +increased availability to take photos from aircraft and satellites, planners and architects were drawn to the newly +available planar perspective afforded by flight as a way to document the formal development of the city. For Le +Corbusier (1935), the eye of the airplane offered a “pitiless”, objective record of reality, and for many planners in +the United States and Europe in the post-war era, aerial photographs served as evidence of urban blight. This +documentation provided the opportunity of seeing the world anew, but also as confirmation for the need to replan +(Hinchcliffe, 2010). In their seminal work, Learning from Las Vegas, Venturi and Scott Brown (1972) experimented +with various visual media to explore ways of representing American urban form in the late-1960s. As a critical +exploration about the automotive orientation and iconographic expressiveness of Las Vegas, their Yale research +group explored the city using photography and film as both mechanisms for data collection and representation. +Notably, their post-modernist approach adopted “drive-by photography” taken from inside moving cars, a technique +borrowed from the artist Ed Ruscha, as a means of contextual representation. + +Taken together, the imagery serving as data also produces meaning as understood by people. In investigating +urbanism, what is learned about cities and their inhabitants is both mediated and shaped by the characteristics and +limitations of the media and the mode of production (Azar et al., 2021). While historic precedent relies on the +curation of the framing and representation, emerging technologies are increasingly less reliant on humans—images +are increasingly being made by machines, for machines (Paglan, 2016). This computational process forms a shift in +how the production of urban knowledge, vis-à-vis enumeration, data, or models, is being generated and understood, +while remaining potentially invisible to the human altogether. + + + + + + +Preprint submitted to +6 +4. IMAGE-BASED ANALYSIS IN THE “SMART CITY” + +From the horse to the automobile to the telephone, the physical city, as a social artifact, has long evolved in response +to prevalent technologies of the day. Today, the rise of the so-called “smart city” introduces two parallel +technological interventions: a digitally connected citizenry possessing connected devices such as mobile phones and +the ubiquitous instrumentation of the physical artifice of the city through the “internet of things.” Pervasive sensing +enabled by both has created the ability to dynamically sense, analyze and understand individual mobility traces more +quickly and accumulate detailed knowledge over time to see patterns and trends. + +This “big data” approach—having access to large volume datasets to study phenomena and their dynamics— +augments the process by which urban space is designed, developed and evaluated, and offers opportunities for data- +driven analysis and design of the built environment. McLuhan foresaw technologies serving as civic thermostats “to +pattern life in ways that will optimize human awareness” (Norden, 1969). He said, “already, it is technologically +feasible to employ the computer to program societies in beneficial ways.” He stressed that “the programming of +societies could actually be conducted quite constructively and humanistically.” Greenfield (2013) comments that +“the final intent of all this... is to make every unfolding process of the city visible […], to render the previously +opaque or indeterminate not merely knowable but actionable.” + +Despite its complexity, the desire to make understandable the complex nature of vibrant cities has been both a dream +of Euclidean modernists and those who pursue an organic method of urban planning. For instance, for urbanist Jane +Jacobs, “the variables are many, but they are not helter-skelter; they are interrelated into an organic whole.” Jacobs +was talking about Hudson Street in 1960s-era Manhattan (J. Jacobs, 1970), but the same could be said of any street +in any city today. The critical difference is that today, digital data allows us to describe and analyze Jacobs’s +“interrelated variables” better than she could ever have imagined. Big Data may lead to the discovery of fascinating +dependencies and intimate knowledge. (Jemielniak, 2020) + +As city officials, urbanists, and urban planners analyze the figurative reams of data available; imagery has also been +leveraged as a means of understanding larger urban dynamics. For instance, Girardin et al. (2008) used individually- +uploaded photographs on the online photograph sharing platform Flickr to assess the different visitation footprints in +Barcelona of domestic, Spanish tourists, and international Britons visiting the country. To do so, the researchers +analyzed the underlying exchangeable image file (EXIF) that stores metadata created by the camera and user- +volunteered information such as locations, profile demographics, including the countries and cities where the +photographer was from, tags and descriptions. The fusion of this information provided insight into how different +groups may use a city differently (Offenhuber et al., 2013): the British tend to stay to primary, famed tourist sites +such as La Ramba, while Spaniards were willing to go farther afield. It also reaffirmed notions of the “imageable +city,”1 exposing locations that attracted photographers’ attention (monuments, churches, public spaces, for instance) +were exposed, and where the absence of images may reveal locales considered more introverted. + +Although the use of volunteered imagery is still commonly used, Street View images have provided a dataset that is +both immensely comprehensive in its coverage (thanks to the deep financial resources of its creator company, +Google) and consistent in its format. This undertaking was driven by co-founder Larry Page’s belief that this type of +street-level imagery contained a tremendous amount of information that could be organized, made available, and +mined (Anguelov et al., 2010). By 2017, the company captured 16 million kilometers of Street View imagery across +83 countries (Ackerman, 2017). This powerful dataset, albeit through restrictive commercial and proprietary means, +has allowed researchers to investigate street-level dynamics and changes of urban space at a massive, aggregated +scale from understanding the prevalence of tree canopies and green space (Li et al., 2015) to pedestrian counts (Yin +et al., 2015). + +1 “Imageable” here is taken to mean a cognitive, memory-based image as was used in the previous reference of Lynch’s work. + + +Preprint submitted to +7 + +However, several projects have sought to use this dataset to assess more ephemeral or culturally specific attributes +about these spaces. Salesses et al. (2013) used the street-level images in an online survey of people’s reactions to +whether a space looked more or less safe than another to index the perceived safety of a specific location in many +cities around the world, based on visible characteristics such as street width, program use, apparent blight, among +many other formal attributes seen in the photograph. Taken together, Naik & Philipoom (2014) use machine +learning algorithms to scale the survey results into an index applicable to many other cities around the world. The +EthinCITY Linguistic Landscape project from the Spatial Analysis Lab (2019) captured 9 million textual signs on +the streets of Los Angeles—ranging from official street names to business signs and advertisements. By extracting +the text and language of the message and geocoding the words and languages, the team assessed the languages used +to the parcel level. What emerged is a dynamic map that reflects the cultural diversity of tableau that traditional +government sources of data cannot collect, including small, ethnic establishments. In addition to macro-scale +findings such as there being not just one “Chinatown” in the city but five smaller ones, they could redefine notions +of ethnic enclaves as static neighborhoods and instead explore the dynamic, overlapping, and interspersed nature of +urban cultural diversity. + +While user-generated imagery comes at extraordinary velocity, volume, and veracity, a general lack of +standardization of what is captured2 and when limits the potential for a deeper inquiry into specific locales. For +instance, while there are over 2 million photographs of “Disneyland” in California on Flickr, there is only one of +“Captain Kidd's Family Dining” located just across the street from the entrance to the theme park. Moreover, while +Street View images remedy the aspects of standardization at a massive scale, the enormous cost of capturing the +images limits the potential to understand the whole dynamic nature of cities in real-time. However, the emergence of +smart city technologies through placed sensors provides opportunities for deeper urban inquiry that addresses the +standardization and volume concerns while providing the opportunity to understand places in real-time and over +longer durations. + +4.1. Automated Eyes on the Street + +With the emergence of the smart cities movement, much investment in sensing has focused on the domain of +computer vision, advanced machine learning (or deep learning), and more specifically, Convolutional Neural +Networks (CNN) to perform analytics on a single image or a series of images. This artificial intelligence is +concerned with the automatic analysis and extraction of useful information from these images or videos. In other +domains, computer vision has opened new frontiers in disease treatment and diagnosis, agricultural and industrial +efficiency, robotic navigation and self-driving vehicles, and more. In cities, this automated technology is being used +in wildlife tracking, intelligent traffic management, pedestrian flow monitoring, public safety, and infrastructure +management (Kwet, 2020). + +This boom is following a concomitant growth in the number of surveillance cameras being placed in the built +environment for various reasons and by a multitude of public and private parties (Lin & Purnell, 2019). +It is estimated that the number of surveillance cameras globally will exceed one billion by the end of 2021, with over +85 million in the United States alone, rivaling China’s per person camera penetration rate (IHS Markit, 2019). While +cameras on their own lack the overall ability to do the computational performance being described, it is estimated +that 350 million of these situated cameras will possess built-in technology to allow for artificial intelligence in 2025, +with more than 65 per cent of cameras shipped in that year will come with at least one AI chipset (ABI Research, +2021). + + +2 These together form the commonly used “four V’s” of big data: velocity, veracity, volume and variety. + + +Preprint submitted to +8 +To enumerate and describe the behaviors of inhabitants, these smart devices apply a domain of vision-based +machine learning called “action recognition,” which seeks to extract peoples’ identifiable physical features, acts, and +behaviors. Generally, all computer vision approaches generate data abstractions from the captured images and +videos, thereby reducing the image to a series of numerical matrices that can be more easily analyzed.3 These +matrices are produced by assessing patterns in the red-green-blue (RBG) pixel values of a digital image (and depth +information, if available). They are compared against pre-trained models developed from the analysis of prior +imagery. The creation of this pre-trained model is vital as it provides the foundation for comparison, much like how +humans learn how to identify objects: we learn by establishing priors—formal characteristic, visual appearance, +defining features, or other attributes—to be able to hypothesize about the nature of something new. To get to the +point of accurate enumeration, enormous data is required to train the statistical and deep learning models to ascertain +human action from noise. + +These processes are often ambivalent to the identification of a human as a specific individual unless they are +specifically focused on that task, such as through the use of facial recognition technology. Generally, the first step in +identifying human action is to identify features in an image that statistically resembles the figure of a person. Here, +one can think of this abstracted figure as a blob, box, or skeleton (Figure 1). Each of these offers different tradeoffs +in the computational resources and statistical strength but represents humans in different ways. + + + +Figure 1. How a computer interprets subjects and actions in an image. (Original photograph credit: Chetan, 2019.) + +The computationally simplest method is blob detection, whereby an algorithm can assess a figure in the foreground +of a video as being distinct from the background by comparing attributes that change between frames. This approach +toward blob detection is very frequently used in describing movement flows. In Figure 1, we simply identify the +hotspot of where the person is or has been as a measure of where the figure has been relative to the camera’s view, +establishing the background frame. The dichotomous nature of this foreground-background comparison can be used +to make the assignment of actions computationally more efficient by focusing the more intense box or skeleton +analysis on a smaller portion of the total image, however. Within this subset, an algorithm can do one of several +things: compare track this boxed subset within the frame, compare the RBG pixel values against the trained models +(classification), or further reduce the figure to the relationships of certain parts as probable limbs or critical physical +features vis-à-vis a digital, topological skeleton. Ultimately, it is the comparison of these two reductions, the boxed +pixel values or the skeleton, that is used to assess human action algorithmically. These actions can be categorized +according to the complexity of activity: primitive, single person, interaction, and group (Al-Faris et al., 2020), and +form the foundation for the computer assessment of activity. The combination of these methods and categories +allows us to identify individuals' activities in urban space. + +3 While this paper will not comprehensively review the technology and its application depth, other papers have sought to categorize various +approaches. See Ibrahim et al., 2020, for instance. + +1p1 83.43 +AL +Preprint submitted to +9 +4.2. The Players and Builders + +The use of the term smart cities has been characterized as incorporating computational tools into the operations and +management of municipalities and regions. However, this also brought a different paradigm of urban improvement +as the private sector played a more significant role in city-building. Many cities saw the implementation of these +technologies as two-fold endeavors: a means of economic development in an increasingly globally competitive +marketplace (Zukin, 2020), and in light of diminished resources, a means of outsourced services management. +Particularly in light of the inability of city bureaucrats to develop or implement new data collection means, greater +attention has been paid to technology companies, startups, and academics to create and implement these new +technologies within the urban domain. + +Fundamental to planning or policymaking is the restructuring of complexity to a discrete set of standardized +measurements that are understandable by those shaping the environment. Normatively, these measures are drawn +from a desire to benchmark the use or inhabitation of public space. Much of the development of these technologies +have operated at two scales: the infrastructural, driven by transportation and mobility efficiency, and the +architectural, driven to quantify the economic productivity of a space. + +Although primarily focused on the level of service or throughput of a street, much attention has been paid toward the +management of road infrastructure. With increased demand on existing and aging infrastructure leading to +congestion and economic tolls, many entities see opportunities to develop technology to “solve” traffic. Many +universities (Yang et al., 2020), startups such as Miovision, CurrexVision, and vivacitylabs, and incumbent +automotive companies such as Honda have developed different algorithms and sensor packages to count and track +automotive traffic on roadways, each with slightly distinct approaches informed by the countries from which the +companies are from. Automotive companies are also seeking to leverage the data aggregated from individual +vehicle’s sensor suites (Massaro et al., 2017) including the onboard, external cameras to analyze for future products +and to use as derivative data that can be monetized. + +Similar technologies have also been used to analyze interior spaces, particularly in retail or commercial +environments. Companies like Indoor Atlas and Brickstream are applying similar approaches as with car traffic, +including understanding common paths and behaviors to understand consumer patterns in discrete spaces. +RetailNext, in addition to algorithmic counting tracking of occupants and heatmap generation, fuses image data with +cellphone tracking to know how these patterns differentiate between men, women, and children and track repeat +visitors to a space. + +The public adoption of online shopping has required better management of these areas of the street and so valuable +is this piece of real estate that cities are investing in systems to increase access to this space. The World Economic +Forum (2020) estimates that the number of commercial delivery vehicles will increase by 36% in inner cities, +globally. Using similar technologies as the transportation measurement companies, companies like Automotus and +curbFlow focus on the management and monetization of the curb to increase the curb’s efficient use for deliveries, +parking, and repair. + +In public, human-oriented spaces such as sidewalks and plazas, companies like Numina and Placemeter have sought +to enumerate individuals on foot or bicycle. Driven by concerns about privacy, Natix and Numina has taken a +different approach than many other companies to intentionally reduce the amount of data it assesses and does it +within the sensor through an edge computing framework, with the intent to put privacy into precedence. Placemeter, +now part of Netgear’s Arlo home monitoring brand, used off-the-shelf, consumer-grade cameras to quantify and +categorize people, bicycles, motorcycles and vehicles. With the City of Paris, Placemeter piloted projects on the +enumeration of a public plaza and pools. While much development into the technology has been done into issues of +crowding dynamics (usually within the rhetoric of public safety) and level of service, little work has been done to + + +Preprint submitted to +10 +precisely understand the complexity of pedestrian behavior. While research is emerging into understanding non- +commuting behavior (Seer et al., 2014; Sun et al., 2020)—that is to say, ambulating or non-intentional travel— +technologies for understanding social behaviors are still nascent. + +With the creation of the LinkNYC network of upwards of 7,500 digital kiosks built by Intersection (a subsidiary of +Sidewalk Labs, which is a sister company to Google, and a subsidiary of Alphabet), many residents were concerned +with the potential invasion of privacy through citywide, digital tracking, and issues with ubiquitous security +surveillance that would result (Kofman, 2018). Originally proposed as a twenty-first-century replacement to the +ubiquitous payphone, the new kiosks would provide free Wi-Fi and digital signage and included a suite of three +cameras and thirty sensors and heightened sight lines above the crowds and onto the streets below. While these +technologies were touted as opportunities to move into infrastructure management and pedestrian counting, concerns +arose around the program’s vast and indefinite data retention and the possibilities for unwarranted NYPD +surveillance (ACLU, 2016). + +While not fundamentally opposed to the potential to understand public behavior, these camera technologies can also +be used for policing and security purposes at large using the same algorithms, whose ethical rationales have +conflicted with civil libertarians. This issue arises when the desire to scale and monetize every facet of the artificial +intelligence and sensing platform manifests itself into multiple products; when services used for the public good can +be easily adapted for surveillance practices. For instance, cities increasingly turn to facial recognition to identify +individuals with outstanding warrants or pose specific threats. The Singaporean police force, for example, has +installed nearly 80,000 cameras in key public areas, with each camera having the capability to run real-time video +analytics to detect potential criminal threats (Lin & Purnell, 2019). Large companies like Verizon, Amazon, IBM, +and Microsoft have readily available, scalable technologies that offer police and security agencies the ability to +capture and fuse various image sources to provide real-time situational awareness and identification of individuals in +a crowd. The New York Police Department, for example, partnered with Microsoft to equip cameras with the +technology to, they claim, identify crime based on “potentially suspicious body language (Stanley, 2019).” + +However, in the contentious public debate about using these technologies, the concerns span public and privately- +owned digital infrastructure. In addition to public networks, cities are also developing so-called “plug-in” platforms +where business owners can connect their privately-owned, internet-connected cameras to city-operated camera +networks. Most notable of these is Project Greenlight in Detroit, although similar networks exist across the United +States. Their automation systems are built upon facial recognition technology from biometrics system company +DataWorks Plus. The project uses the company’s Face Plus video surveillance product. It compares captured faces +to a database of over 500,000 mugshots with additional access to a statewide database that includes drivers’ license +photographs (Garvie & Moy, 2019). In 2021, the network has grown to over 700 cameras, but cities like Chicago +and New York have tens of thousands of networked surveillance devices (Kwet, 2020). Despite concerns from +individual homeowners, Amazon has also made data from its branded doorbell cameras to 400 police forces +(Harwell, 2019). + +The use of these technologies has come with much criticism, including from the industry itself (Shepardson, 2020), +for the lack of transparency around the use of these technologies—including their questionable accuracy (Hill, 2020; +Lee, 2020)—and the ethical concerns around their use. Using algorithms to label people based on race or ethnicity +has become relatively easy from a technology standpoint. Both IBM and Microsoft readily advertise their services to +sort people into broad groups, including by race (Attribute Detection with Body Camera Analytics, 2020). With +reports that this technology has been used for tracking of to track and control Uighurs individuals in China (Mozer, +2019) and Black Lives Matter protesters in the United States (Selinger & Fox Cahn, 2020), many are questioning the +appropriateness of these technologies in cities. + + + +Preprint submitted to +11 +The same technologies used to identify faces are also being used to identify features of the face for advertising and +retail purposes. Emerging technologies read facial characteristics to measure different types of engagement with +media, such as measuring attention (Picard, 1995). Startups like RealEyes, SightCorp, and Kairos offer technologies +to pinpoint the coordinates and directionality of a person’s gaze, relating it to capturing and holding an individual’s +attention. Going further, Affectiva is tuning to the characteristics of the face as it relates to affect, measuring one’s +emotional state. Their algorithms have been tuned through the analysis of 7.5 million faces from 87 countries, +mostly collected from opt-in recordings of people watching TV or driving their daily commute. While their +applications are clear concerning enumerating commercial activity, these datasets can also be tuned toward +understanding facets of the built environment at large, including the emotional landscape of cities. For instance, an +application of affective computing applied to communities explored the overall emotional happiness around the +university campus of MIT by reading the faces of passers-by and finding the variances by department, building, and +time of semester (Hernandez et al., 2012). + +4.3. The Underlying Data and the Underlying Problem + +The generation of identities, tracks, or outputs is incumbent on creating models that can interpret the live stream of +real-world data. Therefore, a dataset is required to both train and validate these models and must include additional +annotations and metadata to ascertain meaning from the results. Due to the emerging nature of these technologies, +only a handful of datasets account for the vast variety of model-creation. However, a growing number of specialized +datasets are also becoming available to the research and development community. + +Among the most commonly used training dataset for computer vision identification are Imagenet (Deng et al., 2009) +which includes approximately 14 million hand-annotated images and 20,000 categories, and Open Images, with its +nine million images and nearly 20,000 human-verified classes (Krasin et al., 2017). At the same time, many datasets +have been created specifically to train models on human action (Ofli et al., 2013), most derived opportunistically +from existing sources such as film and television or online video-sharing platforms (Idrees et al., 2017; Patron-Perez +et al., 2012; Smaira et al., 2020). The Kinetics-700 dataset, for instance, used 650,000 YouTube video clips to +generate a dataset covering 700 human actions. Business interests are also driving the creation of specific datasets to +improve the abilities of computer vision in the marketplace. The Affectiva-MIT Facial Expression Detection dataset +(McDuff, El Kaliouby, Senechal, et al., 2013) is a provocative image collection of facial emotions of people from +around the world, enabling researchers and companies to measure affect, with applications towards retail, media and +advertising (McDuff, El Kaliouby, Demirdjian, et al., 2013). The increased interest in self-driving vehicles led to +creating the Cityscapes dataset (Cordts et al., 2016), which annotated 25,000 images specific to urban cityscapes +taken from the street. + +While the latter shows promise to shape a digital understanding of urban spaces, the focus on vehicles narrowly +focuses the image set on the spatial perspective of the driver and a car’s navigation through this singular aspect of +the city. The Cityscape does identify people and 30 classes ranging from “sky” to recognizing different types of +street objects and vehicles, but only as it informs a vehicles’ ability to navigate a street and not a proper +understanding of how an urban space operates. Additionally, being generated from the imagery of only German +cities, it may lack a diversity of potential, infrastructural histories and standards, and mobility options. + +This point poses a critical question within the space of urbanism: can they account for cities' real experiential and +social diversity? Of course, one cannot assume that the compilation of data with varying levels of detail and depth +will always be problem-free (Brannen, 2005). The challenge in creating these datasets is the embedded and implicit +values and assumptions made by those who both organize and annotate the images. + +Firstly, the orientation towards universal applicability of these datasets creates an ontology that privileges +generalities over specificity where no agreed upon standard may exist. Societal appreciation of architectural styles or + + +Preprint submitted to +12 +the role it plays in broad society, for instance, varies greatly between cultures and context (Berlyn, 1971; Kubo et +al., 2010). + +Secondly, these datasets built from the opportunistic collection of imagery, such as ImageNet and Kinetics-700, may +omit specific populations’ experiences because of more significant societal inequities. The gender and racial +inequities (R. L. Collins, 2011; Desmond & Danilewicz, 2010) and a lack of geographic diversity (Shankar et al., +2017) in television and media may be reinforced by the derivative use these datasets in the training of new +algorithms unless ameliorated. For instance, the largest share of users and content on YouTube originate from the +United States. Similarly, the United States exerts enormous global soft power in the production of television and +media. Any algorithm trained on that dataset will perform worse on non-American contexts with increasing error as +a culture is more distinct from that standard. For digital images to be useful, they must be organized in accordance +with some type of knowledge system (Pasquinelli, 2015). As this data disappears into the black box of the algorithm +and manifests itself as the building block for policy and the physical reshaping of our cities, it is incumbent to +question whether these existing datasets and paradigms adequately describe urbanism. + +5. MEANING IN SPACE + +Innovations in computing avail new avenues for urban research, but critical considerations remain regarding the use +of these technologies within an urbanism discourse. As Duarte and DeSouza (2020) argue, urban technologies must +consider more fundamentally how the epistemologies behind extensive data methodologies as well as how they +shape ontologies and heuristics about cities, as ultimately there will be lasting transformations to both society and +space due to the specific responsibility of urban planning in shaping communities. + +Among the epistemological standards with computer vision technologies is the orientation towards reductivism or +simplified numeric representations (Dreyfus, 1992). In itself, this approach has vast implications for how residents +and policymakers understand the built environment (Winner, 2017). While this reductivism can produce a +generalized intelligence about a location that may lead to prediction, this reductivism is often uninterested in the +ephemeral qualities that make cities unique. Companies that are developing these technologies strive for scalability +or the universal applicability of their products. This efficiency strategy is opposed to considering unique or +distinguishing factors unique to a place, including the inherent complexity of such environments (Gershenson, 2013; +Gill, 2020). It thus prioritizes simple-to-measure factors such as counts or duration over nuanced observation of +sociability or idiosyncratic behavior. As Czarniawska (1992) frames this focus, the current paradigm of urban +observation excludes the “anthropological frame of mind” that more seeks to explain these behaviors. + +This conflict of definition parallel Clifford Geertz’s (1973) arguments for an interpretive approach to understanding +culture. To Geertz, the existing paradigm of anthropological research was through thin descriptions derived from +which includes surface-level observations of behavior. Similar to the superficial enumeration of many urban +technologies, these observations lack the context and interpretive turn that define thick descriptions. He analogizes +the difference through identification of a wink between two children, the thin description. However, the meaning +behind the wink—be it coded communication between them, a random biological twitch, or an attempt at seeking +attention—can only be understood through think descriptions. For urbanism, a similar a framework could be +understood in everyday activities. For instance, someone holding out their hand in a park may be a gesture to a +friend while the same gesture in the street may be the hailing of a taxi. + +The creation of the metrics by which cities are documented can also distort toward one type of description. There is +also a worry that the ease of thin description through both data collection and analytics runs a risk of privileging +infrastructural efficiency over the more complex social promotion. The apparent focus in the mobility domain +toward level of service also bias interpretations of cities toward thin descriptions, as they premise cities on the sole + + +Preprint submitted to +13 +metric of infrastructural efficiency. The relative ease of this type of data collection has implications on how the +cities are described, with the risk that they are seen as optimization challenges versus social environments. + +This conflict is fundamental to the emerging practice of big data research, let alone when embedded within complex +social environments such as cities. Geertz’s position with the derivation of thick descriptions sought to evolve +ethnographic research toward the ascertainment of more significant meaning. Nevertheless, it relied on the existing +models of practice as a foundation for criticism. However, in urban science research, like the computational social +sciences on which it is founded, the canons of practice are still being formed (Jemielniak, 2020; Rossman & Rallis, +2017). Thus, the critical observation of urban phenomena is caught uncomfortably unformed between the data +science discourse—where the prevailing axiom accepts the unknowability of models and analytics performed within +the analogous black box are worth the gains in accuracy from such processes—and that of the interpretive +anthropological work. + +Lessons in performing observations in this liminal space between reductivism and interpretation may be found from +pre-computer research. In the pursuit of documenting public space as part of the Street Life Project, William H. +Whyte (1980) notably used Super 8 film to understand the use of plazas and public spaces. Whyte and his associates +recorded and analyzed hundreds of hours of time-lapse film to assess variation and regularity in the behavior of +anonymous pedestrians in just a handful of cases. His research was performed to provide data critical to measuring +activity patterns in urban spaces, which did not exist at the time. Whyte’s investigations are among the earliest +attempts at using new technologies to surveil and understand human activities at scale. While the process was still +incredibly manual, as researchers needed to comb hours of video manually, it was an attempt to document an entire +public square as a whole and an example of a technology that was not yet readily available to the public. In doing so, +the project revealed both generalized behaviors, such as the conclusion that park use was directly related to the +amount of “sittable space” and not shape or size as previously thought, and idiosyncratic behaviors of individuals, +such as gender differences in seat and location choice. + +The dual nature of his research on ascertaining thin, generalized patterns of behavior and thicker, contextually +derived considerations allowed Whyte to discover practices that would inform zoning regulations and propose +interventions. For instance, a key characteristic identified for plazas that succeeded as popular gathering spots was +the presence of a variety and abundance of places to sit, including benches, movable chairs, ledges, and steps. They +also found that physical characteristics, including tree canopies, water features, public art, and food vendors, all +played a role in attracting people to urban plazas and parks. These attributes played a role in creating positive +feedback loops for bringing people together, while streets observed with blank walls and devoid of shops, windows, +or doors saw little activity. In Whyte’s words, “what attracts people most, it would appear, is other people.” While +these findings are taken for granted in the twenty-first century, Whyte’s findings broke with contemporary +paradigms of modernist urban design. These findings would notably lead to New York City passing zoning +regulations that required plazas and publicly-owned public spaces to no more than three feet above or three feet +below street level to allow for visibility and easy access, and the complete revitalization of Bryant Park, which in the +1980s saw little activity, to increase its visibility and to add more seating, including its now-familiar moveable +furniture. Today, the park is popular year-round with more than 1,000 movable chairs plus several cafe kiosks and +many scheduled events. + +5.1. The Conflicts in Meaning + +There was much optimism for new opportunities for digitally-mediated and more open civic governance (Goldsmith +& Crawford, 2014). However, the mismatched objectives of neo-liberal development and the scale-driven focus of +industry and the slower, discursive, community-oriented democratic process of urban management in many cases +yielded incompatible metrics and conflicting outcomes. While it is not in dispute that the ability to understand and +enumerate how residents used urban spaces could lead to more appropriately responsive interventions, the more + + +Preprint submitted to +14 +considerable utility of these datasets to urbanists is questioned. Fundamental to this conflict is how each party saw +the challenge of enumeration and what would entail defining the “use” of public space. + +Similarly, tradeoffs with computational intensity limit the fidelity by which technologists can process the imagery. +As imagery processing requires immense computing power, and many companies rely on weaker edge computing +platforms to preserve privacy, the scale, detail, and precision of what can be enumerated is directly related to the +number of computational resources available. These limitations are present at all points in the technology: from the +captured image’s pixel resolution, to the storage and transmission of the image, to the amount of processing power +available to convert the image into a useable measurement. As such, many purported benefits of particular +technologies are mitigated by such computation’s cost and physical demands. + +Ultimately a series of tradeoffs, the interrelationship between capacity limitations, societal impacts, and the +thickness of its meaning offers a way to evaluate how citizens can evaluate the appropriateness of computer vision +or any smart cities technology. In Figure 2, we evaluate the companies previously mentioned to organize the +tradeoffs and limitations on each of these three metrics. Taken together, they offer a way for citizens to both assess +the adequacy of various technologies for their community and find ontological clusters of technologies that share +similar opportunities and constraints across domains. + + + +Figure 2. Evaluating the meaning, impacts and computational costs of various technologies. + +This taxonomy of tradeoffs also offers a lens by which citizens can evaluate the potential societal impacts of +computer vision technologies. Where certain activities such as the maintenance of a sidewalk bear few stakes for +society at large, high-fidelity and detail are likely unnecessary, and costs associated with processing high resolution, +real-time images are likely unnecessary. However, these technologies’ use in policing has high implications for the +citizens when questions of justice are involved and requires extraordinary precision and accuracy. For instance, the +high false positive rate of the Detroit facial recognition program, where the police chief points to a 96% error rate + +DataworksPlus +Placemeter +Numina +LinkNYC +Affectiva +Urban Impacts +Kairos +Miovision +CurbFlow +Brickstrearm +RealEyes +RetailNext +Computational Requirements +Thickness +Preprint submitted to +15 +and reflects the inadequacies of the technologies and algorithms (Lee, 2020), should give pause in how readily +citizens should accept the meaning of its outputs. + +6. REFRAMING THE METHODS TOWARD URBAN-SEMANTICS + +Many of the vision-based smart cities technologies that seek to understand how people use space lack specificity for +how people use urban space within urbanism discourse. This presents an opportunity for computer vision research to +develop semantically specific approaches that are appropriate and contextually informed to the nuances of urban +studies. However, what would an urban-sematic computer vision look like? To relate to Geertz’s writing, at present, +the state of computer vision is proficient in identifying winks but does very poorly in identifying what those discrete +objects may mean within a city. + +The state of the art of technology allows for a computer to identify objects and actions in isolation. However, when +accurate, the definitions are thin in their descriptions without a situation to contextualize the assessment. +In Figure 3, four photographs of “people sitting on a bench” were run through the ImSitu object recognition +algorithm (Yatskar et al., 2016) to identify the images through the algorithm’s perspective. The algorithm attempts +situation recognition, by which the algorithm provides a concise summary by including the main activity, the actors, +and roles. The algorithm was trained on FrameNet, which contained over 125,000 images and 200,000 unique +situations. The same images were identified by individuals working in urban design and planning, and the most +frequent description was used in the image, although the human descriptions varied little. + + + + + +Figure 3. Results of ImSitu algorithm versus human identification of four images of people sitting on a bench. +(Photograph credits: hjl, 2012; byronv2, 2019, 2020a, 2020b) + + +The findings showed fairly accurate thin descriptions across the four images, although the confidence varied due to +the image quality and color differences on each image. While it was interesting that the algorithm could identify +individuals sitting on a bench, the nuance of what each group was doing was missed. This could be due to a variety +of reasons, including the lack of lexical data on those specific activities, the quality or perspective of the image, or +improper training of the dataset. In any case, the algorithm could not achieve the definition of the already-reductive +human descriptions and demonstrates the current challenge of assessing thicker descriptions of urban activities. + + +people individually fooking at +person eating alone +their phones +shivering (0.21) +waiting (0.35) +speaking (0.18) +begging (0.17) +mourning (0.11) +peeing (0.09) +people bench +male child sidewalk +Preprint submitted to +16 +Recognizing the limitations of state of the art in computer vision, we highlight potential methodologies by which +technology developers can gain a thicker understanding of activities in urban spaces. The challenge with using +algorithmic black boxes is that the processes from which outcomes are computed are unknown. However, +understanding the general mechanics of how algorithms operate may open opportunities to contextualize computer +vision toward urban-specific contexts. + +Firstly, when we can consider how models are derived and the generalized libraries commonly used for training. We +can derive urban-specific identification through a priori contextualization by creating a spatially specific corpus of +images containing descriptions of and action annotations calibrated to urbanism. That is, an unambiguously urban +dataset must be arranged in a taxonomic form so that an algorithm might deem what is and is not relevant +information specific to a particular context. A disadvantage of this approach is the necessity to plan in advance the +classes and categories of urban-semantic annotations or relying on a large group of human annotators trained toward +specific city-focused labels of footage. + +An alternative is to consider an a posteriori approach. Here, we leverage the strengths of computer vision algorithms +to find common patterns from imagery, and an algorithm clusters similar activity. Humans could then verify and +label tag these clusters with thicker descriptions after the clusters were found. While technically feasible, +unsupervised action identification from the footage is new (O’Hara et al., 2011; Soomro & Shah, 2017) but can +allow urbanists to revisit the world of individuals like Whyte to the computer could find that humans could not. + +Within a societal context of cities, there are also questions about these technologies’ relationship with the public. +Residents are not strictly customers nor users of these services and may not have any option to opt-out of these +technologies. Despite the risk of these black-box processes reinforcing the detrimental status quo, or worse, +furthering pre-existing bias, how these technologies are created and therefore drive planning, policy, and design +decisions often lack the input of those whom they will impact. The Boston Beta Blocks program was created as a +policy-based mechanism to pilot technologies and create a platform for community engagement and empowerment. +In addition to civic experimentation with technologies, the program also organizes educational workshops and +events with the platform providers, whether or not they are considered for procurement (Mayor's Office for New +Urban Mechanics, 2018). + +Under the civic experimentation mandate, new technologies are piloted in the open in pre-selected areas where the +community has mechanisms for feedback, offering transparency and citizen oversight into selecting, testing, and +creating success metrics. The residents are invited to co-generate with the city and the companies the values around +civic and privacy concerns. They also provide oversight into the processes and policies that govern these +experiments. As a result, the social milieu around the implementation of technologies is contextualized to the +people, needs, place, and time of that specific community. + +7. CONCLUSION + +The current orientation of computer vision technologies has been inwardly focused on its own development and +toward broad generalizability of its applications. However, as the adage goes, “when something is good at +everything, it is the best at nothing.” When these technologies are implemented within the complex milieu of cities, +the application thus far has erred toward reductive quantifications at the sacrifice of the dynamic characteristics of +public space that draw billions of people to live, work, and play. Precisely because of these dynamic characteristics, +the conception and development of these tools should be reconsidered to appreciate the idiosyncrasies of these +spaces. + +Drawing from precedent conversations in anthropology and the social sciences, urban technologists must move +algorithmic enumeration away from simplistic ontologies toward thick descriptions that better capture the dynamism + + +Preprint submitted to +17 +of cities. Like Whyte, the interrelationship between enumeration, description, and interpretation is vital to drawing +conclusions about the urban spaces. As the proliferation of these technologies persists, mechanisms to consider bias +in the recording, training, development and use of these datasets and algorithms are vital, especially when the +benefits may be inequitably born by inhabitants, and because the role these analytics may play in the shaping of the +built environment and its policies. + +While imperfect, these proposed approaches allow urbanist to move slightly away from what David Hand (2020) +considers “dark data,” the data that is inaccessible from current tools and do not fit within existing methods, but still +can influence the decisions and policies that may result. 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Theory and Society, 49(5–6), 941–964. +https://doi.org/10.1007/s11186-020-09410-4 + + diff --git a/xdAzT4oBgHgl3EQf7v58/content/tmp_files/load_file.txt b/xdAzT4oBgHgl3EQf7v58/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..295e7318524d1fa4c887b47cec5e8e82ddb4d289 --- /dev/null +++ b/xdAzT4oBgHgl3EQf7v58/content/tmp_files/load_file.txt @@ -0,0 +1,1186 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf,len=1185 +page_content='Preprint submitted to 1 Urban-Semantic Computer Vision: A Framework for Contextual Understanding of People in Urban Spaces Anthony Vanky, Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=', Corresponding author Graduate School of Architecture, Planning and Preservation Columbia University, New York, NY 10025 Email: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content='vanky@columbia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content='edu Ri Le Graduate School of Architecture, Planning and Preservation Columbia University, New York, NY 10025 Email: r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content='le@columbia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content='edu Preprint submitted to 2 KEYWORDS artificial intelligence, computer vision, urban space, urban context, urbanism, semantic meaning, thick description, evaluation ABSTRACT Increasing computational power and improving deep learning methods have made computer vision technologies pervasively common in urban environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Their applications in policing, traffic management, and documenting public spaces are increasingly common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=" Despite the often-discussed biases in the algorithms' training and unequally borne benefits, almost all applications similarly reduce urban experiences to simplistic, reductive, and mechanistic measures." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' There is a lack of context, depth, and specificity in these practices that enables semantic knowledge or analysis within urban contexts, especially within the context of using and occupying urban space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' This paper will critique existing uses of artificial intelligence and computer vision in urban practices to propose a new framework for understanding people, action, and public space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=" This paper revisits Geertz's use of thick descriptions in generating interpretive theories of culture and activity and uses this lens to establish a framework to evaluate the varied uses of computer vision technologies that weigh meaning." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=" We discuss how the framework's positioning may differ (and conflict) between different users of the technology." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' This paper also discusses the current use and training of deep learning algorithms and how this process limits semantic learning and proposes three potential methodologies for gaining a more contextually specific, urban- semantic, description of urban space relevant to urbanists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' This paper contributes to the critical conversations regarding the proliferation of artificial intelligence by challenging the current applications of these technologies in the urban environment by highlighting their failures within this context while also proposing an evolution of these algorithms that may ultimately make them sensitive and useful within this spatial and cultural milieu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Preprint submitted to 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' INTRODUCTION Ever-powerful computational ability, the reduced cost of communications infrastructure, and the increase- diminishing size of sensors have enabled the pervasive placement of technologies into the fabric of urban spaces, birthing a movement of the “smart city.” For many in the so-called smart cities movement, the trend has been towards the instrumentalization of cities, finding greater efficiencies, and the problematization of many facets of urban living (Hollands, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' For technologists working in this field, the enumeration game is being applied to all domains ranging from mobility and infrastructure to public safety and democratic participation (Eagle & Pentland, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Goldsmith & Crawford, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Nevertheless, the defining social characteristics of urban space defy a reduction to a mere optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' These digital technologies and their resultant models and data outcomes have the ability to shape our perspective of the built environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' No different than the well-publicized challenges of bias prevalently found in other algorithmic processes (Crawford, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Kirchner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=', 2016), the black box methodologies and opaque outcomes so too can unduly influence our reading of the places we inhabit (Schwarzer, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Despite this conflict between reductivism and complexity, there is an urgent need to understand through new models and tools may open new avenues for research into public space and urban form in light of rapid urbanization and the increased privatization of urban space (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Talen & Ellis, 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' As if it were an iteration from the modernist use of photography in urban planning, the growing ubiquity of deep learning and computer vision applications have created new opportunities to understand cities through imagery (Lecun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' While there is optimism in how these emerging technologies can allow for a more precise (and perhaps, broad) method for understanding cities, questions remain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Within this computational milieu, this paper focuses specifically on the nascent, but growing role of these algorithmic tools are being applied to urban planning and management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' In one sense, they quantify human behavior in urban space that offers the ability for decision-makers to base policy in more informed ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' In another, this numerical reductivism applied to urban space is blind to the specificity and essential character that makes cities unique places of inhabitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' As such, this paper argues that in addition to situated technologies’ reductivist orientation, there is a need for a distinct approach to their use in the understanding of how people inhabit and use these public urban spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Further, with the increased proliferation of computer-vision and image-based approaches toward the instrumentalization of cities, an urban-specific lexicon to the training, implementation, and adoption of urban technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' While the use of computer vision technologies has spanned many facets of urban management, such as infrastructure utilization and public safety, this paper considers explicitly using these technologies to understand how people inhabit and use public space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' This paper argues that image-based artificial intelligence is a natural progression in the modernist use of photo imagery to capture data on the city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' This paper also reviews, in brief, how artificial intelligence is being applied to image-based data to illustrate the potential weaknesses in creating thicker, domain- specific ontologies about the occupants and the spaces in which they inhabit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Further, this paper discusses how thin data is being marketed by both public and private actors to the potential detriment of planning human-centric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' However, this paper also discusses potential approaches to reconceptualize the methodologies currently used to get toward an urban-semantic description of public space using these algorithmically-based methodologies despite these limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' “URBANISM”, AND THE ISSUE OF CONTEXT A fundamental challenge is defining what characteristics make a space or practice urban, especially when contemporary city building is neo-liberal and capital driven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' In other words, what makes Washington Square Park Preprint submitted to 4 urban while the Hudson Yards feel not, despite being just a few kilometers apart in Manhattan?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Or what makes Seoul’s Namdaemun Night Markets an urban experience versus the nearby, impressively large Lotte Shopping Center?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' By their nature, urban space is where individual experience comes together with strangers, even though they seldom share our values, history, and perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' It is in these spaces where the mingling and contact with individuals and groups differ in their social presentation, appearance, and experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' As urbanist Michael Brill (1989) frames these experiences, it is where inhabitants “can seek and find excitement and extraordinariness in the productions and presentations created by strangers, and in those they create themselves.” The spaces around them influence these social dynamics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' the form and configuration of urban space frame the interactions of those in them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=" Ultimately, public life is uniquely able to be experienced in these spatial commons—the streets and public, urban spaces— because of how they make inhabitants' acute awareness of their dependence on one another, as well as the societal obligations that dependence spawned (Chidster, 1989)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Cities are, in a sense, where individualism intermixes to create collective experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Defining these socio-spatial interrelationships is a difficult challenge, and there exist differing perspectives on the nature of a good public realm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' There exist divergent but parallel theories about the character of these spaces, which take up a significant portion of the physical city: streets, sidewalks, squares, arcades, non-motorized transport, greenways, and the like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Jane Jacobs (1961) describes the importance of streets as public spaces: “Streets are almost always public: owned by the public, and when we speak of the public realm, we are speaking in large measure of streets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' If we can develop and design streets so that they are wonderful, fulfilling places to be, community-building places, attractive public places for all people of cities and neighborhoods, then we will have successfully designed about one-third of the city directly and will have had an immense impact on the rest.” The sidewalk is, as she called them, “the main public places of the city” and “its most vital organ.” In a similar vein, Jan Gehl (1987) calls attention to the life between buildings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=" It is in the spaces between architecture where the social connections of inhabitants are created and reaffirmed and where the space of movement coexists with the city's social life." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' These aspects form inter-related patterns of movement and activity to which buildings, in turn, respond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' To Kevin Lynch (1960), this life also imbues these spaces with meaning, whereby they become “imageable,” containing spatial qualities that strengthen the attention and meaning of urban space to its inhabitants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Such spaces establish memorable relationships among buildings, paths, features, and amenities due to their proximity to one another or the geometric configuration of the space itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Ultimately, these concepts are intended to influence the practice of place-making or the shaping of urban spaces that support the public activities mentioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Designers are often unaware of their own biases and roots as influencing their decision-making (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Jacobs & Appleyard, 1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' The profession of urban planning would be helped by a renewed quest for the elements of “good city form,” where the planning of the physical city is concentrated on people and place (Emily Talen & Ellis, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' For designers and planners, normative limits such as time and budgets are often drivers of the opportunities available to improve urban space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' However, they are often exacerbated by a lack of understanding of how both users and non-users perceive them and the characteristics of users (Chidster, 1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Such an endeavor would have to deal with the complexities of aesthetic, ethical, and political theory to secure its foundations and, as such, require domain-specific ontologies unique to the social and physical milieu of urbanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Preprint submitted to 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' IMAGERY AS URBAN DATA Imagery had enormous agency in being evidence of human activity in public space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Our contemporary era has seen the ubiquity of cameras as tools used to documentation of social ills and injustice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Videos of vegetable seller Mohamed Bouazizi setting himself on fire in protest in Tunisia (Noueihed, 2011) or of George Floyd gasping “I can’t breathe” as police offers aggressively restrained him (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Collins, 2020) have served as evidence for protests demanding justice and change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Yet, the use of photographic imagery as a data source for learning and exploring urban space dates as far back as the technology itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Louis Daguerre’s View of the Boulevard du Temple, created in 1938, was the earliest photograph to include people in a busy street.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Although it depicts two men on a street corner, they appear only because one has his shoes polished by the other and thus required them to be stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' The long exposure required left no trace of moving traffic and other people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' In the 1860s, the nascent art of urban photography became a political tool for social and urban reform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' In Paris, Charles Meville’s commissions to photo-document the modernization efforts of Baron Georges-Eugène Haussmann recorded both dirty and cramped “before” and grand, triumphalist “after” vignettes of the city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Melville’s photographs have been the fodder for debates on whether they served more than an archive of a foregone Paris on behalf of the city by also acting as propagandist justifications for the razing of many neighborhoods (Dahlberg, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Historian Shelley Rice’s (1997) critiques of Meville’s repetitious and “clinical” streetscapes point to the photographer’s “passivity” toward the grand transformations of the city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' On the other side of the Atlantic, Jacob Riis, while working as a police reporter for the New York Tribune, documented the slum conditions on the Lower East Side of Manhattan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Using lantern slide shows, illustrated articles, and printed books, Riis harnessed the power of photographic imagery as evidence of the need for housing and social reform, which galvanized the politically connected gentile and policymakers to action and reform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Where once rare, the twentieth century saw the increased ubiquity of cameras which served in the modernist era as tools that could provide an unblinking empiricism in the documentation of human behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Powered by the increased availability to take photos from aircraft and satellites, planners and architects were drawn to the newly available planar perspective afforded by flight as a way to document the formal development of the city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' For Le Corbusier (1935), the eye of the airplane offered a “pitiless”, objective record of reality, and for many planners in the United States and Europe in the post-war era, aerial photographs served as evidence of urban blight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' This documentation provided the opportunity of seeing the world anew, but also as confirmation for the need to replan (Hinchcliffe, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' In their seminal work, Learning from Las Vegas, Venturi and Scott Brown (1972) experimented with various visual media to explore ways of representing American urban form in the late-1960s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' As a critical exploration about the automotive orientation and iconographic expressiveness of Las Vegas, their Yale research group explored the city using photography and film as both mechanisms for data collection and representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Notably, their post-modernist approach adopted “drive-by photography” taken from inside moving cars, a technique borrowed from the artist Ed Ruscha, as a means of contextual representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Taken together, the imagery serving as data also produces meaning as understood by people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' In investigating urbanism, what is learned about cities and their inhabitants is both mediated and shaped by the characteristics and limitations of the media and the mode of production (Azar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' While historic precedent relies on the curation of the framing and representation, emerging technologies are increasingly less reliant on humans—images are increasingly being made by machines, for machines (Paglan, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' This computational process forms a shift in how the production of urban knowledge, vis-à-vis enumeration, data, or models, is being generated and understood, while remaining potentially invisible to the human altogether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Preprint submitted to 6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' IMAGE-BASED ANALYSIS IN THE “SMART CITY” From the horse to the automobile to the telephone, the physical city, as a social artifact, has long evolved in response to prevalent technologies of the day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Today,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' the rise of the so-called “smart city” introduces two parallel technological interventions: a digitally connected citizenry possessing connected devices such as mobile phones and the ubiquitous instrumentation of the physical artifice of the city through the “internet of things.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Pervasive sensing enabled by both has created the ability to dynamically sense,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' analyze and understand individual mobility traces more quickly and accumulate detailed knowledge over time to see patterns and trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' This “big data” approach—having access to large volume datasets to study phenomena and their dynamics— augments the process by which urban space is designed, developed and evaluated, and offers opportunities for data- driven analysis and design of the built environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' McLuhan foresaw technologies serving as civic thermostats “to pattern life in ways that will optimize human awareness” (Norden, 1969).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' He said, “already, it is technologically feasible to employ the computer to program societies in beneficial ways.” He stressed that “the programming of societies could actually be conducted quite constructively and humanistically.” Greenfield (2013) comments that “the final intent of all this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' is to make every unfolding process of the city visible […], to render the previously opaque or indeterminate not merely knowable but actionable.” Despite its complexity, the desire to make understandable the complex nature of vibrant cities has been both a dream of Euclidean modernists and those who pursue an organic method of urban planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' For instance, for urbanist Jane Jacobs, “the variables are many, but they are not helter-skelter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' they are interrelated into an organic whole.” Jacobs was talking about Hudson Street in 1960s-era Manhattan (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Jacobs, 1970), but the same could be said of any street in any city today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' The critical difference is that today, digital data allows us to describe and analyze Jacobs’s “interrelated variables” better than she could ever have imagined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Big Data may lead to the discovery of fascinating dependencies and intimate knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' (Jemielniak, 2020) As city officials, urbanists, and urban planners analyze the figurative reams of data available;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' imagery has also been leveraged as a means of understanding larger urban dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' For instance, Girardin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' (2008) used individually- uploaded photographs on the online photograph sharing platform Flickr to assess the different visitation footprints in Barcelona of domestic, Spanish tourists, and international Britons visiting the country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' To do so, the researchers analyzed the underlying exchangeable image file (EXIF) that stores metadata created by the camera and user- volunteered information such as locations, profile demographics, including the countries and cities where the photographer was from, tags and descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' The fusion of this information provided insight into how different groups may use a city differently (Offenhuber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=', 2013): the British tend to stay to primary, famed tourist sites such as La Ramba, while Spaniards were willing to go farther afield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' It also reaffirmed notions of the “imageable city,”1 exposing locations that attracted photographers’ attention (monuments, churches, public spaces, for instance) were exposed, and where the absence of images may reveal locales considered more introverted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Although the use of volunteered imagery is still commonly used, Street View images have provided a dataset that is both immensely comprehensive in its coverage (thanks to the deep financial resources of its creator company, Google) and consistent in its format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' This undertaking was driven by co-founder Larry Page’s belief that this type of street-level imagery contained a tremendous amount of information that could be organized, made available, and mined (Anguelov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' By 2017, the company captured 16 million kilometers of Street View imagery across 83 countries (Ackerman, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' This powerful dataset, albeit through restrictive commercial and proprietary means, has allowed researchers to investigate street-level dynamics and changes of urban space at a massive, aggregated scale from understanding the prevalence of tree canopies and green space (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=', 2015) to pedestrian counts (Yin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' 1 “Imageable” here is taken to mean a cognitive, memory-based image as was used in the previous reference of Lynch’s work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Preprint submitted to 7 However, several projects have sought to use this dataset to assess more ephemeral or culturally specific attributes about these spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Salesses et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' (2013) used the street-level images in an online survey of people’s reactions to whether a space looked more or less safe than another to index the perceived safety of a specific location in many cities around the world, based on visible characteristics such as street width, program use, apparent blight, among many other formal attributes seen in the photograph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Taken together, Naik & Philipoom (2014) use machine learning algorithms to scale the survey results into an index applicable to many other cities around the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' The EthinCITY Linguistic Landscape project from the Spatial Analysis Lab (2019) captured 9 million textual signs on the streets of Los Angeles—ranging from official street names to business signs and advertisements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' By extracting the text and language of the message and geocoding the words and languages, the team assessed the languages used to the parcel level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' What emerged is a dynamic map that reflects the cultural diversity of tableau that traditional government sources of data cannot collect, including small, ethnic establishments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' In addition to macro-scale findings such as there being not just one “Chinatown” in the city but five smaller ones, they could redefine notions of ethnic enclaves as static neighborhoods and instead explore the dynamic, overlapping, and interspersed nature of urban cultural diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' While user-generated imagery comes at extraordinary velocity, volume, and veracity, a general lack of standardization of what is captured2 and when limits the potential for a deeper inquiry into specific locales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=" For instance, while there are over 2 million photographs of “Disneyland” in California on Flickr, there is only one of “Captain Kidd's Family Dining” located just across the street from the entrance to the theme park." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Moreover, while Street View images remedy the aspects of standardization at a massive scale, the enormous cost of capturing the images limits the potential to understand the whole dynamic nature of cities in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' However, the emergence of smart city technologies through placed sensors provides opportunities for deeper urban inquiry that addresses the standardization and volume concerns while providing the opportunity to understand places in real-time and over longer durations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Automated Eyes on the Street With the emergence of the smart cities movement, much investment in sensing has focused on the domain of computer vision, advanced machine learning (or deep learning), and more specifically, Convolutional Neural Networks (CNN) to perform analytics on a single image or a series of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' This artificial intelligence is concerned with the automatic analysis and extraction of useful information from these images or videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' In other domains, computer vision has opened new frontiers in disease treatment and diagnosis, agricultural and industrial efficiency, robotic navigation and self-driving vehicles, and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' In cities, this automated technology is being used in wildlife tracking, intelligent traffic management, pedestrian flow monitoring, public safety, and infrastructure management (Kwet, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' This boom is following a concomitant growth in the number of surveillance cameras being placed in the built environment for various reasons and by a multitude of public and private parties (Lin & Purnell, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' It is estimated that the number of surveillance cameras globally will exceed one billion by the end of 2021, with over 85 million in the United States alone, rivaling China’s per person camera penetration rate (IHS Markit, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' While cameras on their own lack the overall ability to do the computational performance being described, it is estimated that 350 million of these situated cameras will possess built-in technology to allow for artificial intelligence in 2025, with more than 65 per cent of cameras shipped in that year will come with at least one AI chipset (ABI Research, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' 2 These together form the commonly used “four V’s” of big data: velocity, veracity, volume and variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Preprint submitted to 8 To enumerate and describe the behaviors of inhabitants, these smart devices apply a domain of vision-based machine learning called “action recognition,” which seeks to extract peoples’ identifiable physical features, acts, and behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Generally, all computer vision approaches generate data abstractions from the captured images and videos, thereby reducing the image to a series of numerical matrices that can be more easily analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content='3 These matrices are produced by assessing patterns in the red-green-blue (RBG) pixel values of a digital image (and depth information, if available).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' They are compared against pre-trained models developed from the analysis of prior imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' The creation of this pre-trained model is vital as it provides the foundation for comparison, much like how humans learn how to identify objects: we learn by establishing priors—formal characteristic, visual appearance, defining features, or other attributes—to be able to hypothesize about the nature of something new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' To get to the point of accurate enumeration, enormous data is required to train the statistical and deep learning models to ascertain human action from noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' These processes are often ambivalent to the identification of a human as a specific individual unless they are specifically focused on that task, such as through the use of facial recognition technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Generally, the first step in identifying human action is to identify features in an image that statistically resembles the figure of a person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Here, one can think of this abstracted figure as a blob, box, or skeleton (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Each of these offers different tradeoffs in the computational resources and statistical strength but represents humans in different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' How a computer interprets subjects and actions in an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' (Original photograph credit: Chetan, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=') The computationally simplest method is blob detection, whereby an algorithm can assess a figure in the foreground of a video as being distinct from the background by comparing attributes that change between frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' This approach toward blob detection is very frequently used in describing movement flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' In Figure 1, we simply identify the hotspot of where the person is or has been as a measure of where the figure has been relative to the camera’s view, establishing the background frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' The dichotomous nature of this foreground-background comparison can be used to make the assignment of actions computationally more efficient by focusing the more intense box or skeleton analysis on a smaller portion of the total image, however.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Within this subset, an algorithm can do one of several things: compare track this boxed subset within the frame, compare the RBG pixel values against the trained models (classification), or further reduce the figure to the relationships of certain parts as probable limbs or critical physical features vis-à-vis a digital, topological skeleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Ultimately, it is the comparison of these two reductions, the boxed pixel values or the skeleton, that is used to assess human action algorithmically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' These actions can be categorized according to the complexity of activity: primitive, single person, interaction, and group (Al-Faris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=', 2020), and form the foundation for the computer assessment of activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=" The combination of these methods and categories allows us to identify individuals' activities in urban space." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' 3 While this paper will not comprehensively review the technology and its application depth, other papers have sought to categorize various approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' See Ibrahim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=', 2020, for instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' 1p1 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content='43 AL Preprint submitted to 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' The Players and Builders The use of the term smart cities has been characterized as incorporating computational tools into the operations and management of municipalities and regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' However, this also brought a different paradigm of urban improvement as the private sector played a more significant role in city-building.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Many cities saw the implementation of these technologies as two-fold endeavors: a means of economic development in an increasingly globally competitive marketplace (Zukin, 2020), and in light of diminished resources, a means of outsourced services management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Particularly in light of the inability of city bureaucrats to develop or implement new data collection means, greater attention has been paid to technology companies, startups, and academics to create and implement these new technologies within the urban domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Fundamental to planning or policymaking is the restructuring of complexity to a discrete set of standardized measurements that are understandable by those shaping the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Normatively, these measures are drawn from a desire to benchmark the use or inhabitation of public space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Much of the development of these technologies have operated at two scales: the infrastructural, driven by transportation and mobility efficiency, and the architectural, driven to quantify the economic productivity of a space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Although primarily focused on the level of service or throughput of a street, much attention has been paid toward the management of road infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' With increased demand on existing and aging infrastructure leading to congestion and economic tolls, many entities see opportunities to develop technology to “solve” traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Many universities (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=', 2020), startups such as Miovision, CurrexVision, and vivacitylabs, and incumbent automotive companies such as Honda have developed different algorithms and sensor packages to count and track automotive traffic on roadways, each with slightly distinct approaches informed by the countries from which the companies are from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Automotive companies are also seeking to leverage the data aggregated from individual vehicle’s sensor suites (Massaro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=', 2017) including the onboard, external cameras to analyze for future products and to use as derivative data that can be monetized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Similar technologies have also been used to analyze interior spaces, particularly in retail or commercial environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Companies like Indoor Atlas and Brickstream are applying similar approaches as with car traffic, including understanding common paths and behaviors to understand consumer patterns in discrete spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' RetailNext, in addition to algorithmic counting tracking of occupants and heatmap generation, fuses image data with cellphone tracking to know how these patterns differentiate between men, women, and children and track repeat visitors to a space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' The public adoption of online shopping has required better management of these areas of the street and so valuable is this piece of real estate that cities are investing in systems to increase access to this space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' The World Economic Forum (2020) estimates that the number of commercial delivery vehicles will increase by 36% in inner cities, globally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Using similar technologies as the transportation measurement companies, companies like Automotus and curbFlow focus on the management and monetization of the curb to increase the curb’s efficient use for deliveries, parking, and repair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' In public, human-oriented spaces such as sidewalks and plazas, companies like Numina and Placemeter have sought to enumerate individuals on foot or bicycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Driven by concerns about privacy, Natix and Numina has taken a different approach than many other companies to intentionally reduce the amount of data it assesses and does it within the sensor through an edge computing framework, with the intent to put privacy into precedence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Placemeter, now part of Netgear’s Arlo home monitoring brand, used off-the-shelf, consumer-grade cameras to quantify and categorize people, bicycles, motorcycles and vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' With the City of Paris, Placemeter piloted projects on the enumeration of a public plaza and pools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' While much development into the technology has been done into issues of crowding dynamics (usually within the rhetoric of public safety) and level of service, little work has been done to Preprint submitted to 10 precisely understand the complexity of pedestrian behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' While research is emerging into understanding non- commuting behavior (Seer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=', 2020)—that is to say, ambulating or non-intentional travel— technologies for understanding social behaviors are still nascent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' With the creation of the LinkNYC network of upwards of 7,500 digital kiosks built by Intersection (a subsidiary of Sidewalk Labs, which is a sister company to Google, and a subsidiary of Alphabet), many residents were concerned with the potential invasion of privacy through citywide, digital tracking, and issues with ubiquitous security surveillance that would result (Kofman, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Originally proposed as a twenty-first-century replacement to the ubiquitous payphone, the new kiosks would provide free Wi-Fi and digital signage and included a suite of three cameras and thirty sensors and heightened sight lines above the crowds and onto the streets below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' While these technologies were touted as opportunities to move into infrastructure management and pedestrian counting, concerns arose around the program’s vast and indefinite data retention and the possibilities for unwarranted NYPD surveillance (ACLU, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' While not fundamentally opposed to the potential to understand public behavior, these camera technologies can also be used for policing and security purposes at large using the same algorithms, whose ethical rationales have conflicted with civil libertarians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' This issue arises when the desire to scale and monetize every facet of the artificial intelligence and sensing platform manifests itself into multiple products;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' when services used for the public good can be easily adapted for surveillance practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' For instance, cities increasingly turn to facial recognition to identify individuals with outstanding warrants or pose specific threats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' The Singaporean police force, for example, has installed nearly 80,000 cameras in key public areas, with each camera having the capability to run real-time video analytics to detect potential criminal threats (Lin & Purnell, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Large companies like Verizon, Amazon, IBM, and Microsoft have readily available, scalable technologies that offer police and security agencies the ability to capture and fuse various image sources to provide real-time situational awareness and identification of individuals in a crowd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' The New York Police Department, for example, partnered with Microsoft to equip cameras with the technology to, they claim, identify crime based on “potentially suspicious body language (Stanley, 2019).” However, in the contentious public debate about using these technologies, the concerns span public and privately- owned digital infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' In addition to public networks, cities are also developing so-called “plug-in” platforms where business owners can connect their privately-owned, internet-connected cameras to city-operated camera networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Most notable of these is Project Greenlight in Detroit, although similar networks exist across the United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Their automation systems are built upon facial recognition technology from biometrics system company DataWorks Plus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' The project uses the company’s Face Plus video surveillance product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' It compares captured faces to a database of over 500,000 mugshots with additional access to a statewide database that includes drivers’ license photographs (Garvie & Moy, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' In 2021, the network has grown to over 700 cameras, but cities like Chicago and New York have tens of thousands of networked surveillance devices (Kwet, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Despite concerns from individual homeowners, Amazon has also made data from its branded doorbell cameras to 400 police forces (Harwell, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' The use of these technologies has come with much criticism, including from the industry itself (Shepardson, 2020), for the lack of transparency around the use of these technologies—including their questionable accuracy (Hill, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Lee, 2020)—and the ethical concerns around their use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Using algorithms to label people based on race or ethnicity has become relatively easy from a technology standpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Both IBM and Microsoft readily advertise their services to sort people into broad groups, including by race (Attribute Detection with Body Camera Analytics, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' With reports that this technology has been used for tracking of to track and control Uighurs individuals in China (Mozer, 2019) and Black Lives Matter protesters in the United States (Selinger & Fox Cahn, 2020), many are questioning the appropriateness of these technologies in cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Preprint submitted to 11 The same technologies used to identify faces are also being used to identify features of the face for advertising and retail purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Emerging technologies read facial characteristics to measure different types of engagement with media, such as measuring attention (Picard, 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Startups like RealEyes, SightCorp, and Kairos offer technologies to pinpoint the coordinates and directionality of a person’s gaze, relating it to capturing and holding an individual’s attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Going further, Affectiva is tuning to the characteristics of the face as it relates to affect, measuring one’s emotional state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Their algorithms have been tuned through the analysis of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content='5 million faces from 87 countries, mostly collected from opt-in recordings of people watching TV or driving their daily commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' While their applications are clear concerning enumerating commercial activity, these datasets can also be tuned toward understanding facets of the built environment at large, including the emotional landscape of cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' For instance, an application of affective computing applied to communities explored the overall emotional happiness around the university campus of MIT by reading the faces of passers-by and finding the variances by department, building, and time of semester (Hernandez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' The Underlying Data and the Underlying Problem The generation of identities, tracks, or outputs is incumbent on creating models that can interpret the live stream of real-world data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Therefore, a dataset is required to both train and validate these models and must include additional annotations and metadata to ascertain meaning from the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Due to the emerging nature of these technologies, only a handful of datasets account for the vast variety of model-creation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' However, a growing number of specialized datasets are also becoming available to the research and development community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Among the most commonly used training dataset for computer vision identification are Imagenet (Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=', 2009) which includes approximately 14 million hand-annotated images and 20,000 categories, and Open Images, with its nine million images and nearly 20,000 human-verified classes (Krasin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' At the same time, many datasets have been created specifically to train models on human action (Ofli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=', 2013), most derived opportunistically from existing sources such as film and television or online video-sharing platforms (Idrees et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Patron-Perez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Smaira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' The Kinetics-700 dataset, for instance, used 650,000 YouTube video clips to generate a dataset covering 700 human actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Business interests are also driving the creation of specific datasets to improve the abilities of computer vision in the marketplace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' The Affectiva-MIT Facial Expression Detection dataset (McDuff, El Kaliouby, Senechal, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=', 2013) is a provocative image collection of facial emotions of people from around the world, enabling researchers and companies to measure affect, with applications towards retail, media and advertising (McDuff, El Kaliouby, Demirdjian, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' The increased interest in self-driving vehicles led to creating the Cityscapes dataset (Cordts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=', 2016), which annotated 25,000 images specific to urban cityscapes taken from the street.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' While the latter shows promise to shape a digital understanding of urban spaces, the focus on vehicles narrowly focuses the image set on the spatial perspective of the driver and a car’s navigation through this singular aspect of the city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' The Cityscape does identify people and 30 classes ranging from “sky” to recognizing different types of street objects and vehicles, but only as it informs a vehicles’ ability to navigate a street and not a proper understanding of how an urban space operates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Additionally, being generated from the imagery of only German cities, it may lack a diversity of potential, infrastructural histories and standards, and mobility options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=" This point poses a critical question within the space of urbanism: can they account for cities' real experiential and social diversity?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Of course, one cannot assume that the compilation of data with varying levels of detail and depth will always be problem-free (Brannen, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' The challenge in creating these datasets is the embedded and implicit values and assumptions made by those who both organize and annotate the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Firstly, the orientation towards universal applicability of these datasets creates an ontology that privileges generalities over specificity where no agreed upon standard may exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Societal appreciation of architectural styles or Preprint submitted to 12 the role it plays in broad society, for instance, varies greatly between cultures and context (Berlyn, 1971;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Kubo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Secondly, these datasets built from the opportunistic collection of imagery, such as ImageNet and Kinetics-700, may omit specific populations’ experiences because of more significant societal inequities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' The gender and racial inequities (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Collins, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Desmond & Danilewicz, 2010) and a lack of geographic diversity (Shankar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=', 2017) in television and media may be reinforced by the derivative use these datasets in the training of new algorithms unless ameliorated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' For instance, the largest share of users and content on YouTube originate from the United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Similarly, the United States exerts enormous global soft power in the production of television and media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Any algorithm trained on that dataset will perform worse on non-American contexts with increasing error as a culture is more distinct from that standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' For digital images to be useful, they must be organized in accordance with some type of knowledge system (Pasquinelli, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' As this data disappears into the black box of the algorithm and manifests itself as the building block for policy and the physical reshaping of our cities, it is incumbent to question whether these existing datasets and paradigms adequately describe urbanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' MEANING IN SPACE Innovations in computing avail new avenues for urban research, but critical considerations remain regarding the use of these technologies within an urbanism discourse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' As Duarte and DeSouza (2020) argue, urban technologies must consider more fundamentally how the epistemologies behind extensive data methodologies as well as how they shape ontologies and heuristics about cities, as ultimately there will be lasting transformations to both society and space due to the specific responsibility of urban planning in shaping communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Among the epistemological standards with computer vision technologies is the orientation towards reductivism or simplified numeric representations (Dreyfus, 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' In itself, this approach has vast implications for how residents and policymakers understand the built environment (Winner, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' While this reductivism can produce a generalized intelligence about a location that may lead to prediction, this reductivism is often uninterested in the ephemeral qualities that make cities unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Companies that are developing these technologies strive for scalability or the universal applicability of their products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' This efficiency strategy is opposed to considering unique or distinguishing factors unique to a place, including the inherent complexity of such environments (Gershenson, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Gill, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' It thus prioritizes simple-to-measure factors such as counts or duration over nuanced observation of sociability or idiosyncratic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' As Czarniawska (1992) frames this focus, the current paradigm of urban observation excludes the “anthropological frame of mind” that more seeks to explain these behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' This conflict of definition parallel Clifford Geertz’s (1973) arguments for an interpretive approach to understanding culture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' To Geertz, the existing paradigm of anthropological research was through thin descriptions derived from which includes surface-level observations of behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Similar to the superficial enumeration of many urban technologies, these observations lack the context and interpretive turn that define thick descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' He analogizes the difference through identification of a wink between two children, the thin description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' However, the meaning behind the wink—be it coded communication between them, a random biological twitch, or an attempt at seeking attention—can only be understood through think descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' For urbanism, a similar a framework could be understood in everyday activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' For instance, someone holding out their hand in a park may be a gesture to a friend while the same gesture in the street may be the hailing of a taxi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' The creation of the metrics by which cities are documented can also distort toward one type of description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' There is also a worry that the ease of thin description through both data collection and analytics runs a risk of privileging infrastructural efficiency over the more complex social promotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' The apparent focus in the mobility domain toward level of service also bias interpretations of cities toward thin descriptions, as they premise cities on the sole Preprint submitted to 13 metric of infrastructural efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' The relative ease of this type of data collection has implications on how the cities are described, with the risk that they are seen as optimization challenges versus social environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' This conflict is fundamental to the emerging practice of big data research, let alone when embedded within complex social environments such as cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Geertz’s position with the derivation of thick descriptions sought to evolve ethnographic research toward the ascertainment of more significant meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Nevertheless, it relied on the existing models of practice as a foundation for criticism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' However, in urban science research, like the computational social sciences on which it is founded, the canons of practice are still being formed (Jemielniak, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Rossman & Rallis, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Thus, the critical observation of urban phenomena is caught uncomfortably unformed between the data science discourse—where the prevailing axiom accepts the unknowability of models and analytics performed within the analogous black box are worth the gains in accuracy from such processes—and that of the interpretive anthropological work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Lessons in performing observations in this liminal space between reductivism and interpretation may be found from pre-computer research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' In the pursuit of documenting public space as part of the Street Life Project, William H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Whyte (1980) notably used Super 8 film to understand the use of plazas and public spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Whyte and his associates recorded and analyzed hundreds of hours of time-lapse film to assess variation and regularity in the behavior of anonymous pedestrians in just a handful of cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' His research was performed to provide data critical to measuring activity patterns in urban spaces, which did not exist at the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Whyte’s investigations are among the earliest attempts at using new technologies to surveil and understand human activities at scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' While the process was still incredibly manual, as researchers needed to comb hours of video manually, it was an attempt to document an entire public square as a whole and an example of a technology that was not yet readily available to the public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' In doing so, the project revealed both generalized behaviors, such as the conclusion that park use was directly related to the amount of “sittable space” and not shape or size as previously thought, and idiosyncratic behaviors of individuals, such as gender differences in seat and location choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' The dual nature of his research on ascertaining thin, generalized patterns of behavior and thicker, contextually derived considerations allowed Whyte to discover practices that would inform zoning regulations and propose interventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' For instance, a key characteristic identified for plazas that succeeded as popular gathering spots was the presence of a variety and abundance of places to sit, including benches, movable chairs, ledges, and steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' They also found that physical characteristics, including tree canopies, water features, public art, and food vendors, all played a role in attracting people to urban plazas and parks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' These attributes played a role in creating positive feedback loops for bringing people together, while streets observed with blank walls and devoid of shops, windows, or doors saw little activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' In Whyte’s words, “what attracts people most, it would appear, is other people.” While these findings are taken for granted in the twenty-first century, Whyte’s findings broke with contemporary paradigms of modernist urban design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' These findings would notably lead to New York City passing zoning regulations that required plazas and publicly-owned public spaces to no more than three feet above or three feet below street level to allow for visibility and easy access, and the complete revitalization of Bryant Park, which in the 1980s saw little activity, to increase its visibility and to add more seating, including its now-familiar moveable furniture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Today, the park is popular year-round with more than 1,000 movable chairs plus several cafe kiosks and many scheduled events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' The Conflicts in Meaning There was much optimism for new opportunities for digitally-mediated and more open civic governance (Goldsmith & Crawford, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' However, the mismatched objectives of neo-liberal development and the scale-driven focus of industry and the slower, discursive, community-oriented democratic process of urban management in many cases yielded incompatible metrics and conflicting outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' While it is not in dispute that the ability to understand and enumerate how residents used urban spaces could lead to more appropriately responsive interventions, the more Preprint submitted to 14 considerable utility of these datasets to urbanists is questioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Fundamental to this conflict is how each party saw the challenge of enumeration and what would entail defining the “use” of public space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Similarly, tradeoffs with computational intensity limit the fidelity by which technologists can process the imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' As imagery processing requires immense computing power, and many companies rely on weaker edge computing platforms to preserve privacy, the scale, detail, and precision of what can be enumerated is directly related to the number of computational resources available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' These limitations are present at all points in the technology: from the captured image’s pixel resolution, to the storage and transmission of the image, to the amount of processing power available to convert the image into a useable measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' As such, many purported benefits of particular technologies are mitigated by such computation’s cost and physical demands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Ultimately a series of tradeoffs, the interrelationship between capacity limitations, societal impacts, and the thickness of its meaning offers a way to evaluate how citizens can evaluate the appropriateness of computer vision or any smart cities technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' In Figure 2, we evaluate the companies previously mentioned to organize the tradeoffs and limitations on each of these three metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Taken together, they offer a way for citizens to both assess the adequacy of various technologies for their community and find ontological clusters of technologies that share similar opportunities and constraints across domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Evaluating the meaning, impacts and computational costs of various technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' This taxonomy of tradeoffs also offers a lens by which citizens can evaluate the potential societal impacts of computer vision technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Where certain activities such as the maintenance of a sidewalk bear few stakes for society at large, high-fidelity and detail are likely unnecessary, and costs associated with processing high resolution, real-time images are likely unnecessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' However, these technologies’ use in policing has high implications for the citizens when questions of justice are involved and requires extraordinary precision and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' For instance, the high false positive rate of the Detroit facial recognition program, where the police chief points to a 96% error rate DataworksPlus Placemeter Numina LinkNYC Affectiva Urban Impacts Kairos Miovision CurbFlow Brickstrearm RealEyes RetailNext Computational Requirements Thickness Preprint submitted to 15 and reflects the inadequacies of the technologies and algorithms (Lee, 2020), should give pause in how readily citizens should accept the meaning of its outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' REFRAMING THE METHODS TOWARD URBAN-SEMANTICS Many of the vision-based smart cities technologies that seek to understand how people use space lack specificity for how people use urban space within urbanism discourse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' This presents an opportunity for computer vision research to develop semantically specific approaches that are appropriate and contextually informed to the nuances of urban studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' However, what would an urban-sematic computer vision look like?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' To relate to Geertz’s writing, at present, the state of computer vision is proficient in identifying winks but does very poorly in identifying what those discrete objects may mean within a city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' The state of the art of technology allows for a computer to identify objects and actions in isolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' However, when accurate, the definitions are thin in their descriptions without a situation to contextualize the assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' In Figure 3, four photographs of “people sitting on a bench” were run through the ImSitu object recognition algorithm (Yatskar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=', 2016) to identify the images through the algorithm’s perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' The algorithm attempts situation recognition, by which the algorithm provides a concise summary by including the main activity, the actors, and roles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' The algorithm was trained on FrameNet, which contained over 125,000 images and 200,000 unique situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' The same images were identified by individuals working in urban design and planning, and the most frequent description was used in the image, although the human descriptions varied little.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Results of ImSitu algorithm versus human identification of four images of people sitting on a bench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' (Photograph credits: hjl, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' byronv2, 2019, 2020a, 2020b) The findings showed fairly accurate thin descriptions across the four images, although the confidence varied due to the image quality and color differences on each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' While it was interesting that the algorithm could identify individuals sitting on a bench, the nuance of what each group was doing was missed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' This could be due to a variety of reasons, including the lack of lexical data on those specific activities, the quality or perspective of the image, or improper training of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' In any case, the algorithm could not achieve the definition of the already-reductive human descriptions and demonstrates the current challenge of assessing thicker descriptions of urban activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' people individually fooking at person eating alone their phones shivering (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content='21) waiting (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content='35) speaking (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content='18) begging (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content='17) mourning (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content='11) peeing (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content='09) people bench male child sidewalk Preprint submitted to 16 Recognizing the limitations of state of the art in computer vision, we highlight potential methodologies by which technology developers can gain a thicker understanding of activities in urban spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' The challenge with using algorithmic black boxes is that the processes from which outcomes are computed are unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' However, understanding the general mechanics of how algorithms operate may open opportunities to contextualize computer vision toward urban-specific contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Firstly, when we can consider how models are derived and the generalized libraries commonly used for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' We can derive urban-specific identification through a priori contextualization by creating a spatially specific corpus of images containing descriptions of and action annotations calibrated to urbanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' That is, an unambiguously urban dataset must be arranged in a taxonomic form so that an algorithm might deem what is and is not relevant information specific to a particular context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' A disadvantage of this approach is the necessity to plan in advance the classes and categories of urban-semantic annotations or relying on a large group of human annotators trained toward specific city-focused labels of footage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' An alternative is to consider an a posteriori approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Here, we leverage the strengths of computer vision algorithms to find common patterns from imagery, and an algorithm clusters similar activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Humans could then verify and label tag these clusters with thicker descriptions after the clusters were found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' While technically feasible, unsupervised action identification from the footage is new (O’Hara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Soomro & Shah, 2017) but can allow urbanists to revisit the world of individuals like Whyte to the computer could find that humans could not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Within a societal context of cities, there are also questions about these technologies’ relationship with the public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Residents are not strictly customers nor users of these services and may not have any option to opt-out of these technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Despite the risk of these black-box processes reinforcing the detrimental status quo, or worse, furthering pre-existing bias, how these technologies are created and therefore drive planning, policy, and design decisions often lack the input of those whom they will impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' The Boston Beta Blocks program was created as a policy-based mechanism to pilot technologies and create a platform for community engagement and empowerment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=" In addition to civic experimentation with technologies, the program also organizes educational workshops and events with the platform providers, whether or not they are considered for procurement (Mayor's Office for New Urban Mechanics, 2018)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Under the civic experimentation mandate, new technologies are piloted in the open in pre-selected areas where the community has mechanisms for feedback, offering transparency and citizen oversight into selecting, testing, and creating success metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' The residents are invited to co-generate with the city and the companies the values around civic and privacy concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' They also provide oversight into the processes and policies that govern these experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' As a result, the social milieu around the implementation of technologies is contextualized to the people, needs, place, and time of that specific community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' CONCLUSION The current orientation of computer vision technologies has been inwardly focused on its own development and toward broad generalizability of its applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' However, as the adage goes, “when something is good at everything, it is the best at nothing.” When these technologies are implemented within the complex milieu of cities, the application thus far has erred toward reductive quantifications at the sacrifice of the dynamic characteristics of public space that draw billions of people to live, work, and play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Precisely because of these dynamic characteristics, the conception and development of these tools should be reconsidered to appreciate the idiosyncrasies of these spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Drawing from precedent conversations in anthropology and the social sciences, urban technologists must move algorithmic enumeration away from simplistic ontologies toward thick descriptions that better capture the dynamism Preprint submitted to 17 of cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Like Whyte, the interrelationship between enumeration, description, and interpretation is vital to drawing conclusions about the urban spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' As the proliferation of these technologies persists, mechanisms to consider bias in the recording, training, development and use of these datasets and algorithms are vital, especially when the benefits may be inequitably born by inhabitants, and because the role these analytics may play in the shaping of the built environment and its policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' While imperfect, these proposed approaches allow urbanist to move slightly away from what David Hand (2020) considers “dark data,” the data that is inaccessible from current tools and do not fit within existing methods, but still can influence the decisions and policies that may result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' These approaches allow urbanists to “hard work of theory” to critically examine the ontological and epistemological frameworks that exist with the use of these technologies, and reorient the practice toward metrics that relate to the social life of cities and away from reductive, service- oriented quantifications (Pickles, 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Preprint submitted to 18 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' BIBLIOGRAPHY ABI Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Deep Learning-Based Machine Vision in Smart Cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' https://www.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Google street view: Capturing the world at street level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' Computer, 43(6), 32–38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content='1109/MC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} +page_content='2010.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQf7v58/content/2301.01894v1.pdf'} diff --git a/xtE3T4oBgHgl3EQf_gt3/content/tmp_files/2301.04835v1.pdf.txt b/xtE3T4oBgHgl3EQf_gt3/content/tmp_files/2301.04835v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b089dd3ffb9e9adff950d1570b16838810608d48 --- /dev/null +++ b/xtE3T4oBgHgl3EQf_gt3/content/tmp_files/2301.04835v1.pdf.txt @@ -0,0 +1,1759 @@ +PySAGES: fexible, advanced sampling methods +accelerated with GPUs +Pablo F. Zubieta Ricoa, Ludwig Schneidera, Gustavo Perez-Lemusa, +Riccardo Alessandria, Siva Dasettya, Cintia A. Menéndeza, Yiheng Wua, +Yezhi Jina, Trung Nguyenb, John Parkerb, Andrew L. Fergusona, Juan J. +de Pabloa +aPritzker School of Molecular Engineering, The University of Chicago, 5640 South Ellis +Avenue, Chicago, 60637, IL, USA +bResearch Computing Center, The University of Chicago, 6054 S. Drexel +Avenue, Chicago, 60637, IL, USA +Abstract +Molecular dynamics simulations are a core element of research in physics, +chemistry and biology. A key aspect for extending the capability of simula- +tion tools is providing access to advanced sampling methods and techniques +that permit calculation of the relevant, underlying free energy landscapes. +In this sense, sofware tools that can be seamlessly adapted to a broad range +of complex systems are essential. Building on past eforts to provide an +open-source community supported sofware for advanced sampling, we +introduce PySAGES, a Python implementation of the Sofware Suite for Ad- +vanced General Ensemble Simulations (SSAGES) that provides full support +of GPUs for massively parallel applications of enhanced sampling methods +such as adaptive biasing forces, harmonic bias, and forward fux sampling +in the context of molecular dynamics simulations. By providing an intuitive +interface that facilitates the treatment of the confguration of the system, +the inclusion of new collective variables, and the implementation of sophis- +ticated free energy methods, the PySAGES library will serve as a general +platform for development and implementation of emerging simulation algo- +rithms. The capabilities and core features of this new tool are demonstrated +with clear and concise examples pertaining to diferent classes of molecular +systems. +Keywords: Enhanced sampling methods, GPU acceleration +Preprint +January 13, 2023 +arXiv:2301.04835v1 [physics.comp-ph] 12 Jan 2023 + +1. Introduction +Molecular simulations are used extensively in a wide range of science +and engineering disciplines [1]. As their use has grown for discovery of +new phenomena, or for interpretation of sophisticated experimental mea- +surements, so have the complexity of the systems under consideration and +the ambition of simulators. Classical atomistic molecular dynamics (MD) +simulations however, continue to be limited to microsecond time scales +and tens of nanometers length scales. For systems that are characterized by +rugged free energy length scales, such time scales are insufcient to ensure +sufcient sampling of the relevant phase space, and advanced methods must +therefore be adopted to overcome free energy barriers. In that regard, it +is useful and increasingly common to rely on the use of properly chosen +collective variables (CVs) +which are generally diferentiable functions of +the atomic coordinates of the system. +The rapid growth of hardware accelerators such as GPUs or TPUs, or +specialized hardware designed for fast MD computations [2, 3] has provided +researchers with increased opportunities for longer simulations of larger +systems. GPUs, in particular, provide a widely accessible option, and several +simulation packages, such as HOOMD-blue [4], OpenMM [5], JAX MD [6, 7], +and Gromacs, are now available for MD simulations on such devices. +In general, enhanced sampling methods can be used to overcome the +high energy barriers that separate multiple metastable states in a system, +while still allowing for the recovery of relevant thermodynamic and kinetic +quantities as functions of diferent CVs such as reaction rates or free energy +surfaces (FES). Several libraries, such as PLUMED [8], Colvars [9], and our +previously in-house developed SSAGES package [10], provide out-of-the-box +solutions for performing enhanced sampling MD simulations. +Among the various enhanced sampling methods, some of the most re- +cently devised schemes rely on machine learning (ML) strategies to ap- +proximate free energy surfaces and their gradients (generalized forces) [11, +12, 13, 14]. Similarly, algorithms for identifying meaningful CVs that cor- +relate with the slowest degrees of freedoms (DOFs) are based on autoen- +coders [15, 16, 17, 18]. These advances serve to highlight the need for seam- +less integration of ML frameworks with existing MD sofware libraries. +To date, there are no solutions that combine enhanced sampling tech- +niques, hardware acceleration, and ML frameworks to facilitate enhanced- +sampling MD simulations on GPUs. While some MD libraries that sup- +port GPUs provide access to a limited set of enhanced sampling meth- +ods [5, 19, 20, 21, 22], there are currently no packages that enable users +2 + +to take advantage of all of these features within the same platform and in +the same backend-agnostic fashion that tools such as PLUMED and SSAGES +have provided for CPU-based MD simulations. +Here we present PySAGES, a Python Suite for Advanced General Ensem- +ble Simulations. It is a free, open-source sofware package written in Python +and based on JAX that follows the design ideas of SSAGES and enables users +to easily perform enhanced-sampling MD simulations on CPUs, GPUs, and +TPUs. PySAGES can currently be coupled with HOOMD-blue, OpenMM, +JAX MD and, Atomic Simulation Environment (ASE) +and, by extension +from the latter, to CP2K, Quantum ESPRESSO, VASP, and Gaussian, among +others. At this time, PySAGES ofers the following enhanced sampling meth- +ods: Umbrella Sampling, Metadynamics, Well-tempered Metadynamics, +Forward Flux Sampling, String Method, Adaptive Biasing Force, Artifcial +neural network sampling, Adaptive Biasing Force using neural networks, +Combined Force Frequency, and Spectral Adaptive Biasing Force. PySAGES +also includes some of the most commonly used CVs but, importantly, defn- +ing new ones is relatively simple as long as they can be expressed in terms +of the NumPy [23] interface provided by JAX. All CVs can be automatically +diferentiated through JAX functional transforms. PySAGES is highly modu- +lar, thereby allowing for the easy implementation of new methods as the +emerge, even as part of a user-facing script. +In the following sections, we provide a general overview of the design +and implementation of PySAGES and present a series of examples to show- +case its fexibility for tackling problems in diferent application areas. We +also discuss its performance on GPUs and present some perspectives on +how to grow and improve the package to cover more research use cases. +2. Implementation +We begin by briefy outlining the core components of PySAGES, how they +play together, and how communication with each backend allows PySAGES +to bias a simulation during its run course. A summary of the execution +workfow of PySAGES along with a mapping of the user interface with the +main stages of the simulation and the interaction with the backends, are +illustrated in Figure 1. +To provide a uniform user interface while minimizing disruption to pre- +existing workfows, PySAGES only requires the user to wrap their traditional +backend scripting code into simulation generator functions. This approach +accommodates the heterogeneity of Python interfaces across the diferent +simulation backends that PySAGES supports. An example of a simulation +3 + +pysages.run( , , ) +pysages.analyze( ) +generate_simulation +function +Collective variable (CV) +User +Backend +Sampling method +generate_simulation +is called on every rank +Device query and +Snapshot creation +Creation of replicas of + the simulation system +Automatic differentiation +of the CV +Functional specialization +of the sampling method +Launch simulation +Simulation time step +Simulation ends +Computation of forces +Calculation of +free energy +Transparent (zero-copying) +wrapping of particle data +Sampling method update, +computation of biasing forces +Addition of biasing forces +to the backend forces +repeats until +stopping criterion +is reached +CVs and Sampling Methods +can be user defined or +imported from pysages +Figure 1: The PySAGES simulation fowchart. For a simulation, a user sets up a script that +declares the CV and sampling methods to be used. +4 + +import openmm +import openmm.unit as unit +import openmm.app as app +pdb = app.PDBFile("adp-vacuum.pdb") +ff = app.ForceField("amber99sb.xml") +positions = pdb.getPositions(asNumpy=True) +system = ff.createSystem( + pdb.topology, constraints=app.HBonds, + nonbondedMethod=app.PME, nonbondedCutoff=1.0 * unit.nanometer +) +integrator = openmm.LangevinIntegrator( + 298.15 * unit.kelvin, 1 / unit.picosecond, 2.0 * unit.femtoseconds +) +simulation = app.Simulation(pdb.topology, system, integrator) +simulation.context.setPositions(positions) +simulation.minimizeEnergy() +simulation.run(int(1e6)) +def generate_simulation(): + pdb = app.PDBFile("adp-vacuum.pdb") + ff = app.ForceField("amber99sb.xml") + positions = pdb.getPositions(asNumpy=True) + system = ff.createSystem( + pdb.topology, constraints=app.HBonds, + nonbondedMethod=app.PME, nonbondedCutoff=1.0 * unit.nanometer + ) + integrator = openmm.LangevinIntegrator( + 298.15 * unit.kelvin, 1 / unit.picosecond, 2.0 * unit.femtoseconds + ) + simulation = app.Simulation(pdb.topology, system, integrator) + simulation.context.setPositions(positions) + simulation.minimizeEnergy() + return simulation +import openmm +import openmm.unit as unit +import openmm.app as app +import pysages +from numpy import pi +from pysages import ABF, DihedralAngle, Grid +cvs = [DihedralAngle([4, 6, 8, 14]), DihedralAngle([6, 8, 14, 16])] +grid = Grid(lower=(-pi, -pi), upper=(pi, pi), shape=(32, 32), periodic=True) +method = ABF(cvs, grid) +raw_results = pysages.run(method, generate_simulation, int(1e6)) +result = pysages.analyze(raw_result) +Lines removed +Lines added for pysages +Preserved user code +Figure 2: Example of how to use the Python interface for PySAGES. It is easy to extend +existing MD scripts with PySAGES to perform enhanced-sampling MD, without many changes +to the code. In general, the only requirement is for the user to wrap the code that defnes +the simulation system into a simulation generator function. +generator function and how a traditional OpenMM script can be modifed +to perform an enhanced-sampling MD simulation is depicted in Figure 2. +At the start of a simulation, the simulation generator function is called +to instantiate as many replicas of the simulation as needed. Then, for each +replica, PySAGES queries the particle information and the device that the +backend will be using. In addition, during this initial stage PySAGES also +performs automatic diferentiation of the collective variables via JAX’s grad +transform +required to estimate the biasing forces, and generates special- +ized initialization and updating routines for the user-declared sampling +method. +Like SSAGES, PySAGES wraps the simulation information into an object +called a Snapshot. This object exposes the most important simulation infor- +mation, such as particle positions, velocities, and forces in a backend- and +device-agnostic format. To achieve this, PySAGES uses DLPack [24] +for +C++ based MD libraries +to directly access the contents of the backend- +allocated bufers for the diferent particle properties without creating data +copies whenever possible. +Once the setup of both the simulation and sampling method is completed, +PySAGES hands control back to the backend, which will run for a given +number of time steps or until some other stopping criteria is reached. In +order to exchange information back and forth, PySAGES adds a force-like +object or function to the backend which gets called as part of the time +5 + +integration routine. Here, the sampling method state gets updated and the +computed biasing forces are added to the backend net forces. +Finally, the information collected by the sampling method is returned +and can be used for calculating the free energy as function of the selected +CVs. Unlike SSAGES, PySAGES ofers a user-friendly analyze interface that +simplifes the process of performing post simulation analysis, including the +automatic calculation of free energies based the chosen sampling method. +Thus, reducing the time and efort required to gain valuable insights from +simulations. +PySAGES ofers an easy way to leverage diferent parallelism frameworks +including MPI with the same uniform fronted available to run enhanced +sampling simulations. This is achieved via Python’s concurrent.futures +interface. In particular, for MPI parallelism, the user only needs to pass an +additional MPIPoolExecutor (from mpi4py) to PySAGES’ run method. If +the user selects a method such as UmbrellaSampling, the workload for +each image will be distributed across available MPI nodes. On the other +hand, for most of the sampling methods, the parallelization interface allows +the user to run multiple replicas of the same system to enable, for instance, +analysis of the uncertainties associated to computing the free energy of a +given system. +To ensure the reproducibility and correctness of our implementation +and to follow sofware engineering best practices, we have implemented a +comprehensive unit tests suite, and leverage GitHub’s continuous integra- +tion services. In addition, we use trunk.io [25] to adhere to some quality +standards as well as to ease the collaboration of developers. +2.1. Enhanced Sampling Methods +While we assume the reader has some basic understanding of enhanced +sampling methods, here we provide a more detailed overview of these tech- +niques. In addition, we discuss the general structure of how enhanced +sampling methods are implemented within PySAGES, and also present a +summary of the various methods already available in the library. +Enhanced sampling methods are a class of simulation techniques that +manipulate regular MD simulations in order to more efectively sample the +confguration space. In MD a collective variable, 𝜉, is typically a function of +the positions of all particles, ˆ𝜉({𝑟𝑖}). +For a given statistical ensemble (such as the canonical, NVT), the cor- +responding free energy can be written as 𝐴 = −𝑘B𝑇 ln(𝑍), where 𝐴 is the +Helmholtz free energy and 𝑍 is the canonical partition function. To make ex- +plicit the dependency of the free energy on 𝜉, let us write down the partition +6 + +function: +𝑍(𝜉) ∝ +∫ +d𝑁𝑟𝑖 𝛿( ˆ𝜉({𝑟𝑖}) − 𝜉) 𝑒−𝑈({𝑟𝑖})/𝑘B𝑇 +(1) +Normalizing this partition function gives us the probability of occur- +rence, 𝑝(𝜉), for confgurations in the CV subspace. Substituting this proba- +bility into the expression for the free energy, we get: +𝐴(𝜉) = −𝑘B𝑇 ln(𝑝(𝜉)) + 𝐶 +(2) +where 𝐶 is a constant. +If we take the derivative of the free energy with respect to 𝜉 we get +𝑑𝐴(𝜉) +𝑑𝜉 += +∫ +d𝑁𝑟𝑖 +𝑑𝑈 +𝑑𝜉 𝛿( ˆ𝜉({𝑟𝑖}) − 𝜉) 𝑒−𝑈({𝑟𝑖 })/𝑘B𝑇 +∫ +d𝑁𝑟𝑖 𝛿( ˆ𝜉({𝑟𝑖}) − 𝜉) 𝑒−𝑈({𝑟𝑖 })/𝑘B𝑇 += +�𝑑𝑈 +𝑑𝜉 +� +𝜉 +, +(3) +where ⟨. . .⟩𝜉 denotes the conditional average. +The goal of enhanced sampling methods is to accurately determine +either 𝑝(𝜉) or 𝑑𝐴(𝜉)/𝑑𝜉 +from which 𝐴(𝜉) can be recovered +in a compu- +tationally tractable manner. +In PySAGES, the implementation of sampling methods follows the JAX +functional style programming model. New methods are implemented as +subclasses of theSamplingMethod class, and are required to defne abuild +method. This method returns two methods, initialize and update, used +as part of the process of biasing the simulation. For readers familiar with +JAX MD, these could be thought of as analogues to the higher level functions +returned by JAX MD’s simulate integration methods. The initialize +method allocates all the necessary helper objects and stores them in a +State data structure, while the update method uses the information from +the simulation at any given time to update the State. +While PySAGES allows new methods to be written seamlessly as part +of Python scripts used to set up molecular dynamics simulations, it also +provides out-of-the-box implementations of several of the most important +known sampling methods. We list and briefy detail them next. +2.1.1. Harmonic Biasing +One simple way to sample a specifc region of the phase space is to bias +the simulation around a point 𝜉0 with harmonic bias. This adds a quadratic +potential energy term to the Hamiltonian that increases the potential energy +quadratically as a system moves away from the target point: H𝑏 = H + +7 + +𝑘/2(𝜉 − 𝜉0)2, where 𝑘 > 0 is the spring constant. The unbiased probability +distribution 𝑝(𝜉) can be recovered by dividing the biased distribution by +the known weight of the bias 𝑝(𝜉) = 𝑝𝑏(𝜉)/𝑒−𝑘/2(𝜉−𝜉0)2/𝑘B𝑇. +The disadvantage of this approach is that it can only be used to explore +the free energy landscape near a well-know point in phase space. This may +not be sufcient for many systems, where the free energy landscape is +complex. +2.1.2. Umbrella Integration +Umbrella sampling is a technique that builds of harmonic biasing by +combining multiple harmonically-biased simulations. It is a well-known +method for exploring a known path in phase space to obtain a free energy +profle along that path [26, 27] Typically, a path between to point of inter- +est is described by 𝑁 points in phase space, 𝜉𝑖. At each of these points, a +harmonically biased simulation is performed, and the resulting occurrence +histograms are combined to obtain a single free energy profle. +In PySAGES, we implement umbrella integration for multi-dimensional +CVs. This method approximates the forces acting on the biasing points and +integrates these forces to fnd the free energy profle 𝐴(𝜉), and allows to +explore complex high-dimensional free energy landscapes. +2.1.3. Improved String Method +When only the endpoints are known, but not the path itself, the improved +(spline-based) string method can be used to fnd the mean free energy +pathway (MFEP) between these two end-points. The spline-based string +method improves the original string method by interpolating the MFEP +using cubic-splines. In this method, the intermediate points of the path are +moved according to the recorded mean forces acting on them, but only in +the direction perpendicular to the contour of the path. This ensures that +distances between the points along the path remain constant. +This method has been widely used and has been shown to be an efective +way to fnd the MFEP between two points in the phase space [28]. +2.1.4. Adaptive Biasing Force sampling +The adaptive biasing force (ABF) sampling method is a technique used +to map complex free-energy landscapes. It can be applied without prior +knowledge of the potential energy of the system, as it generates on-the-fy +estimates of the derivative of the free energy at each point along the integra- +tion pathway. ABF works by introducing an additional force to the system +that biases the motion of the atoms, with the strength and direction of the +8 + +bias is continuously updated during the simulation. In the long-time limit, +this yields a Hamiltonian with no average force acting along the transition +coordinate of interest, resulting in a fat free-energy surface and allowing +the system to display accelerated dynamics, thus providing reliable free- +energy estimates [29, 30]. Similarly to SSAGES, PySAGES implementation of +ABF is based on the algorithm described in [30]. +2.1.5. Metadynamics +Metadynamics is another popular approach for enhancing sampling of +complex systems. In metadynamics [31], a bias potential is applied along +one or more CVs in the form of Gaussian functions. The height and width (𝜎) +of these Gaussians are controlled by the user. The Gaussian bias potentials +are cumulatively deposited at user-defned intervals during the simulation. +In standard metadynamics, the height of the Gaussian bias potentials is +fxed. +In contrast, for well-tempered metadynamics (WTMD) [32] simulations, +the height of the Gaussian bias potentials is adjusted at each timestep using +a preset temperature based bias factor. This scaling of Gaussian heights in +WTMD leads to faster convergence compared to standard metadynamics, as +it restricts the range of free energy explored to a range defned by the bias +factor. +In PySAGES, we have implemented both standard metadynamics and +WTMD. The well-tempered variant is activated when a user sets a value for +the bias factor. To improve the computational performance, we have added +optional support for storing the bias potentials in both on a pre-defned grid. +This allows users to trade-of accuracy for faster simulations, depending on +their needs. +2.1.6. Forward Flux Sampling +Forward fux sampling (FFS) belongs to a diferent family of enhanced +sampling methods than the ones described above. In the previously de- +scribed methods, the free energy change from a region in the phase space +(𝐴) to the region of interest (𝐵) is calculated by applying a bias to the system. +In FFS no bias is added and instead an efcient selection of trajectories that +crosses the phase space from 𝐴 to 𝐵 is performed. Since no bias is used, +the intrinsic dynamics of the system is conserved and therefore kinetic +and microscopic information of the transition path can be studied [33]. In +PySAGES we have implemented the direct version of FFS [34, 35]. +9 + +2.1.7. Artifcial neural networks sampling +Artifcial neural networks sampling (ANN) [11] employs regularized neu- +ral networks to directly approximate the free energy from the histogram +of visits to each region of the CV space, and generates a biasing force that +avoids ringing and boundary artifacts [11], which are commonly observed +in methods such as metadynamics or basis functions sampling [36]. This +approach is efective at quickly adapting to diverse free energy landscapes +by interpolating undersampled regions and extrapolating bias into new, +unexplored areas. +The implementation on PySAGES ofers more fexible approaches to +network regularization than SSAGES, which uses Bayesian regularization. +2.1.8. Force-biasing using neural networks +Force-biasing using neural networks (FUNN) [12] is based upon the same +idea as ANN, that is, relying on artifcial neural networks to provide con- +tinuous functions to bias a simulation, but instead of using the histogram +to visits to CV space it updates its network parameters by training on the +ABF estimates for the mean forces as the simulation advances. This method +shares all of the features of ABF, but the smooth approximation of the gen- +eralized mean force it produces enables much faster convergence to the +free energy of a system compared to ABF. +2.1.9. Combined Force Frequency sampling +The combined force frequency sampling (CFF) method [13] combines +the speed of generalized-force based techniques such as ABF or FUNN with +the advantages of frequency-based methods like metadynamics or ANN. No- +table improvements over earlier force-based methods include eliminating +the need for hyperparameters to dampen early-time estimates, automating +the integration of forces to generate the free energy, and providing an ex- +plicit expression for the free energy at all times, enabling the use of replica +exchange or reweighing. +In principle, by using sparse storage of histograms, it should be possible +to scale the method to higher dimensions without encountering memory +limitations, such optimization is however not yet implemented in PySAGES. +2.1.10. Spectral Adaptive Biasing Force +Spectral ABF [37] is a method that follows the same principle as neural- +network-based sampling methods, in that it builds a continuous approxima- +tion to the free energy. However, in contrast to methods like FUNN it does so +by ftting exponentially convergent basis functions expansions, and could +10 + +be thought as a generalization of the Basis Functions Sampling Method. In +contrast to the latter, and similar to CFF, it allows for the recovery of an +explicit expression for the free energy of a system. It is an extremely fast +method in terms of both runtime and convergence. +2.2. Collective variables +As previously mentioned, enhanced sampling calculations commonly +involve the selection of a CV. An appropriate CV for a given system could +simply be the distance between the centers of mass of two groups of atoms, +but could be a complex specialized quantity. +Below, we list a set of CVs predefned in PySAGES, sorted by the number +of groups of atoms coordinates necessary for their use: +1. TwoPointCV. This subclass is for CVs that need two groups for their +defnition. This includes Distance and Displacement (vector). +2. ThreePointCV. Subclass of CVs with three groups of atoms, such as +Angle. +3. FourPointCV. Subclass of CVs with four groups of atoms, such as +DihedralAngle. +4. AxisCV. Subclass of CVs that are projected on a determinated axis. +This includes Component and PrincipalMoment. +5. CollectiveVariable General base class for all CVs. In PySAGES, +CVs that directly derive from this class, and do not belong to the +previous groups, include: RingPhaseAngle, RingAmplitude, +RadiusofGyration, Asphericity, Acylindricity, +ShapeAnisotropy, RingPuckeringCoordinates [38] (vector). +In PySAGES we provide users with a simple framework for defning +CVs, which are automatically diferentiated with JAX. To illustrate this, we +compare how to write the calculation of a CV that measures the projection +of the vector between two groups of atoms over the axis that passes by other +two groups, in both SSAGES and PySAGES (see Figure 3). In PySAGES the +gradient calculation is done automatically whereas in SSAGES it has to be +coded explicitly. +The following second example shows the power of diferential program- +ming for CV declaration in PySAGES. +2.2.1. Case study: A collective variable for interfaces +When the two immiscible liquids are in contact with each other, the +density of one liquid experiences a gradual change. This transition region +11 + +Gradient calculation +CV calculation +// Preamble ommited +class ParallelProjectionCV : public CollectiveVariable { +public: + ParallelProjectionCV(int atomid1, int atomid2, int atomid3) : + atomids_({atomid1, atomid2, atomid3}) + { } + void Initialize(const Snapshot& snapshot) override + { + // Code ommited for brievity + } + void Evaluate(const Snapshot& snapshot) override + { + auto n = snapshot.GetNumAtoms(); + auto idx_i = snapshot.GetLocalIndex(atomids_[0]); + auto idx_j = snapshot.GetLocalIndex(atomids_[1]); + auto idx_k = snapshot.GetLocalIndex(atomids_[2]); + + auto com_i = snapshot.CenterOfMass(idx_i); + auto com_j = snapshot.CenterOfMass(idx_j); + auto com_k = snapshot.CenterOfMass(idx_k); + auto rik = com_i - com_k; + auto rjk = com_j - com_k; + auto rij = rjk - rik; + auto nrij = rij.norm(); + auto nij = (1.0 / nrij) * rij; + val_ = nij.dot(-rik); // This writes the CV value + // Manual computation of the gradient + std��fill(grad_.begin(), grad_.end(), Vector3{0, 0, 0}); + grad_.resize(n, Vector3{0, 0, 0}); + Matrix3 dij = Matrix3��Zero(); + dij(0, 0) = ( -(nij[1] * nij[1] + nij[2] * nij[2]) / nrij ); // dx/dx + dij(0, 1) = ( nij[0] * nij[1] / nrij ); // dx/dy + dij(0, 2) = ( nij[0] * nij[2] / nrij ); // dx/dz + dij(1, 0) = ( nij[1] * nij[0] / nrij ); // dy/dx + dij(1, 1) = ( -(nij[0] * nij[0] + nij[2] * nij[2]) / nrij ); // dy/dy + dij(1, 2) = ( nij[1] * nij[2] / nrij ); + dij(2, 0) = ( nij[2] * nij[0] / nrij ); + dij(2, 1) = ( nij[2] * nij[1] / nrij ); + dij(2, 2) = ( -(nij[1] * nij[1] + nij[0] * nij[0]) / nrij ); + grad_[idx_i] = dij * (-rik) - nij; + grad_[idx_j] = dij * rik; + grad_[idx_k] = nij; + } + static ParallelProjectionCV* Build( + const Json��Value& json, const std��string& path + ) { + // Code ommited for brievity + } +private: + Label atomids_; +}; +SSAGES +// Preamble ommited +class ParallelProjection(ThreePointCV): + @property + def function(self): + return parallel_projection +def parallel_projection(p1, p2, p3): + r1 = barycenter(p1) + r2 = barycenter(p2) + r3 = barycenter(p3) + a = r3 - r1 + b = r2 - r1 + return np.dot(a, b) / norm(b) +PySAGES +Apart from the usual +overhead involved +in writting C++ +code in comparison +to Python, the gradients +of a �� need to be +manually implemented +in ������, whereas in + �y����� these are + automatically +computed with ���. +Figure 3: Example of how to write a CV in PySAGES. On the lef is the same CVs written in +SSAGES and on the right the PySAGES version. In general, the only requirement is for the +user to write the CV as a diferentiable function in JAX. +12 + +is the liquid-liquid interface and its position has high importance in many +studies (see section 3.1.3). However, the location of such interface is not a +trivial task since it generally fuctuates as the simulation progresses. As a +representative CV for the interface, we can utilize the position of the point +where the gradient of the density is maximized. More formally, let 𝜌(𝑥) de- +note the density of a liquid of interest at a coordinate 𝑥 on the perpendicular +axis. We would like to fnd the location of the interface: +𝐼 = arg max +𝑥 +|𝜌′(𝑥)| +(4) +However, the density function 𝜌(𝑥) is not directly measurable in a molec- +ular simulation, as the coordinates of atoms are discrete. To obtain an ap- +proximation of 𝜌(𝑥), we divide the coordinates into multiple bins, each with +a width of 𝛿, and create a histogram 𝑝(𝑥) that records the number of atoms +falling into the bin around position 𝑥. In other words, +𝑝(𝑥) = +∑︁ +𝑖=1...𝑛 +1(𝑥𝑖 − 𝑥) < 𝛿 +2 +(5) +in which 𝑥𝑖 denotes the coordinate of atom 𝑖. As written above, 𝑝(𝑥) is non- +diferentiable. Therefore, as in other works [39], we utilize the kernel density +trick with a Gaussian kernel to modify 𝑝(𝑥). The modifed ˜𝑝(𝑥), is defned +as: +˜𝑝(𝑥) = +∑︁ +𝑖=1...𝑛 +exp − (𝑥𝑖 − 𝑥)2 +2𝜎2 +(6) +in which 𝜎 is a hyperparameter that decides the width of the Gaussian kernel. +Then, the gradient of the density can be approximated as: +˜𝑝 ′(𝑥) = ˜𝑝(𝑥 + 𝛿/2) − ˜𝑝(𝑥 − 𝛿/2) +𝛿 +(7) +and we calculate the location of the interface as 𝐼 = arg max𝑥 | ˜𝑝 ′(𝑥)|. The +arg max operator is also non-diferentiable. As a result, we replace it with a +sofmax function that transforms the raw input into a probability. Denote the +𝑚 bins as 𝑗 = 1 . . . 𝑚, and fnally we calculate the location of the interface +as: +𝐼 = +� +𝑗 𝑥 𝑗 exp | ˜𝑝 ′(𝑥 𝑗)| +� +𝑗 exp | ˜𝑝 ′(𝑥 𝑗)| +(8) +As demonstrated in the code snippet for this CV, provided in Appendix A, +PySAGES allows for the concise and straightforward implementation of +complex CVs such as this one. +13 + +3. Results and Discussion +To evaluate a sofware package like PySAGES, we must consider at least +two factors: physical correctness and computational performance. +First, to assess the correctness of the enhanced sampling methods imple- +mented in PySAGES, we present in Appendix B.1 the free-energy landscape +for the dihedral angles 𝜙 and 𝜓 of alanine dipeptide (ADP). This example is +commonly used to benchmark new enhanced sampling algorithms. Sim- +ilarly, we also show in Appendix B.2 the free-energy as a function of the +dihedral angle of butane. Our results show that PySAGES reproduces the +expected free-energy landscapes using diferent methods and backends. +In section 3.1, we further investigate the applicability and correctness of +PySAGES beyond these simple model systems. +Second, we demonstrate the performance of PySAGES on GPUs with +two diferent backends in section 3.2. In particular, we compare the perfor- +mance of enhanced sampling simulations to the performance of pure MD +simulations, as well as other enhanced sampling implementations. +3.1. Example applications of enhanced sampling with PySAGES +To demonstrate the versatility and efectiveness of PySAGES in diferent +contexts, we present several examples of how enhanced sampling methods +can be used to gain valuable insights in various felds including biology, drug +design, materials engineering, polymer physics, and ab-initio simulations. +These examples showcase how PySAGES can be used in diverse research +areas and the utility of diferent enhanced sampling methods and backends. +Overall, these examples confrm that the enhanced sampling methods +implemented in PySAGES work as intended and provide results consistent +with existing literature. +3.1.1. Structural Stability of Protein–Ligand Complexes for Drug Discovery +High-throughput docking techniques are a widely-used computational +technique in drug lead discovery. However, these techniques are limited +by the lack of information about protein conformations and the stability +of ligands in the docked region [40]. To address this issue, the Dynamical +Undocking (DuCK) method was developed to evaluate the stability of the +ligand binding by calculating the work required to break the most impor- +tant native contact (hydrogen bond interactions) in the protein-ligand com- +plex [41]. This method has been shown to be complementary and orthogonal +to classical docking, making both techniques work parallel in drug discover- +ing [42, 43]. However, DuCK can be slow to converge when combined with +14 + +traditional enhanced sampling techniques [41], making it unsuitable for +high-throughput drug discovery protocols. +Here, we demonstrate how PySAGES with OpenMM can be used ef- +ciently in drug discovery applications, where the user-friendly interface, +native parallel capabilities, and new enhanced sampling methods with fast +convergence are synergistically combined to accelerate the virtual screen- +ing of ligand databases. In this example, we study the main protease (Mpro) +of Sars-CoV-2 virus (PDB: 7JU7 [44]), where the ligands were removed and +the monomer A was selected as the docking receptor. A ligand with SMILES +string CCCCOCC(=O)c1ccc(C)cc1N[C@H]1N[C@@H](c2cccnc2)CS1 was +docked using RDock [45]. The best scoring pose was used to initialize the sys- +tem, which was simulated using the ff14SB [46], TIP3P [47], and GAFF [48] +force felds. A 10 ns equilibration procedure was carried out to fnd the most +stable hydrogen bond between the ligand and the protein. The last frame of +this equilibration was then used to initialize the enhanced sampling calcula- +tions in PySAGES with ABF, metadynamics, FUNN, ANN, and Spectral ABF. +These methods were compared against the same system simulated using +Amber20 [49] with Steered Molecular Dynamics (see Figure 4b). Our results +suggest that we can reduce the simulation time by an order of magnitude +using new enhanced sampling methods like Spectral ABF or FUNN. This +can greatly accelerate the drug discovery process and help identify potential +drug leads more quickly. +3.1.2. Fission of a Diblock Copolymer Spherical Domain +We now investigate the fssion of a single spherical domain of a diblock +copolymer using a coarse-grained model. We use a sof, coarse-grained +dissipative particle dynamics (DPD) model published in previous studies [50, +51, 52]. The model consists of 𝑛 = 200 chains with 𝑁 = 256 beads each, +representing a liquid polymer melt. The frst 𝑁𝐴 = 16 beads in each chain +are type A, while the remaining 𝑁𝐵 = 240 are type B. +A standard DPD potential is used to enforce incompressibility with a +repulsion parameter of 𝐴𝑖𝑖 = 5𝑘𝐵𝑇/𝜎2. However, a higher interaction of +𝐴𝐴𝐵 = 𝐴𝑖𝑖 + Δ𝐴𝑘𝐵𝑇/𝜎2, with Δ𝐴 ∈ [0.1, 0.4] is applied between unlike +particles to create a repulsion that leads to a microphase separation. A Flory- +Huggins parameter Δ𝐴 ∝ 𝜒𝑁 > 0 can characterize this phase separation. +The interaction range of this non-bonded potential is 1𝜎, as well as the range +of the DPD thermostat that keeps the temperature at 𝑇 = 1𝑘𝐵𝑇 = 1𝜖. +In addition, a harmonic spring force with zero resting length is used +to connect the beads to polymer chains with a spring constant of 𝑘 = +16/3𝑘𝐵/𝜎2, resulting in an average bond length of 𝑏0 = 0.75𝜎. The equi- +15 + +Figure 4: Dynamical Undocking (DuCK) method in detail. For a proposed binding mode +obtained from classical docking, a short run using MD simulations is carried out and the +most stable receptor-ligand native contact is selected from that run. In this case, it is the +hydrogen bond between the red and blue atoms highlighted in panel a). b) Comparison +between diferent methods in PySAGES for DuCK calculations averaged over 5 diferent +replicas for each method. The reference, a Steered MD simulations simulations of 2 ns +is in red. In comparison, diferent methods in PySAGES are used considering simulation +period 10 times shorter: only ANN [11] provides inferior performance against the reference; +Spectral ABF [37] or FUNN [12] give the best performance. +librium phase for this polymer melt is a body-centered cubic (BCC) phase of +spherical A droplets inside a B melt. [53] However, we confne the polymer +to a tight cubic simulation box of length 𝐿0 = 10𝜎, which results in a single +A spherical domain in the B matrix. We integrate the simulation with a +time step of Δ𝑡 = 10−3𝜏 and each simulation is equilibrated for 𝑡 = 1000𝜏, +followed by a production run of 𝑡 = 1000𝜏 as well. A discussion of the GPU +performance of this system with and without PySAGES can be found in +section 3.2.1. +Afer defning the diblock copolymer system, the next step is to defne +a CV within the system. In this case, we are interested in the fssion of the +single spherical A domain into two equally sized smaller A domains. To +achieve this, we divide the polymer chains into two groups: the frst 𝑛 = 100 +chains are going to form the frst small domain (blue in Figure 5) and the +second 𝑛 = 100 chains form the second spherical domain (red in Figure 5). +To defne and enforce the separation of the two groups, we defne our CV as +the distance, 𝑅, between the center of mass of the blue A-tails and the center +of mass of the red A-tails. Initially, without biasing, the two groups form a +single spherical domain and blue and red polymer tails are well mixed, as +shown at small 𝑅 < 1𝜎 in Figure 5. +To study the separation of the spherical domain, we use harmonic bias- +16 + +b) +(e +time=200ps +12 +SMD-2ns +ABF +: energy (kcal/mol) +10 +ANN +FUNN +S-ABF +8 +WT-MetaD +6 +2 +Free +0 +2 +0.25 +0.3 +0.35 +0.4 +0.45 +0.5 +Distance (nm)0 +1 +2 +3 +4 +5 +6 +distance R [ ] +0 +200 +400 +600 +800 +free energy [ ] +A = 0.1 +A = 0.2 +A = 0.3 +A = 0.4 +Figure 5: Free energy landscape of the fssion of a spherical diblock-copolymer domain. The +chain ends forming the spherical domain are split into two groups (blue) and (red), the other +chain ends not visible for clarity except for a single chain (grey). Initially, a single spherical +domain is formed, but as we constraint the center of mass between the blue and red groups +further, the domain frst elongates and then separates completely. During this separation, the +free energy continuously increases and the increase is steeper for high repulsion between +unlike type Δ𝐴. As soon as the domain is separated, the free energy plateaus. +17 + +ing (see section 2.1.1) to enforce a separation distance 𝑅0 between the two +groups. The high density in the system +√ +¯N = 𝜌0 +𝑁 𝑅3 +𝑒0 ≈ 344, leads to low fuc- +tuations and suppression of unfavorable conformations. Therefore, we use +a high spring force constant of 𝑘𝐶𝑉 = 1500𝜖/𝜎2 to facilitate the separation. +We investigate a separation of 𝑅 ∈ [0, 6]𝜎 with 14 replicas and use um- +brella integration (see section 2.1.2) to determine the free energy profle, as +shown in Figure 5. As we increase the external separation distance 𝑅0, we +observe how the single domain splits into two. At a low separation distance +𝑅 < 2𝜎, the single domain is mostly undeformed, but the two groups sepa- +rate inside the single spherical domain. Increasing the separation distance +further goes beyond the dimensions of the spherical domain, leading to +the deformation of the domain into an elongated rod-like shape. The two +groups still maintain a connection to minimize the AB interface. +At a separation between 4𝜎 and 5𝜎 the deformation becomes so strong, +that the penalty of forming another AB interface between the two groups, +and hence forming two spherical domains, is lower than the entropic penalty +of the domain deformation and elongated AB interface of the droplet. Af- +ter the separation, the free energy landscape remains indiferent to the +separation, since there is no interaction between the two domains lef. +The free energy profle of separation is controlled by the repulsion of +unlike types 𝜒𝑁 ∝ Δ𝐴. The stronger the repulsion, the more energy is +necessary to enlarge the AB surface area for the fssion. For the strongest +interaction Δ𝐴 = 0.4𝜖, the total free energy barrier reaches about 800𝜖, while +for the lowest Δ𝐴 = 0.1𝜖 it remains below 400𝜖. Both barriers are orders of +magnitude larger than thermal fuctuations 1𝑘𝐵𝑇 = 1𝜖, so a spontaneous +separation is not expected and the fssion can only be studied via enhanced +sampling. +It is interesting to note that at the lowest separation distance 𝑅0 = 0 it is +not the lowest free energy state. Enforcing perfect mixing is not favorable, +as the two groups naturally want to separate slightly optimizing the entropy +of the chain end-tails. +3.1.3. Liquid Crystal Anchoring in Aqueous Interfaces +Liquid crystals (LCs), materials that fow like liquids but have anisotropic +properties as crystals, have been used lately as prototypes for molecular +sensors at interfaces given the high sensitivity in their anchoring behavior +relative to small concentration of molecules at aqueous interfaces [54]. The +presence of molecules at the interface changes drastically the free energy +surface of LC molecules relative to their orientation and distance to such in- +terface. In this example, we are revisiting some canonical interfaces for LC; +18 + +4-cyano-4’-pentylbiphenyl (5CB) at the interface of pure water and sodium +lauryl sulfate (SDS). For 5CB and water, previous work has focused on ob- +taining the free energy surface of a 5CB at the water interface [55]. In our +case, hybrid anchoring conditions have been imposed on a 16 nm slab of +1000 5CB molecules in the nematic phase (300 K) interacting with a 3 nm +slab of water with 62 molecules of SDS at one of the interfaces. The force +felds used are: united atom for 5CB [56], TIP3P [47] for water, GAFF [48] and +Lipid 17 for SDS. The CVs chosen to study this system are the distance of +the center of mass of one molecule of 5CB at each one of the interfaces +(see Appendix A), and the tilt orientation of the same molecule with respect +to the z axis of the box. The free energy surfaces for the pure water and with +SDS at the interface are both displayed in Figure 6. We can observe that the +free energy surface of pure water shows a minimum corresponding to a +parallel orientation to the surface with a similar shape that one calculated +in [55]. On the contrary, the presence of SDS transforms the minimum to +a maximum in the same relative position and orientation to the interface +(Figure 6 top lef), moving now the minima to a perpendicular orientation +of 5CB to the interface, in agreement to the experimental observation of +change from planar to homeotropic anchoring in the presence of SDS in +water. +Figure 6: Free energy surface of 5CB in a hybrid anchoring slab with SDS and water. Right: +Snapshot of the system with water molecules in red, 5CB in purple, SDS in green and sodium +ions in yellow. Top Lef: FES of 5CB molecule near the water–SDS interface. Bottom Lef: FES +of 5CB near a pure water interface. Both FES were obtained with PySAGES and OpenMM +using the FUNN method. +19 + +0.5 +0.0 +0.5 +Distance to interface (nm) +-1.0 +20 +-1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +2.0 +100 +1.5 +08 +1.0 +60 +0.5 +0° +20 +0.5 +1.0 +0.5 +0.0 +0.5 +1.0 +Tilt relativeto zaxis3.1.4. Ab Initio Enhanced Sampling Simulations +In the feld of ab initio simulations of heterogenous catalysis, capturing +the dynamic and entropic efects is crucial for an accurate description of +the phenomena [57]. Classical force felds are inadequate for capturing the +essential bond breaking events involved in catalysis, so MD simulations +based on frst-principles calculations are necessary. Given that reactive +events are ofen limited by large free energy barriers, enhanced sampling +techniques are a crucial part of these simulations. Coupling PySAGES to +ASE, provides access to a wide range of frst-principle calculators. +As an example, we have used CP2K [58] as a calculator for a simple ab +initio enhanced sampling simulation. The CV is the separation distance +between a sodium and chlorine atom using the LDA functional. The results +are shown in Figure 7, where the minimum in the free energy profle along +the Na–Cl distance corresponds to the equilibrium distance between Na and +Cl atoms in vacuum. +Figure 7: Free energy calculation of Na–Cl distance with ASE+CP2K using Spectral ABF in +PySAGES. +3.1.5. Enhanced Sampling with Machine Learning Force Fields +Deep neural network (NN) force felds can retain the accuracy of ab initio +MD while allowing for computational costs similar to those of classical MD. +Through ASE it is possible to access NN potentials such as DeepMD [59], and +the Gaussian Approximation Potential (GAP). Additionally, JAX MD allows to +20 + +2.5 +300K +2 +(eV) +Energy +1.5 +1 +Free +0.5 +0 +2 +2.2 +2.4 +2.6 +2.8 +3 +3.2 +3.4 +Distance(Angstroms)leverage more general NN potentials that can be used in enhanced sampling +calculations. Coupling of PySAGES with ASE or JAX MD can be used in +active learning of NN force felds by efciently sampling rare events using +any of the enhanced sampling methods provided by PySAGES as described +in Ref. [60] where parallel tempering metadynamics was used to generate +accurate NN force feld in urea decomposition in water. +To test the capabilities of PySAGES to handle diferent NN force felds, +we have selected three diferent systems trained with the methods men- +tioned above. For DeepMD, we use a pre-trained model for water, where +the enhanced sampling system is one single water molecule in vacuum and +the collective variable is the internal angle of the molecule. The results in +Figure 8 show that the minimum for this free energy profle is around 105 +degrees, which is within the range of the experimental value. +Next, in Figure 8b, a GAP potential was used for Si–H amorphous mix- +tures [61]. In this case, a system of 244 atoms was used, and the collective +variable is the bond angle between a triad of Si–Si–H atoms in the mixture. +The global minimum in free energy agrees with the histogram taken from +unbiased simulations reported in [61]. +Lastly, we studied a Graph neural network (GNN) model of a Si crystal [62] +with PySAGES and JAX MD. In this case, a crystalline Si system of 64 atoms +was used, and the CV was the Si–Si distance for the the crystal. The results +of Figure 8c show that for this model, the minimum in the free energy +corresponds almost exactly to the experimental value for the Si–Si nearest +distance of 2.35 Å. +3.2. Performance +Our analysis revealed that PySAGES is at least ∼14–15 times faster than +SSAGES on a GPU machine containing four V100 GPUs. To obtain this esti- +mate, we ran enhanced sampling using umbrella sampling along the center +of mass distance between two spherical polymer domains to measure the +free energy landscape of the fssion of a spherical diblock-copolymer blend +(Figure 5) described in section 3.1.2. For support and compatibility across +the libraries and MD engine versions, we estimated the performance with +SSAGES v0.9.2-alpha and PySAGES v0.3.0 using HOOMD-blue v2.6.0 and +HOOMD-blue v2.9.7, respectively. +3.2.1. GPU utilization analysis +PySAGES is designed to execute every compute-intensive step of a sim- +ulation on the GPU and have zero copy instruction between GPU device +and host CPU memory for its explicit backends for HOOMD-blue [4] and +21 + +Figure 8: Free energy calculation of: a) Water internal angle from a DeepMD model with +ASE, b) Si–Si–H angle of GAP model with ASE and c) Si–Si distance of a GNN model with +JAX MD. +OpenMM [5], while still providing Python code for the user through JAX +[63]. In this section, we investigate the calculation efciency of PySAGES by +examining two example systems, one for each backend. +For HOOMD-blue, we are investigating a system of highly coarse-grained +DPD diblock-copolymers as discussed in section 3.1.2. The simulation box +contains a total of 𝑛𝑁 = 51 200 particles at a density of 𝜌 = 51.2/𝜎3, which +we use for benchmarking purposes with an Nvidia V100 GPU hosted on an +Intel Xeon Gold 6248R CPU @ 3.00GHz. Running only with HOOMD-blue +v2.9.7 we achieve an average time steps per second (TPS) of 754, which is +the expected high performance of HOOMD-blue on GPUs. +Figure 9 shows a detailed profled timeline during the execution of a +single time step. During 1.8 ms, HOOMD-blue spends the most computa- +tional efort on the calculation of pairwise DPD forces. It can be noted that +HOOMD-blue is designed to have almost no idle time of the GPU during a +time step. As soon as PySAGES is added computation part, we observe that +an additional part is added to calculate the CV and add the forces to every +particle. This causes a small period of idle of the GPU, since the execution +also requires action of the Python runtime interface with JAX. In the future, +we plan to launch the calculation of CV asynchronously with the regular +22 + +a +a +H-O-H angle +Si-Si-H angle +2.5 +2.5 +(eV) +2 +2 +energy +1.5 +1.5 +Free +1 +1 +0.5 +0.5 +0 +0 +60 +80 +100 +120 +140 +160 +180 +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +Cv (degrees) +cv (degrees) +c) +3 +Si-Si distance +2.5 +(eV) +energy +1.5 +Free +1 +0.5 +0 +2 +2.2 +2.4 +2.6 +2.8 +3 +3.2 +3.4 +CV (Angstroms)Hoomd-blue, DPD simulation +PySAGES + Hoomd-blue, COM harmonic biasing +a b c d e f +a b c f +247μs +Figure 9: The fgure shows a 1.8ms section of profled timeline recorded with Nvidia Nsight +systems on an Nvidia V100 GPU. The top row shows a vanilla HOOMD-blue simulation step, +while the bottom row shows a PySAGES/HOOMD-blue simulation with harmonic biasing +of a center of mass CV. Light-blue represents the GPU activity while dark-blue represents +individual CUDA compute kernels. The maroon letters show case the same compute steps in +both simulations: a) First half-step of integration, b) compute of bond forces, c) pair-forces, +d) calculation of the CV, e) addition of the harmonic biasing force to the HOOMD-blue +simulation, and f) the second integration step. Sections d) and e) are PySAGES only and are +executed on the GPU. We observe GPU idle time during the PySAGES Python coordination +with GPU–JAX/CuPy (green bar), but note that there is no memory copies even within the +GPU memory. The additional time for CV biasing per time step is 247 μs (teal bar). +force calculation, which would hide this small CPU-intensive GPU idle time. +However, we measure that the total delay due to the extra computation is +about 247 μs only: an acceptable overhead for the user-friendly defnition +of CVs. +In order to connect multiple points in CV space we can use enhanced +sampling methods such as umbrella sampling (see section 2.1.2) or the +improved string method (see section 2.1.3) to calculate the MFEP. Common +for these advanced sampling methods that multiple replica of the system +are simulations. With PySAGES we easily parallelize their execution using +the Python module mpi4py and its MPIPoolExecutor. This enables us +to execute replica of the simulations on multiple GPUs even as they span +diferent host machines. In our example, we used 14 replicas for umbrella +integration with 7 Nvidia V100 GPUs. The use of a single V100 GPU to execute +the simulations with 5 · 105 time steps for all replicas takes 2 hours and 59 +minutes. Ideal scaling with 7 GPUs reduces the time to solution to about +26 minutes. With our MPI-parallel implementation, we achieve a time-to- +solution of 28 minutes. Synchronization overhead and nonparallel aspects +like fnal analysis sum up to 2 minutes or about 9% overhead. This multi-GPU +implementation via MPI enables automatically efcient enhanced sampling +in high performance computing (HPC) environments. +For enhanced sampling methods that are designed for single replica sim- +ulations, we ofer an implementation that allows multiple replicas to run in +parallel, known as embarrassingly parallel computing. In this situation, the +build-in analysis averages the results from multiple replicas and estimates +23 + ++104.8ms ++105ms ++105.2ms ++105.4ms ++105.6ms ++105.8ms ++106ms ++106.2ms ++106.4ms ++106.6ms +..2ms ++435.4ms ++435.6ms ++435.8ms ++436ms ++436.2ms ++436.4ms ++436.6ms ++436.8ms ++437ms +void gpu_compute_dpd_forces_kernel(double4 *, double *, unsigned int, unsi.. +void gpu_computeidp.OpenMM OPLS all-atom simulation +636μs +b c af +b c af d e +PySAGES + OpenMM 31 COM harmonic biasing +Figure 10: 1.6ms profled time line of an OpenMM OPLS simulation of 40, 981 particles as +polymers with particle mesh Ewald (PME) summation for long-range Coulomb forces. The +colors and labels are identical to Figure 9. OpenMM works with asynchronous GPU kernel +execution, which leads to less linearly sorted timelines, compared with HOOMD-blue, but +we can still identify the CV calculation d) and force biasing e) and the synchronization idle of +the GPU (green). Overall, the performance degradation is more pronounced with OpenMM +compared to HOOMD-blue. +uncertainties. +In the previous section, we have demonstrated the fast GPU interoper- +ability between PySAGES and HOOMD-blue via JAX. However, the concept +of PySAGES is to develop enhanced sampling methods independently of the +simulation backend, so here we demonstrate that similar performance can +be achieved with OpenMM. Since OpenMM focuses on all-atom simulations, +we simulate an all-atom model of a polymer with the BigSMILES [64] no- +tation {[$]CC([$])(C)C(OCC(O)CSC1=CC=C(F)C(F)=C1)=O} with an +OPLS-AA force feld [65, 66] including long-range Coulomb forces via particle +mesh Ewald (PME). We simulate a bulk system of 40mers with 31 macro- +molecules present, adding up to 40 981 atoms. As a proof of concept, we +calculated the center of mass for every polymer chain and biased it harmon- +ically via PySAGES. As a performance metric, we evaluate the nano-seconds +per day (NS/DAY) executed on the same hardware confguration as a the +HOOMD-blue example above. For the unbiased, pure OpenMM simulation +we achieve a performance of ≈ 136 NS/DAY. For the PySAGES biased sim- +ulation, we achieve a performance of ≈ 75 NS/DAY, equating to a biasing +overhead of approximately 50%. Figure 10 shows a similar time series anal- +ysis as for HOOMD-blue. +It is notable that OpenMM’s execution model makes more use of parallel +execution of independent kernels, which also changes the order of execu- +tion compared to HOOMD-blue. As a result, the same CPU synchronization +changes the execution more drastically than in HOOMD-blue. Additionally, +a single time step for this system is faster executed compared to HOOMD- +blue, making the synchronization overhead more noticeable. In this case, +24 + ++176.8ms ++177ms ++177.2ms ++177.4ms ++177.6ms ++177.8ms ++178ms ++178.2ms +computeBondedForces +computeNonbonded +gridSpreadchargefi... +computeNonbonded +mm +gridSpreadCharge +void vector fft<(unsi... +50... +find... +computeBonded...+200.6ms ++200.8ms ++201ms ++201.2ms ++201.4ms ++201.6ms ++201.8ms ++202ms ++202.2 +L +comput... +computeNonbonded + so...find... +g... +gridSpread... +comput... +computeNonbonded +so...find... +gridspread..g... +...+200.6ms ++200.8ms ++201ms ++201.2ms ++201.4ms ++201.6ms ++201.8ms ++202ms ++202.2 +L +comput... +computeNonbonded + so...find... +g... +gridSpread... +comput... +computeNonbonded +so...find... +gridspread..g... +...parallelization of PySAGES and OpenMM is projected to have a bigger perfor- +mance advantage. Furthermore, we notice that the calculation of the center +of mass and the biasing of all 31 polymer chains is more costly than the +single CV in the previous example. The combination of these factors explain +the higher PySAGES overhead for this OpenMM simulation, but overall per- +formance is good and signifcantly better for alternative implementations +that require CV calculations on the CPU. +4. Conclusion +We have introduced PySAGES, a library for enhanced sampling in molec- +ular dynamics simulations, which allows users to utilize a variety of en- +hanced sampling methods and Collective Variables, as well as to implement +new ones via a simple Python and JAX-based interface. +We showed how PySAGES can be used through a number of example +applications in diferent felds such as drug design, materials engineering, +polymer physics, and ab-initio MD simulations. We hope that these convey +for the reader the fexibility and potential of the library for addressing a +diverse set of problems in a high-performance manner. +As our analysis showcased, for large problems, PySAGES can perform +biased simulation well over one order of magnitude faster than a library +such as SSAGES even when the backend already performs computations on +a GPU. +Overall, we believe that PySAGES provides a useful tool for researchers +interested in performing molecular and ab-initio simulations in multiple +felds, due to its user-friendly framework for defning and utilizing sampling +methods and collective variables, as well as its high performance on GPU +devices. +4.1. Outlook +Being a more recently developed library PySAGES might not be as fea- +tureful library as some of the more mature enhanced sampling libraries. In +concrete, we plan to add the ability to allow the user to perform restarts. +The analysis conducted on how the GPU is used also reveals some optimiza- +tion opportunities, such as performing PySAGES-side computations fully +asynchronously with the computation of the forces of the backend. +PySAGES also ofers an exciting platform to develop fully end-to-end +diferentiable free energy calculations, and we expect that in the future this +serves as a starting point to develop newer strategies for force-feld and +materials design. +25 + +Appendix A. Collective variable for the distance to an interface +Implementation of the CV described in section 2.2.1, that is, the distance +between a group of atoms to an interface defned by another group of atoms. +class DistanceToInterface(TwoPointCV): +def __init__(self, indices, axis, sigma, scope, bins=100, coeff=1): +super().__init__(indices) +self.axis = axis +self.sigma = sigma +self.scope = scope +self.bins = bins +self.coeff = coeff +@property +def function(self): +return lambda r1, r2: distance_to_interface( +r1, r2, axis=self.axis, +sigma=self.sigma, scope=self.scope, +bins=self.bins, coeff=self.coeff +) +def distance_to_interface(p1, p2, axis, sigma, scope, bins, coeff): +mobile_axis = barycenter(p1)[axis] +positions_axis = p2.flatten()[axis::3] +centers = np.linspace(scope[0], scope[1], bins) +centers = np.expand_dims(centers, 1) +positions_axis = np.expand_dims(positions_axis, 0) +diff = positions_axis - centers +mass = np.exp(-0.5 * (diff / sigma) ** 2) +mass = np.sum(mass, axis=1) +mass_diff = np.abs(mass[1:] - mass[:-1]) +centers = np.squeeze(centers) +centers_mean = (centers[1:] + centers[:-1]) / 2 +probability = nn.softmax(mass_diff * coeff) +interface = np.sum(probability * centers_mean) +return mobile_axis - interface +26 + +Appendix B. Benchmark test systems +In the following sections, we present the results of the free energy cal- +culation for the benchmark test systems of alanine dipeptide and butane. +The details of all the parameters chosen to perform the enhanced sampling +simulation of these are summarized in Appendix B.3. +Appendix B.1. Alanine Dipeptide +The frst test system involves alanine dipeptide in vacuum (Figure B.11), +a benchmark system for enhanced sampling methods that is frequently +used in the literature. +3 +2 +1 +0 +1 +2 +3 +3 +2 +1 +0 +1 +2 +3 +SpectralABF - 2.0 ns +0 +20 +40 +60 +80 +A (kJ mol +1) +3 +2 +1 +0 +1 +2 +3 +3 +2 +1 +0 +1 +2 +3 +Metadynamics - 12.0 ns +0 +20 +40 +60 +80 +A (kJ mol +1) +3 +2 +1 +0 +1 +2 +3 +3 +2 +1 +0 +1 +2 +3 +ANN - 4.0 ns +0 +20 +40 +60 +80 +A (kJ mol +1) +3 +2 +1 +0 +1 +2 +3 +3 +2 +1 +0 +1 +2 +3 +CFF - 16.0 ns +0 +20 +40 +60 +80 +A (kJ mol +1) +Figure B.11: Free energy landscape of alanine dipeptide (Amber ff99SB [67]) in vacuum as a +function of the dihedral angles 𝜙 and 𝜓 obtained with PySAGES and OpenMM via diferent +enhanced sampling methods: ABF, Metadynamics, Spectral ABF, ANN, FUNN, CFF. Each +panel also indicates the length of the simulation necessary for the free energy to converge. +Appendix B.2. Butane +As a second test system, we compute the free energy profle along the C- +C-C-C dihedral angle, 𝜙𝐶𝐶𝐶𝐶, of a butane molecule (in vacuum), Figure B.12. +Appendix B.3. Example System Details +27 + +FUNN - 4.0 ns +3 +80 +2 +1- +60 +A (kJ mol-1) +0 +40 +-1 - +20 +-2 +-3 +0 +0 +2 +-3 +-2 +-1 +1 +3ABF - 40.0 ns +3 +80 +2 +1 - +60 +A (kj mol-1) +40 +-1 - +20 +-21 +-3 +-2 +-3 +-1 +0 +1 +2 +33 +2 +1 +0 +1 +2 +3 +CCCC +0 +1 +2 +3 +4 +5 +6 +A (kcal mol +1) +ANN (2.0 ns) +CFF (1.0 ns) +FUNN (2.0 ns) +WTMD (16.0 ns) +SpectralABF (1.0 ns) +Figure B.12: Free energy profle along the dihedral angle of a butane molecule (using an OPLS- +based force feld [65]) obtained via diferent enhanced sampling methods with PySAGES and +HOOMD-blue: ANN, CFF, FUNN, Spectral ABF, WTMD. The legend also indicates the length +of the simulation. +Table B.1: Parameters and methods details for the various examples. 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Simmer- +ling, Comparison of multiple amber force felds and development of +improved protein backbone parameters, Proteins: Structure, Func- +tion, and Bioinformatics 65 (3) (2006) 712–725. +36 + diff --git a/xtE3T4oBgHgl3EQf_gt3/content/tmp_files/load_file.txt b/xtE3T4oBgHgl3EQf_gt3/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..09523031439f8e08c0bc4bb8f16df47d34dd20a2 --- /dev/null +++ b/xtE3T4oBgHgl3EQf_gt3/content/tmp_files/load_file.txt @@ -0,0 +1,1631 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf,len=1630 +page_content='PySAGES: fexible, advanced sampling methods accelerated with GPUs Pablo F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Zubieta Ricoa, Ludwig Schneidera, Gustavo Perez-Lemusa, Riccardo Alessandria, Siva Dasettya, Cintia A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Menéndeza, Yiheng Wua, Yezhi Jina, Trung Nguyenb, John Parkerb, Andrew L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Fergusona, Juan J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' de Pabloa aPritzker School of Molecular Engineering, The University of Chicago, 5640 South Ellis Avenue, Chicago, 60637, IL, USA bResearch Computing Center, The University of Chicago, 6054 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Drexel Avenue, Chicago, 60637, IL, USA Abstract Molecular dynamics simulations are a core element of research in physics, chemistry and biology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' A key aspect for extending the capability of simula- tion tools is providing access to advanced sampling methods and techniques that permit calculation of the relevant, underlying free energy landscapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' In this sense, sofware tools that can be seamlessly adapted to a broad range of complex systems are essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Building on past eforts to provide an open-source community supported sofware for advanced sampling, we introduce PySAGES, a Python implementation of the Sofware Suite for Ad- vanced General Ensemble Simulations (SSAGES) that provides full support of GPUs for massively parallel applications of enhanced sampling methods such as adaptive biasing forces, harmonic bias, and forward fux sampling in the context of molecular dynamics simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' By providing an intuitive interface that facilitates the treatment of the confguration of the system, the inclusion of new collective variables, and the implementation of sophis- ticated free energy methods, the PySAGES library will serve as a general platform for development and implementation of emerging simulation algo- rithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' The capabilities and core features of this new tool are demonstrated with clear and concise examples pertaining to diferent classes of molecular systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Keywords: Enhanced sampling methods, GPU acceleration Preprint January 13, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='04835v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='comp-ph] 12 Jan 2023 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Introduction Molecular simulations are used extensively in a wide range of science and engineering disciplines [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' As their use has grown for discovery of new phenomena, or for interpretation of sophisticated experimental mea- surements, so have the complexity of the systems under consideration and the ambition of simulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Classical atomistic molecular dynamics (MD) simulations however, continue to be limited to microsecond time scales and tens of nanometers length scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' For systems that are characterized by rugged free energy length scales, such time scales are insufcient to ensure sufcient sampling of the relevant phase space, and advanced methods must therefore be adopted to overcome free energy barriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' In that regard, it is useful and increasingly common to rely on the use of properly chosen collective variables (CVs) which are generally diferentiable functions of the atomic coordinates of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' The rapid growth of hardware accelerators such as GPUs or TPUs, or specialized hardware designed for fast MD computations [2, 3] has provided researchers with increased opportunities for longer simulations of larger systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' GPUs, in particular, provide a widely accessible option, and several simulation packages, such as HOOMD-blue [4], OpenMM [5], JAX MD [6, 7], and Gromacs, are now available for MD simulations on such devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' In general, enhanced sampling methods can be used to overcome the high energy barriers that separate multiple metastable states in a system, while still allowing for the recovery of relevant thermodynamic and kinetic quantities as functions of diferent CVs such as reaction rates or free energy surfaces (FES).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Several libraries, such as PLUMED [8], Colvars [9], and our previously in-house developed SSAGES package [10], provide out-of-the-box solutions for performing enhanced sampling MD simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Among the various enhanced sampling methods, some of the most re- cently devised schemes rely on machine learning (ML) strategies to ap- proximate free energy surfaces and their gradients (generalized forces) [11, 12, 13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Similarly, algorithms for identifying meaningful CVs that cor- relate with the slowest degrees of freedoms (DOFs) are based on autoen- coders [15, 16, 17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' These advances serve to highlight the need for seam- less integration of ML frameworks with existing MD sofware libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' To date, there are no solutions that combine enhanced sampling tech- niques, hardware acceleration, and ML frameworks to facilitate enhanced- sampling MD simulations on GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' While some MD libraries that sup- port GPUs provide access to a limited set of enhanced sampling meth- ods [5, 19, 20, 21, 22], there are currently no packages that enable users 2 to take advantage of all of these features within the same platform and in the same backend-agnostic fashion that tools such as PLUMED and SSAGES have provided for CPU-based MD simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Here we present PySAGES, a Python Suite for Advanced General Ensem- ble Simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' It is a free, open-source sofware package written in Python and based on JAX that follows the design ideas of SSAGES and enables users to easily perform enhanced-sampling MD simulations on CPUs, GPUs, and TPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' PySAGES can currently be coupled with HOOMD-blue, OpenMM, JAX MD and, Atomic Simulation Environment (ASE) and, by extension from the latter, to CP2K, Quantum ESPRESSO, VASP, and Gaussian, among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' At this time, PySAGES ofers the following enhanced sampling meth- ods: Umbrella Sampling, Metadynamics, Well-tempered Metadynamics, Forward Flux Sampling, String Method, Adaptive Biasing Force, Artifcial neural network sampling, Adaptive Biasing Force using neural networks, Combined Force Frequency, and Spectral Adaptive Biasing Force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' PySAGES also includes some of the most commonly used CVs but, importantly, defn- ing new ones is relatively simple as long as they can be expressed in terms of the NumPy [23] interface provided by JAX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' All CVs can be automatically diferentiated through JAX functional transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' PySAGES is highly modu- lar, thereby allowing for the easy implementation of new methods as the emerge, even as part of a user-facing script.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' In the following sections, we provide a general overview of the design and implementation of PySAGES and present a series of examples to show- case its fexibility for tackling problems in diferent application areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' We also discuss its performance on GPUs and present some perspectives on how to grow and improve the package to cover more research use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Implementation We begin by briefy outlining the core components of PySAGES, how they play together, and how communication with each backend allows PySAGES to bias a simulation during its run course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' A summary of the execution workfow of PySAGES along with a mapping of the user interface with the main stages of the simulation and the interaction with the backends, are illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' To provide a uniform user interface while minimizing disruption to pre- existing workfows, PySAGES only requires the user to wrap their traditional backend scripting code into simulation generator functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' This approach accommodates the heterogeneity of Python interfaces across the diferent simulation backends that PySAGES supports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' An example of a simulation 3 pysages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='run( , , ) pysages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='analyze( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=') ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='generate_simulation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='function ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='Collective variable (CV) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='User ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='Backend ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='Sampling method ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='generate_simulation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='is called on every rank ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='Device query and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='Snapshot creation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='Creation of replicas of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='the simulation system ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='Automatic differentiation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='of the CV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='Functional specialization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='of the sampling method ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='Launch simulation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='Simulation time step ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='Simulation ends ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='Computation of forces ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='Calculation of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='free energy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='Transparent (zero-copying) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='wrapping of particle data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='Sampling method update,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' computation of biasing forces Addition of biasing forces to the backend forces repeats until stopping criterion is reached CVs and Sampling Methods can be user defined or imported from pysages Figure 1: The PySAGES simulation fowchart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' For a simulation, a user sets up a script that declares the CV and sampling methods to be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' 4 import openmm import openmm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='unit as unit import openmm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='app as app pdb = app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='PDBFile("adp-vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='pdb") ff = app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='ForceField("amber99sb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='xml") positions = pdb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='getPositions(asNumpy=True) system = ff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='createSystem( pdb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='topology, constraints=app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='HBonds, nonbondedMethod=app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='PME, nonbondedCutoff=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='0 * unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='nanometer ) integrator = openmm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='LangevinIntegrator( 298.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='15 * unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='kelvin, 1 / unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='picosecond, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='0 * unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='femtoseconds ) simulation = app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='Simulation(pdb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='topology, system, integrator) simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='setPositions(positions) simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='minimizeEnergy() simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='run(int(1e6)) def generate_simulation(): pdb = app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='PDBFile("adp-vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='pdb") ff = app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='ForceField("amber99sb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='xml") positions = pdb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='getPositions(asNumpy=True) system = ff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='createSystem( pdb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='topology, constraints=app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='HBonds, nonbondedMethod=app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='PME, nonbondedCutoff=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='0 * unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='nanometer ) integrator = openmm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='LangevinIntegrator( 298.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='15 * unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='kelvin, 1 / unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='picosecond, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='0 * unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='femtoseconds ) simulation = app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='Simulation(pdb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='topology, system, integrator) simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='setPositions(positions) simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='minimizeEnergy() return simulation import openmm import openmm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='unit as unit import openmm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='app as app import pysages from numpy import pi from pysages import ABF, DihedralAngle, Grid cvs = [DihedralAngle([4, 6, 8, 14]), DihedralAngle([6, 8, 14, 16])] grid = Grid(lower=(-pi, -pi), upper=(pi, pi), shape=(32, 32), periodic=True) method = ABF(cvs, grid) raw_results = pysages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='run(method, generate_simulation, int(1e6)) result = pysages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='analyze(raw_result) Lines removed Lines added for pysages Preserved user code Figure 2: Example of how to use the Python interface for PySAGES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' It is easy to extend existing MD scripts with PySAGES to perform enhanced-sampling MD, without many changes to the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' In general, the only requirement is for the user to wrap the code that defnes the simulation system into a simulation generator function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' generator function and how a traditional OpenMM script can be modifed to perform an enhanced-sampling MD simulation is depicted in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' At the start of a simulation, the simulation generator function is called to instantiate as many replicas of the simulation as needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Then, for each replica, PySAGES queries the particle information and the device that the backend will be using.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' In addition, during this initial stage PySAGES also performs automatic diferentiation of the collective variables via JAX’s grad transform required to estimate the biasing forces, and generates special- ized initialization and updating routines for the user-declared sampling method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Like SSAGES, PySAGES wraps the simulation information into an object called a Snapshot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' This object exposes the most important simulation infor- mation, such as particle positions, velocities, and forces in a backend- and device-agnostic format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' To achieve this, PySAGES uses DLPack [24] for C++ based MD libraries to directly access the contents of the backend- allocated bufers for the diferent particle properties without creating data copies whenever possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Once the setup of both the simulation and sampling method is completed, PySAGES hands control back to the backend, which will run for a given number of time steps or until some other stopping criteria is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' In order to exchange information back and forth, PySAGES adds a force-like object or function to the backend which gets called as part of the time 5 integration routine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Here, the sampling method state gets updated and the computed biasing forces are added to the backend net forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Finally, the information collected by the sampling method is returned and can be used for calculating the free energy as function of the selected CVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Unlike SSAGES, PySAGES ofers a user-friendly analyze interface that simplifes the process of performing post simulation analysis, including the automatic calculation of free energies based the chosen sampling method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Thus, reducing the time and efort required to gain valuable insights from simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' PySAGES ofers an easy way to leverage diferent parallelism frameworks including MPI with the same uniform fronted available to run enhanced sampling simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' This is achieved via Python’s concurrent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='futures interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' In particular, for MPI parallelism, the user only needs to pass an additional MPIPoolExecutor (from mpi4py) to PySAGES’ run method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' If the user selects a method such as UmbrellaSampling, the workload for each image will be distributed across available MPI nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' On the other hand, for most of the sampling methods, the parallelization interface allows the user to run multiple replicas of the same system to enable, for instance, analysis of the uncertainties associated to computing the free energy of a given system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' To ensure the reproducibility and correctness of our implementation and to follow sofware engineering best practices, we have implemented a comprehensive unit tests suite, and leverage GitHub’s continuous integra- tion services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' In addition, we use trunk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='io [25] to adhere to some quality standards as well as to ease the collaboration of developers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Enhanced Sampling Methods While we assume the reader has some basic understanding of enhanced sampling methods, here we provide a more detailed overview of these tech- niques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' In addition, we discuss the general structure of how enhanced sampling methods are implemented within PySAGES, and also present a summary of the various methods already available in the library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Enhanced sampling methods are a class of simulation techniques that manipulate regular MD simulations in order to more efectively sample the confguration space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' In MD a collective variable, 𝜉, is typically a function of the positions of all particles, ˆ𝜉({𝑟𝑖}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' For a given statistical ensemble (such as the canonical, NVT), the cor- responding free energy can be written as 𝐴 = −𝑘B𝑇 ln(𝑍), where 𝐴 is the Helmholtz free energy and 𝑍 is the canonical partition function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' To make ex- plicit the dependency of the free energy on 𝜉, let us write down the partition 6 function: 𝑍(𝜉) ∝ ∫ d𝑁𝑟𝑖 𝛿( ˆ𝜉({𝑟𝑖}) − 𝜉) 𝑒−𝑈({𝑟𝑖})/𝑘B𝑇 (1) Normalizing this partition function gives us the probability of occur- rence, 𝑝(𝜉), for confgurations in the CV subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Substituting this proba- bility into the expression for the free energy, we get: 𝐴(𝜉) = −𝑘B𝑇 ln(𝑝(𝜉)) + 𝐶 (2) where 𝐶 is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' If we take the derivative of the free energy with respect to 𝜉 we get 𝑑𝐴(𝜉) 𝑑𝜉 = ∫ d𝑁𝑟𝑖 𝑑𝑈 𝑑𝜉 𝛿( ˆ𝜉({𝑟𝑖}) − 𝜉) 𝑒−𝑈({𝑟𝑖 })/𝑘B𝑇 ∫ d𝑁𝑟𝑖 𝛿( ˆ𝜉({𝑟𝑖}) − 𝜉) 𝑒−𝑈({𝑟𝑖 })/𝑘B𝑇 = �𝑑𝑈 𝑑𝜉 � 𝜉 , (3) where ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='⟩𝜉 denotes the conditional average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' The goal of enhanced sampling methods is to accurately determine either 𝑝(𝜉) or 𝑑𝐴(𝜉)/𝑑𝜉 from which 𝐴(𝜉) can be recovered in a compu- tationally tractable manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' In PySAGES, the implementation of sampling methods follows the JAX functional style programming model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' New methods are implemented as subclasses of theSamplingMethod class, and are required to defne abuild method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' This method returns two methods, initialize and update, used as part of the process of biasing the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' For readers familiar with JAX MD, these could be thought of as analogues to the higher level functions returned by JAX MD’s simulate integration methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' The initialize method allocates all the necessary helper objects and stores them in a State data structure, while the update method uses the information from the simulation at any given time to update the State.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' While PySAGES allows new methods to be written seamlessly as part of Python scripts used to set up molecular dynamics simulations, it also provides out-of-the-box implementations of several of the most important known sampling methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' We list and briefy detail them next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Harmonic Biasing One simple way to sample a specifc region of the phase space is to bias the simulation around a point 𝜉0 with harmonic bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' This adds a quadratic potential energy term to the Hamiltonian that increases the potential energy quadratically as a system moves away from the target point: H𝑏 = H + 7 𝑘/2(𝜉 − 𝜉0)2, where 𝑘 > 0 is the spring constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' The unbiased probability distribution 𝑝(𝜉) can be recovered by dividing the biased distribution by the known weight of the bias 𝑝(𝜉) = 𝑝𝑏(𝜉)/𝑒−𝑘/2(𝜉−𝜉0)2/𝑘B𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' The disadvantage of this approach is that it can only be used to explore the free energy landscape near a well-know point in phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' This may not be sufcient for many systems, where the free energy landscape is complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Umbrella Integration Umbrella sampling is a technique that builds of harmonic biasing by combining multiple harmonically-biased simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' It is a well-known method for exploring a known path in phase space to obtain a free energy profle along that path [26, 27] Typically, a path between to point of inter- est is described by 𝑁 points in phase space, 𝜉𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' At each of these points, a harmonically biased simulation is performed, and the resulting occurrence histograms are combined to obtain a single free energy profle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' In PySAGES, we implement umbrella integration for multi-dimensional CVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' This method approximates the forces acting on the biasing points and integrates these forces to fnd the free energy profle 𝐴(𝜉), and allows to explore complex high-dimensional free energy landscapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Improved String Method When only the endpoints are known, but not the path itself, the improved (spline-based) string method can be used to fnd the mean free energy pathway (MFEP) between these two end-points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' The spline-based string method improves the original string method by interpolating the MFEP using cubic-splines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' In this method, the intermediate points of the path are moved according to the recorded mean forces acting on them, but only in the direction perpendicular to the contour of the path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' This ensures that distances between the points along the path remain constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' This method has been widely used and has been shown to be an efective way to fnd the MFEP between two points in the phase space [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Adaptive Biasing Force sampling The adaptive biasing force (ABF) sampling method is a technique used to map complex free-energy landscapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' It can be applied without prior knowledge of the potential energy of the system, as it generates on-the-fy estimates of the derivative of the free energy at each point along the integra- tion pathway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' ABF works by introducing an additional force to the system that biases the motion of the atoms, with the strength and direction of the 8 bias is continuously updated during the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' In the long-time limit, this yields a Hamiltonian with no average force acting along the transition coordinate of interest, resulting in a fat free-energy surface and allowing the system to display accelerated dynamics, thus providing reliable free- energy estimates [29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Similarly to SSAGES, PySAGES implementation of ABF is based on the algorithm described in [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Metadynamics Metadynamics is another popular approach for enhancing sampling of complex systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' In metadynamics [31], a bias potential is applied along one or more CVs in the form of Gaussian functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' The height and width (𝜎) of these Gaussians are controlled by the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' The Gaussian bias potentials are cumulatively deposited at user-defned intervals during the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' In standard metadynamics, the height of the Gaussian bias potentials is fxed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' In contrast, for well-tempered metadynamics (WTMD) [32] simulations, the height of the Gaussian bias potentials is adjusted at each timestep using a preset temperature based bias factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' This scaling of Gaussian heights in WTMD leads to faster convergence compared to standard metadynamics, as it restricts the range of free energy explored to a range defned by the bias factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' In PySAGES, we have implemented both standard metadynamics and WTMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' The well-tempered variant is activated when a user sets a value for the bias factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' To improve the computational performance, we have added optional support for storing the bias potentials in both on a pre-defned grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' This allows users to trade-of accuracy for faster simulations, depending on their needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Forward Flux Sampling Forward fux sampling (FFS) belongs to a diferent family of enhanced sampling methods than the ones described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' In the previously de- scribed methods, the free energy change from a region in the phase space (𝐴) to the region of interest (𝐵) is calculated by applying a bias to the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' In FFS no bias is added and instead an efcient selection of trajectories that crosses the phase space from 𝐴 to 𝐵 is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Since no bias is used, the intrinsic dynamics of the system is conserved and therefore kinetic and microscopic information of the transition path can be studied [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' In PySAGES we have implemented the direct version of FFS [34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Artifcial neural networks sampling Artifcial neural networks sampling (ANN) [11] employs regularized neu- ral networks to directly approximate the free energy from the histogram of visits to each region of the CV space, and generates a biasing force that avoids ringing and boundary artifacts [11], which are commonly observed in methods such as metadynamics or basis functions sampling [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' This approach is efective at quickly adapting to diverse free energy landscapes by interpolating undersampled regions and extrapolating bias into new, unexplored areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' The implementation on PySAGES ofers more fexible approaches to network regularization than SSAGES, which uses Bayesian regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Force-biasing using neural networks Force-biasing using neural networks (FUNN) [12] is based upon the same idea as ANN, that is, relying on artifcial neural networks to provide con- tinuous functions to bias a simulation, but instead of using the histogram to visits to CV space it updates its network parameters by training on the ABF estimates for the mean forces as the simulation advances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' This method shares all of the features of ABF, but the smooth approximation of the gen- eralized mean force it produces enables much faster convergence to the free energy of a system compared to ABF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Combined Force Frequency sampling The combined force frequency sampling (CFF) method [13] combines the speed of generalized-force based techniques such as ABF or FUNN with the advantages of frequency-based methods like metadynamics or ANN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' No- table improvements over earlier force-based methods include eliminating the need for hyperparameters to dampen early-time estimates, automating the integration of forces to generate the free energy, and providing an ex- plicit expression for the free energy at all times, enabling the use of replica exchange or reweighing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' In principle, by using sparse storage of histograms, it should be possible to scale the method to higher dimensions without encountering memory limitations, such optimization is however not yet implemented in PySAGES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Spectral Adaptive Biasing Force Spectral ABF [37] is a method that follows the same principle as neural- network-based sampling methods, in that it builds a continuous approxima- tion to the free energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' However, in contrast to methods like FUNN it does so by ftting exponentially convergent basis functions expansions, and could 10 be thought as a generalization of the Basis Functions Sampling Method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' In contrast to the latter, and similar to CFF, it allows for the recovery of an explicit expression for the free energy of a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' It is an extremely fast method in terms of both runtime and convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Collective variables As previously mentioned, enhanced sampling calculations commonly involve the selection of a CV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' An appropriate CV for a given system could simply be the distance between the centers of mass of two groups of atoms, but could be a complex specialized quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Below, we list a set of CVs predefned in PySAGES, sorted by the number of groups of atoms coordinates necessary for their use: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' TwoPointCV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' This subclass is for CVs that need two groups for their defnition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' This includes Distance and Displacement (vector).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' ThreePointCV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Subclass of CVs with three groups of atoms, such as Angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' FourPointCV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Subclass of CVs with four groups of atoms, such as DihedralAngle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' AxisCV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Subclass of CVs that are projected on a determinated axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' This includes Component and PrincipalMoment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' CollectiveVariable General base class for all CVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' In PySAGES, CVs that directly derive from this class, and do not belong to the previous groups, include: RingPhaseAngle, RingAmplitude, RadiusofGyration, Asphericity, Acylindricity, ShapeAnisotropy, RingPuckeringCoordinates [38] (vector).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' In PySAGES we provide users with a simple framework for defning CVs, which are automatically diferentiated with JAX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' To illustrate this, we compare how to write the calculation of a CV that measures the projection of the vector between two groups of atoms over the axis that passes by other two groups, in both SSAGES and PySAGES (see Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' In PySAGES the gradient calculation is done automatically whereas in SSAGES it has to be coded explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' The following second example shows the power of diferential program- ming for CV declaration in PySAGES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Case study: A collective variable for interfaces When the two immiscible liquids are in contact with each other, the density of one liquid experiences a gradual change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' This transition region 11 Gradient calculation CV calculation // Preamble ommited class ParallelProjectionCV : public CollectiveVariable { public: ParallelProjectionCV(int atomid1, int atomid2, int atomid3) : atomids_({atomid1, atomid2, atomid3}) { } void Initialize(const Snapshot& snapshot) override { // Code ommited for brievity } void Evaluate(const Snapshot& snapshot) override { auto n = snapshot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='GetNumAtoms();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' auto idx_i = snapshot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='GetLocalIndex(atomids_[0]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' auto idx_j = snapshot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='GetLocalIndex(atomids_[1]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' auto idx_k = snapshot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='GetLocalIndex(atomids_[2]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' auto com_i = snapshot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='CenterOfMass(idx_i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' auto com_j = snapshot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='CenterOfMass(idx_j);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' auto com_k = snapshot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='CenterOfMass(idx_k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' auto rik = com_i - com_k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' auto rjk = com_j - com_k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' auto rij = rjk - rik;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' auto nrij = rij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='norm();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' auto nij = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='0 / nrij) * rij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' val_ = nij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='dot(-rik);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' // This writes the CV value // Manual computation of the gradient std��fill(grad_.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='begin(), grad_.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='end(), Vector3{0, 0, 0});' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' grad_.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='resize(n, Vector3{0, 0, 0});' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Matrix3 dij = Matrix3��Zero();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' dij(0, 0) = ( -(nij[1] * nij[1] + nij[2] * nij[2]) / nrij );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' // dx/dx dij(0, 1) = ( nij[0] * nij[1] / nrij );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' // dx/dy dij(0, 2) = ( nij[0] * nij[2] / nrij );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' // dx/dz dij(1, 0) = ( nij[1] * nij[0] / nrij );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' // dy/dx dij(1, 1) = ( -(nij[0] * nij[0] + nij[2] * nij[2]) / nrij );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' // dy/dy dij(1, 2) = ( nij[1] * nij[2] / nrij );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' dij(2, 0) = ( nij[2] * nij[0] / nrij );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' dij(2, 1) = ( nij[2] * nij[1] / nrij );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' dij(2, 2) = ( -(nij[1] * nij[1] + nij[0] * nij[0]) / nrij );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' grad_[idx_i] = dij * (-rik) - nij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' grad_[idx_j] = dij * rik;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' grad_[idx_k] = nij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' } static ParallelProjectionCV* Build( const Json��Value& json, const std��string& path ) { // Code ommited for brievity } private: Label atomids_;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' };' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' SSAGES // Preamble ommited class ParallelProjection(ThreePointCV): @property def function(self): return parallel_projection def parallel_projection(p1, p2, p3): r1 = barycenter(p1) r2 = barycenter(p2) r3 = barycenter(p3) a = r3 - r1 b = r2 - r1 return np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='dot(a, b) / norm(b) PySAGES Apart from the usual overhead involved in writting C++ code in comparison to Python, the gradients of a �� need to be manually implemented in ������, whereas in �y����� these are automatically computed with ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Figure 3: Example of how to write a CV in PySAGES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' On the lef is the same CVs written in SSAGES and on the right the PySAGES version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' In general, the only requirement is for the user to write the CV as a diferentiable function in JAX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' 12 is the liquid-liquid interface and its position has high importance in many studies (see section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' However, the location of such interface is not a trivial task since it generally fuctuates as the simulation progresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' As a representative CV for the interface, we can utilize the position of the point where the gradient of the density is maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' More formally, let 𝜌(𝑥) de- note the density of a liquid of interest at a coordinate 𝑥 on the perpendicular axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' We would like to fnd the location of the interface: 𝐼 = arg max 𝑥 |𝜌′(𝑥)| (4) However, the density function 𝜌(𝑥) is not directly measurable in a molec- ular simulation, as the coordinates of atoms are discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' To obtain an ap- proximation of 𝜌(𝑥), we divide the coordinates into multiple bins, each with a width of 𝛿, and create a histogram 𝑝(𝑥) that records the number of atoms falling into the bin around position 𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' In other words, 𝑝(𝑥) = ∑︁ 𝑖=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='𝑛 1(𝑥𝑖 − 𝑥) < 𝛿 2 (5) in which 𝑥𝑖 denotes the coordinate of atom 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' As written above, 𝑝(𝑥) is non- diferentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Therefore, as in other works [39], we utilize the kernel density trick with a Gaussian kernel to modify 𝑝(𝑥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' The modifed ˜𝑝(𝑥), is defned as: ˜𝑝(𝑥) = ∑︁ 𝑖=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='𝑛 exp − (𝑥𝑖 − 𝑥)2 2𝜎2 (6) in which 𝜎 is a hyperparameter that decides the width of the Gaussian kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Then, the gradient of the density can be approximated as: ˜𝑝 ′(𝑥) = ˜𝑝(𝑥 + 𝛿/2) − ˜𝑝(𝑥 − 𝛿/2) 𝛿 (7) and we calculate the location of the interface as 𝐼 = arg max𝑥 | ˜𝑝 ′(𝑥)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' The arg max operator is also non-diferentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' As a result, we replace it with a sofmax function that transforms the raw input into a probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Denote the 𝑚 bins as 𝑗 = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' 𝑚, and fnally we calculate the location of the interface as: 𝐼 = � 𝑗 𝑥 𝑗 exp | ˜𝑝 ′(𝑥 𝑗)| � 𝑗 exp | ˜𝑝 ′(𝑥 𝑗)| (8) As demonstrated in the code snippet for this CV, provided in Appendix A, PySAGES allows for the concise and straightforward implementation of complex CVs such as this one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' 13 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Results and Discussion To evaluate a sofware package like PySAGES, we must consider at least two factors: physical correctness and computational performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' First, to assess the correctness of the enhanced sampling methods imple- mented in PySAGES, we present in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='1 the free-energy landscape for the dihedral angles 𝜙 and 𝜓 of alanine dipeptide (ADP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' This example is commonly used to benchmark new enhanced sampling algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Sim- ilarly, we also show in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='2 the free-energy as a function of the dihedral angle of butane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Our results show that PySAGES reproduces the expected free-energy landscapes using diferent methods and backends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' In section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='1, we further investigate the applicability and correctness of PySAGES beyond these simple model systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Second, we demonstrate the performance of PySAGES on GPUs with two diferent backends in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' In particular, we compare the perfor- mance of enhanced sampling simulations to the performance of pure MD simulations, as well as other enhanced sampling implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Example applications of enhanced sampling with PySAGES To demonstrate the versatility and efectiveness of PySAGES in diferent contexts, we present several examples of how enhanced sampling methods can be used to gain valuable insights in various felds including biology, drug design, materials engineering, polymer physics, and ab-initio simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' These examples showcase how PySAGES can be used in diverse research areas and the utility of diferent enhanced sampling methods and backends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Overall, these examples confrm that the enhanced sampling methods implemented in PySAGES work as intended and provide results consistent with existing literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Structural Stability of Protein–Ligand Complexes for Drug Discovery High-throughput docking techniques are a widely-used computational technique in drug lead discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' However, these techniques are limited by the lack of information about protein conformations and the stability of ligands in the docked region [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' To address this issue, the Dynamical Undocking (DuCK) method was developed to evaluate the stability of the ligand binding by calculating the work required to break the most impor- tant native contact (hydrogen bond interactions) in the protein-ligand com- plex [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' This method has been shown to be complementary and orthogonal to classical docking, making both techniques work parallel in drug discover- ing [42, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' However, DuCK can be slow to converge when combined with 14 traditional enhanced sampling techniques [41], making it unsuitable for high-throughput drug discovery protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Here, we demonstrate how PySAGES with OpenMM can be used ef- ciently in drug discovery applications, where the user-friendly interface, native parallel capabilities, and new enhanced sampling methods with fast convergence are synergistically combined to accelerate the virtual screen- ing of ligand databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' In this example, we study the main protease (Mpro) of Sars-CoV-2 virus (PDB: 7JU7 [44]), where the ligands were removed and the monomer A was selected as the docking receptor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' A ligand with SMILES string CCCCOCC(=O)c1ccc(C)cc1N[C@H]1N[C@@H](c2cccnc2)CS1 was docked using RDock [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' The best scoring pose was used to initialize the sys- tem, which was simulated using the ff14SB [46], TIP3P [47], and GAFF [48] force felds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' A 10 ns equilibration procedure was carried out to fnd the most stable hydrogen bond between the ligand and the protein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' The last frame of this equilibration was then used to initialize the enhanced sampling calcula- tions in PySAGES with ABF, metadynamics, FUNN, ANN, and Spectral ABF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' These methods were compared against the same system simulated using Amber20 [49] with Steered Molecular Dynamics (see Figure 4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Our results suggest that we can reduce the simulation time by an order of magnitude using new enhanced sampling methods like Spectral ABF or FUNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' This can greatly accelerate the drug discovery process and help identify potential drug leads more quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Fission of a Diblock Copolymer Spherical Domain We now investigate the fssion of a single spherical domain of a diblock copolymer using a coarse-grained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' We use a sof, coarse-grained dissipative particle dynamics (DPD) model published in previous studies [50, 51, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' The model consists of 𝑛 = 200 chains with 𝑁 = 256 beads each, representing a liquid polymer melt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' The frst 𝑁𝐴 = 16 beads in each chain are type A, while the remaining 𝑁𝐵 = 240 are type B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' A standard DPD potential is used to enforce incompressibility with a repulsion parameter of 𝐴𝑖𝑖 = 5𝑘𝐵𝑇/𝜎2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' However, a higher interaction of 𝐴𝐴𝐵 = 𝐴𝑖𝑖 + Δ𝐴𝑘𝐵𝑇/𝜎2, with Δ𝐴 ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='4] is applied between unlike particles to create a repulsion that leads to a microphase separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' A Flory- Huggins parameter Δ𝐴 ∝ 𝜒𝑁 > 0 can characterize this phase separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' The interaction range of this non-bonded potential is 1𝜎, as well as the range of the DPD thermostat that keeps the temperature at 𝑇 = 1𝑘𝐵𝑇 = 1𝜖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' In addition, a harmonic spring force with zero resting length is used to connect the beads to polymer chains with a spring constant of 𝑘 = 16/3𝑘𝐵/𝜎2, resulting in an average bond length of 𝑏0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='75𝜎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' The equi- 15 Figure 4: Dynamical Undocking (DuCK) method in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' For a proposed binding mode obtained from classical docking, a short run using MD simulations is carried out and the most stable receptor-ligand native contact is selected from that run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' In this case, it is the hydrogen bond between the red and blue atoms highlighted in panel a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' b) Comparison between diferent methods in PySAGES for DuCK calculations averaged over 5 diferent replicas for each method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' The reference, a Steered MD simulations simulations of 2 ns is in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' In comparison, diferent methods in PySAGES are used considering simulation period 10 times shorter: only ANN [11] provides inferior performance against the reference;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Spectral ABF [37] or FUNN [12] give the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' librium phase for this polymer melt is a body-centered cubic (BCC) phase of spherical A droplets inside a B melt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' [53] However, we confne the polymer to a tight cubic simulation box of length 𝐿0 = 10𝜎, which results in a single A spherical domain in the B matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' We integrate the simulation with a time step of Δ𝑡 = 10−3𝜏 and each simulation is equilibrated for 𝑡 = 1000𝜏, followed by a production run of 𝑡 = 1000𝜏 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' A discussion of the GPU performance of this system with and without PySAGES can be found in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Afer defning the diblock copolymer system, the next step is to defne a CV within the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' In this case, we are interested in the fssion of the single spherical A domain into two equally sized smaller A domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' To achieve this, we divide the polymer chains into two groups: the frst 𝑛 = 100 chains are going to form the frst small domain (blue in Figure 5) and the second 𝑛 = 100 chains form the second spherical domain (red in Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' To defne and enforce the separation of the two groups, we defne our CV as the distance, 𝑅, between the center of mass of the blue A-tails and the center of mass of the red A-tails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Initially, without biasing, the two groups form a single spherical domain and blue and red polymer tails are well mixed, as shown at small 𝑅 < 1𝜎 in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' To study the separation of the spherical domain, we use harmonic bias- 16 b) (e time=200ps 12 SMD-2ns ABF : energy (kcal/mol) 10 ANN FUNN S-ABF 8 WT-MetaD 6 2 Free 0 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='5 Distance (nm)0 1 2 3 4 5 6 distance R [ ] 0 200 400 600 800 free energy [ ] A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='1 A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='2 A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='3 A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='4 Figure 5: Free energy landscape of the fssion of a spherical diblock-copolymer domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' The chain ends forming the spherical domain are split into two groups (blue) and (red), the other chain ends not visible for clarity except for a single chain (grey).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Initially, a single spherical domain is formed, but as we constraint the center of mass between the blue and red groups further, the domain frst elongates and then separates completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' During this separation, the free energy continuously increases and the increase is steeper for high repulsion between unlike type Δ𝐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' As soon as the domain is separated, the free energy plateaus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' 17 ing (see section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='1) to enforce a separation distance 𝑅0 between the two groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' The high density in the system √ ¯N = 𝜌0 𝑁 𝑅3 𝑒0 ≈ 344, leads to low fuc- tuations and suppression of unfavorable conformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Therefore, we use a high spring force constant of 𝑘𝐶𝑉 = 1500𝜖/𝜎2 to facilitate the separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' We investigate a separation of 𝑅 ∈ [0, 6]𝜎 with 14 replicas and use um- brella integration (see section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='2) to determine the free energy profle, as shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' As we increase the external separation distance 𝑅0, we observe how the single domain splits into two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' At a low separation distance 𝑅 < 2𝜎, the single domain is mostly undeformed, but the two groups sepa- rate inside the single spherical domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Increasing the separation distance further goes beyond the dimensions of the spherical domain, leading to the deformation of the domain into an elongated rod-like shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' The two groups still maintain a connection to minimize the AB interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' At a separation between 4𝜎 and 5𝜎 the deformation becomes so strong, that the penalty of forming another AB interface between the two groups, and hence forming two spherical domains, is lower than the entropic penalty of the domain deformation and elongated AB interface of the droplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Af- ter the separation, the free energy landscape remains indiferent to the separation, since there is no interaction between the two domains lef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' The free energy profle of separation is controlled by the repulsion of unlike types 𝜒𝑁 ∝ Δ𝐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' The stronger the repulsion, the more energy is necessary to enlarge the AB surface area for the fssion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' For the strongest interaction Δ𝐴 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='4𝜖, the total free energy barrier reaches about 800𝜖, while for the lowest Δ𝐴 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='1𝜖 it remains below 400𝜖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Both barriers are orders of magnitude larger than thermal fuctuations 1𝑘𝐵𝑇 = 1𝜖, so a spontaneous separation is not expected and the fssion can only be studied via enhanced sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' It is interesting to note that at the lowest separation distance 𝑅0 = 0 it is not the lowest free energy state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Enforcing perfect mixing is not favorable, as the two groups naturally want to separate slightly optimizing the entropy of the chain end-tails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Liquid Crystal Anchoring in Aqueous Interfaces Liquid crystals (LCs), materials that fow like liquids but have anisotropic properties as crystals, have been used lately as prototypes for molecular sensors at interfaces given the high sensitivity in their anchoring behavior relative to small concentration of molecules at aqueous interfaces [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' The presence of molecules at the interface changes drastically the free energy surface of LC molecules relative to their orientation and distance to such in- terface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' In this example, we are revisiting some canonical interfaces for LC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' 18 4-cyano-4’-pentylbiphenyl (5CB) at the interface of pure water and sodium lauryl sulfate (SDS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' For 5CB and water, previous work has focused on ob- taining the free energy surface of a 5CB at the water interface [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' In our case, hybrid anchoring conditions have been imposed on a 16 nm slab of 1000 5CB molecules in the nematic phase (300 K) interacting with a 3 nm slab of water with 62 molecules of SDS at one of the interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' The force felds used are: united atom for 5CB [56], TIP3P [47] for water, GAFF [48] and Lipid 17 for SDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' The CVs chosen to study this system are the distance of the center of mass of one molecule of 5CB at each one of the interfaces (see Appendix A), and the tilt orientation of the same molecule with respect to the z axis of the box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' The free energy surfaces for the pure water and with SDS at the interface are both displayed in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' We can observe that the free energy surface of pure water shows a minimum corresponding to a parallel orientation to the surface with a similar shape that one calculated in [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' On the contrary, the presence of SDS transforms the minimum to a maximum in the same relative position and orientation to the interface (Figure 6 top lef), moving now the minima to a perpendicular orientation of 5CB to the interface, in agreement to the experimental observation of change from planar to homeotropic anchoring in the presence of SDS in water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Figure 6: Free energy surface of 5CB in a hybrid anchoring slab with SDS and water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Right: Snapshot of the system with water molecules in red, 5CB in purple, SDS in green and sodium ions in yellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Top Lef: FES of 5CB molecule near the water–SDS interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Bottom Lef: FES of 5CB near a pure water interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Both FES were obtained with PySAGES and OpenMM using the FUNN method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' 19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='5 Distance to interface (nm) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='0 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='0 100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='5 08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='0 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='5 0° 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='0 Tilt relativeto zaxis3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Ab Initio Enhanced Sampling Simulations In the feld of ab initio simulations of heterogenous catalysis, capturing the dynamic and entropic efects is crucial for an accurate description of the phenomena [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Classical force felds are inadequate for capturing the essential bond breaking events involved in catalysis, so MD simulations based on frst-principles calculations are necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Given that reactive events are ofen limited by large free energy barriers, enhanced sampling techniques are a crucial part of these simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Coupling PySAGES to ASE, provides access to a wide range of frst-principle calculators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' As an example, we have used CP2K [58] as a calculator for a simple ab initio enhanced sampling simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' The CV is the separation distance between a sodium and chlorine atom using the LDA functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' The results are shown in Figure 7, where the minimum in the free energy profle along the Na–Cl distance corresponds to the equilibrium distance between Na and Cl atoms in vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Figure 7: Free energy calculation of Na–Cl distance with ASE+CP2K using Spectral ABF in PySAGES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Enhanced Sampling with Machine Learning Force Fields Deep neural network (NN) force felds can retain the accuracy of ab initio MD while allowing for computational costs similar to those of classical MD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Through ASE it is possible to access NN potentials such as DeepMD [59], and the Gaussian Approximation Potential (GAP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Additionally, JAX MD allows to 20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='5 300K 2 (eV) Energy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='5 1 Free 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='5 0 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='8 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='4 Distance(Angstroms)leverage more general NN potentials that can be used in enhanced sampling calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Coupling of PySAGES with ASE or JAX MD can be used in active learning of NN force felds by efciently sampling rare events using any of the enhanced sampling methods provided by PySAGES as described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' [60] where parallel tempering metadynamics was used to generate accurate NN force feld in urea decomposition in water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' To test the capabilities of PySAGES to handle diferent NN force felds, we have selected three diferent systems trained with the methods men- tioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' For DeepMD, we use a pre-trained model for water, where the enhanced sampling system is one single water molecule in vacuum and the collective variable is the internal angle of the molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' The results in Figure 8 show that the minimum for this free energy profle is around 105 degrees, which is within the range of the experimental value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Next, in Figure 8b, a GAP potential was used for Si–H amorphous mix- tures [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' In this case, a system of 244 atoms was used, and the collective variable is the bond angle between a triad of Si–Si–H atoms in the mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' The global minimum in free energy agrees with the histogram taken from unbiased simulations reported in [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Lastly, we studied a Graph neural network (GNN) model of a Si crystal [62] with PySAGES and JAX MD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' In this case, a crystalline Si system of 64 atoms was used, and the CV was the Si–Si distance for the the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' The results of Figure 8c show that for this model, the minimum in the free energy corresponds almost exactly to the experimental value for the Si–Si nearest distance of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='35 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Performance Our analysis revealed that PySAGES is at least ∼14–15 times faster than SSAGES on a GPU machine containing four V100 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' To obtain this esti- mate, we ran enhanced sampling using umbrella sampling along the center of mass distance between two spherical polymer domains to measure the free energy landscape of the fssion of a spherical diblock-copolymer blend (Figure 5) described in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' For support and compatibility across the libraries and MD engine versions, we estimated the performance with SSAGES v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='2-alpha and PySAGES v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='0 using HOOMD-blue v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='0 and HOOMD-blue v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='7, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' GPU utilization analysis PySAGES is designed to execute every compute-intensive step of a sim- ulation on the GPU and have zero copy instruction between GPU device and host CPU memory for its explicit backends for HOOMD-blue [4] and 21 Figure 8: Free energy calculation of: a) Water internal angle from a DeepMD model with ASE, b) Si–Si–H angle of GAP model with ASE and c) Si–Si distance of a GNN model with JAX MD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' OpenMM [5], while still providing Python code for the user through JAX [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' In this section, we investigate the calculation efciency of PySAGES by examining two example systems, one for each backend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' For HOOMD-blue, we are investigating a system of highly coarse-grained DPD diblock-copolymers as discussed in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' The simulation box contains a total of 𝑛𝑁 = 51 200 particles at a density of 𝜌 = 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='2/𝜎3, which we use for benchmarking purposes with an Nvidia V100 GPU hosted on an Intel Xeon Gold 6248R CPU @ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='00GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Running only with HOOMD-blue v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='7 we achieve an average time steps per second (TPS) of 754, which is the expected high performance of HOOMD-blue on GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Figure 9 shows a detailed profled timeline during the execution of a single time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' During 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='8 ms, HOOMD-blue spends the most computa- tional efort on the calculation of pairwise DPD forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' It can be noted that HOOMD-blue is designed to have almost no idle time of the GPU during a time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' As soon as PySAGES is added computation part, we observe that an additional part is added to calculate the CV and add the forces to every particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' This causes a small period of idle of the GPU, since the execution also requires action of the Python runtime interface with JAX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' In the future, we plan to launch the calculation of CV asynchronously with the regular 22 a a H-O-H angle Si-Si-H angle 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='5 (eV) 2 2 energy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='5 Free 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='5 0 0 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180 Cv (degrees) cv (degrees) c) 3 Si-Si distance 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='5 (eV) energy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='5 Free 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='5 0 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='8 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='4 CV (Angstroms)Hoomd-blue, DPD simulation PySAGES + Hoomd-blue, COM harmonic biasing a b c d e f a b c f 247μs Figure 9: The fgure shows a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='8ms section of profled timeline recorded with Nvidia Nsight systems on an Nvidia V100 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' The top row shows a vanilla HOOMD-blue simulation step, while the bottom row shows a PySAGES/HOOMD-blue simulation with harmonic biasing of a center of mass CV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Light-blue represents the GPU activity while dark-blue represents individual CUDA compute kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' The maroon letters show case the same compute steps in both simulations: a) First half-step of integration, b) compute of bond forces, c) pair-forces, d) calculation of the CV, e) addition of the harmonic biasing force to the HOOMD-blue simulation, and f) the second integration step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Sections d) and e) are PySAGES only and are executed on the GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' We observe GPU idle time during the PySAGES Python coordination with GPU–JAX/CuPy (green bar), but note that there is no memory copies even within the GPU memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' The additional time for CV biasing per time step is 247 μs (teal bar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' force calculation, which would hide this small CPU-intensive GPU idle time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' However, we measure that the total delay due to the extra computation is about 247 μs only: an acceptable overhead for the user-friendly defnition of CVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' In order to connect multiple points in CV space we can use enhanced sampling methods such as umbrella sampling (see section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='2) or the improved string method (see section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='3) to calculate the MFEP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Common for these advanced sampling methods that multiple replica of the system are simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' With PySAGES we easily parallelize their execution using the Python module mpi4py and its MPIPoolExecutor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' This enables us to execute replica of the simulations on multiple GPUs even as they span diferent host machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' In our example, we used 14 replicas for umbrella integration with 7 Nvidia V100 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' The use of a single V100 GPU to execute the simulations with 5 · 105 time steps for all replicas takes 2 hours and 59 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Ideal scaling with 7 GPUs reduces the time to solution to about 26 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' With our MPI-parallel implementation, we achieve a time-to- solution of 28 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Synchronization overhead and nonparallel aspects like fnal analysis sum up to 2 minutes or about 9% overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' This multi-GPU implementation via MPI enables automatically efcient enhanced sampling in high performance computing (HPC) environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' For enhanced sampling methods that are designed for single replica sim- ulations, we ofer an implementation that allows multiple replicas to run in parallel, known as embarrassingly parallel computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' In this situation, the build-in analysis averages the results from multiple replicas and estimates 23 +104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='8ms +105ms +105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='2ms +105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='4ms +105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='6ms +105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='8ms +106ms +106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='2ms +106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='4ms +106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='6ms .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='.2ms +435.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='4ms +435.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='6ms +435.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='8ms +436ms +436.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='2ms +436.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='4ms +436.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='6ms +436.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='8ms +437ms void gpu_compute_dpd_forces_kernel(double4 *, double *, unsigned int, unsi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='. void gpu_computeidp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='OpenMM OPLS all-atom simulation 636μs b c af b c af d e PySAGES + OpenMM 31 COM harmonic biasing Figure 10: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='6ms profled time line of an OpenMM OPLS simulation of 40, 981 particles as polymers with particle mesh Ewald (PME) summation for long-range Coulomb forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' The colors and labels are identical to Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' OpenMM works with asynchronous GPU kernel execution, which leads to less linearly sorted timelines, compared with HOOMD-blue, but we can still identify the CV calculation d) and force biasing e) and the synchronization idle of the GPU (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Overall, the performance degradation is more pronounced with OpenMM compared to HOOMD-blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' In the previous section, we have demonstrated the fast GPU interoper- ability between PySAGES and HOOMD-blue via JAX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' However, the concept of PySAGES is to develop enhanced sampling methods independently of the simulation backend, so here we demonstrate that similar performance can be achieved with OpenMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Since OpenMM focuses on all-atom simulations, we simulate an all-atom model of a polymer with the BigSMILES [64] no- tation {[$]CC([$])(C)C(OCC(O)CSC1=CC=C(F)C(F)=C1)=O} with an OPLS-AA force feld [65, 66] including long-range Coulomb forces via particle mesh Ewald (PME).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' We simulate a bulk system of 40mers with 31 macro- molecules present, adding up to 40 981 atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' As a proof of concept, we calculated the center of mass for every polymer chain and biased it harmon- ically via PySAGES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' As a performance metric, we evaluate the nano-seconds per day (NS/DAY) executed on the same hardware confguration as a the HOOMD-blue example above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' For the unbiased, pure OpenMM simulation we achieve a performance of ≈ 136 NS/DAY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' For the PySAGES biased sim- ulation, we achieve a performance of ≈ 75 NS/DAY, equating to a biasing overhead of approximately 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Figure 10 shows a similar time series anal- ysis as for HOOMD-blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' It is notable that OpenMM’s execution model makes more use of parallel execution of independent kernels, which also changes the order of execu- tion compared to HOOMD-blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' As a result, the same CPU synchronization changes the execution more drastically than in HOOMD-blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Additionally, a single time step for this system is faster executed compared to HOOMD- blue, making the synchronization overhead more noticeable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' In this case, 24 +176.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='8ms +177ms +177.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='2ms +177.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='4ms +177.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='6ms +177.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='8ms +178ms +178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='2ms computeBondedForces computeNonbonded gridSpreadchargefi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' computeNonbonded mm gridSpreadCharge void vector fft<(unsi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' find.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' computeBonded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='+200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='6ms +200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='8ms +201ms +201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='2ms +201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='4ms +201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='6ms +201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='8ms +202ms +202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='2 L comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' computeNonbonded so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='find.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' gridSpread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' computeNonbonded so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='find.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' gridspread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='.g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='+200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='6ms +200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='8ms +201ms +201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='2ms +201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='4ms +201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='6ms +201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='8ms +202ms +202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='2 L comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' computeNonbonded so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='find.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' gridSpread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' computeNonbonded so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='find.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' gridspread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='.g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='parallelization of PySAGES and OpenMM is projected to have a bigger perfor- mance advantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Furthermore, we notice that the calculation of the center of mass and the biasing of all 31 polymer chains is more costly than the single CV in the previous example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' The combination of these factors explain the higher PySAGES overhead for this OpenMM simulation, but overall per- formance is good and signifcantly better for alternative implementations that require CV calculations on the CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Conclusion We have introduced PySAGES, a library for enhanced sampling in molec- ular dynamics simulations, which allows users to utilize a variety of en- hanced sampling methods and Collective Variables, as well as to implement new ones via a simple Python and JAX-based interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' We showed how PySAGES can be used through a number of example applications in diferent felds such as drug design, materials engineering, polymer physics, and ab-initio MD simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' We hope that these convey for the reader the fexibility and potential of the library for addressing a diverse set of problems in a high-performance manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' As our analysis showcased, for large problems, PySAGES can perform biased simulation well over one order of magnitude faster than a library such as SSAGES even when the backend already performs computations on a GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Overall, we believe that PySAGES provides a useful tool for researchers interested in performing molecular and ab-initio simulations in multiple felds, due to its user-friendly framework for defning and utilizing sampling methods and collective variables, as well as its high performance on GPU devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Outlook Being a more recently developed library PySAGES might not be as fea- tureful library as some of the more mature enhanced sampling libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' In concrete, we plan to add the ability to allow the user to perform restarts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' The analysis conducted on how the GPU is used also reveals some optimiza- tion opportunities, such as performing PySAGES-side computations fully asynchronously with the computation of the forces of the backend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' PySAGES also ofers an exciting platform to develop fully end-to-end diferentiable free energy calculations, and we expect that in the future this serves as a starting point to develop newer strategies for force-feld and materials design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' 25 Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Collective variable for the distance to an interface Implementation of the CV described in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='1, that is, the distance between a group of atoms to an interface defned by another group of atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' class DistanceToInterface(TwoPointCV): def __init__(self, indices, axis, sigma, scope, bins=100, coeff=1): super().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='__init__(indices) self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='axis = axis self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='sigma = sigma self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='scope = scope self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='bins = bins self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='coeff = coeff @property def function(self): return lambda r1, r2: distance_to_interface( r1, r2, axis=self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='axis, sigma=self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='sigma, scope=self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='scope, bins=self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='bins, coeff=self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='coeff ) def distance_to_interface(p1, p2, axis, sigma, scope, bins, coeff): mobile_axis = barycenter(p1)[axis] positions_axis = p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='flatten()[axis::3] centers = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='linspace(scope[0], scope[1], bins) centers = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='expand_dims(centers, 1) positions_axis = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='expand_dims(positions_axis, 0) diff = positions_axis - centers mass = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='exp(-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='5 * (diff / sigma) ** 2) mass = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='sum(mass, axis=1) mass_diff = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='abs(mass[1:] - mass[:-1]) centers = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='squeeze(centers) centers_mean = (centers[1:] + centers[:-1]) / 2 probability = nn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='softmax(mass_diff * coeff) interface = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='sum(probability * centers_mean) return mobile_axis - interface 26 Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Benchmark test systems In the following sections, we present the results of the free energy cal- culation for the benchmark test systems of alanine dipeptide and butane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' The details of all the parameters chosen to perform the enhanced sampling simulation of these are summarized in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Alanine Dipeptide The frst test system involves alanine dipeptide in vacuum (Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='11), a benchmark system for enhanced sampling methods that is frequently used in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' 3 2 1 0 1 2 3 3 2 1 0 1 2 3 SpectralABF - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='0 ns 0 20 40 60 80 A (kJ mol 1) 3 2 1 0 1 2 3 3 2 1 0 1 2 3 Metadynamics - 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='0 ns 0 20 40 60 80 A (kJ mol 1) 3 2 1 0 1 2 3 3 2 1 0 1 2 3 ANN - 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='0 ns 0 20 40 60 80 A (kJ mol 1) 3 2 1 0 1 2 3 3 2 1 0 1 2 3 CFF - 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='0 ns 0 20 40 60 80 A (kJ mol 1) Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='11: Free energy landscape of alanine dipeptide (Amber ff99SB [67]) in vacuum as a function of the dihedral angles 𝜙 and 𝜓 obtained with PySAGES and OpenMM via diferent enhanced sampling methods: ABF, Metadynamics, Spectral ABF, ANN, FUNN, CFF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Each panel also indicates the length of the simulation necessary for the free energy to converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Butane As a second test system, we compute the free energy profle along the C- C-C-C dihedral angle, 𝜙𝐶𝐶𝐶𝐶, of a butane molecule (in vacuum), Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Example System Details 27 FUNN - 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='0 ns 3 80 2 1- 60 A (kJ mol-1) 0 40 1 - 20 2 3 0 0 2 3 2 1 1 3ABF - 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='0 ns 3 80 2 1 - 60 A (kj mol-1) 40 1 - 20 21 3 2 3 1 0 1 2 33 2 1 0 1 2 3 CCCC 0 1 2 3 4 5 6 A (kcal mol 1) ANN (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='0 ns) CFF (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='0 ns) FUNN (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='0 ns) WTMD (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='0 ns) SpectralABF (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='0 ns) Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='12: Free energy profle along the dihedral angle of a butane molecule (using an OPLS- based force feld [65]) obtained via diferent enhanced sampling methods with PySAGES and HOOMD-blue: ANN, CFF, FUNN, Spectral ABF, WTMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' The legend also indicates the length of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='1: Parameters and methods details for the various examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' For all methods but Metadynamics, we used a grid with 50 points along each CV for ADP and with 64 points along the CV for butane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' 𝑁 (ABF) = Threshold parameter before accounting for the full average of the adaptive biasing force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' ADP = alanine dipeptide System Backend CV Method Settings Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' ADP OpenMM 𝜙 and 𝜓 ABF 𝑁 = 500 (default) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='11 ANN topology = (8, 8) CFF topology = (14, ) FUNN topology = (14, ) Metadynamics 𝜎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='35 rad ℎ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='2 kJ/mol stride = 500 Spectral ABF — butane HOOMD-blue 𝜙𝐶𝐶𝐶𝐶 ANN topology = (8, 8) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='12 CFF topology = (8, ) FUNN topology = (8, ) WTMD 𝜎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='10 rad ℎ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content='01 kJ/mol stride = 50 Δ𝑇 = 5000 Spectral ABF — 28 References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} +page_content=' nobelprize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtE3T4oBgHgl3EQf_gt3/content/2301.04835v1.pdf'} 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b/ydAzT4oBgHgl3EQftP1k/content/tmp_files/2301.01672v1.pdf.txt @@ -0,0 +1,1486 @@ +arXiv:2301.01672v1 [math.FA] 4 Jan 2023 +ON ALMOST CONVERGENCE ON LOCALLY COMPACT ABELIAN +GROUPS +RYOICHI KUNISADA +Abstract. We study a summability method called almost convergence for bounded +measurable functions defined on a locally compact abelian group. We define almost +convergence using topologically invariant means and exhibit two different kinds of +necessary and sufficient conditions, one is analytic and the other is functional analytic, +for a given function to be almost convergent. As an application, we show complex +Tauberian theorems for almost convergence on the integers and the real numbers. +In particular, the latter one can be viewed as an analogue of the Wiener-Ikehara +theorem. +1. Introduction +For a locally compact abelian group G, let L1(G) be the group algebra of G and +L∞(G) be the set of all essentially bounded measurable functions on G. Let Cbu(G) +be the set of all bounded, uniformly continuous functions on G. The uniformity U of +G is the set of all subsetes of G2 given by +{(x, y) ∈ G2 : x − y ∈ U}, +where U is a neighborhood of 0. Note that both spaces L∞(G) and Cbu(G) are Ba- +nach spaces with respect to the supremum norm ∥ · ∥∞. Here we consider complex- +valued functions in general and we denote by L∞ +R (G) the space of real-valued essentially +bounded measurable functions. +A general element of L1(G) is denotedy by the symbol f(we also use g, h if necessary) +and that of L∞(G) is denoted by ψ. For each s ∈ R, we use the symbols fs(x) := f(x+s) +and ψs(x) := ψ(x + s), the translates of f and ψ by s, respectively. +Let L∞(G)∗ and Cbu(G)∗ be the dual spaces of L∞(G) and Cbu(G), respectively. An +element ϕ of L∞(G)∗ (Cbu(G)∗) is said to be a mean on L∞(G) (Cbu(G)) if it satisfies +(1) ϕ ≥ 0 (i.e., ϕ(ψ) ≥ 0 for every positive ψ ∈ L∞(G) (Cbu(G))); +(2) ϕ(1) = 1, where 1 is the constant function taking the value 1 everywhere. +Further, ϕ is called an invariant mean on L∞(G) (Cbu(G)) if it is a mean such that +(3) ϕ(ψs) = ϕ(ψ) for every ψ ∈ L∞(G) (Cbu(G)) and s ∈ G. +Let us denote by I(G) (I0(G)) the set of all invariant means on L∞(G) (Cbu(G)). +2010 Mathematics Subject Classification. Primary 40H05; Secondary 22B05. +Key words and phrases. Almost convergence, topologically invariant means, tauberian theorem. +1 + +Now, we introduce another class of means on L∞(G) which is more important for +our purpose. For f ∈ L1(G) and ψ ∈ L∞(G), their convolution f ∗ ψ is defined by +f ∗ ψ(x) = +� +G +f(t)ψ(x − t)dm(t), +x ∈ G, +where m is the Haar measure of G. Let P(G) be the set of positive elements f in +L1(G) such that +� +G f(x)dm(x) = 1. A mean ϕ on L∞(G) is said to be a topologically +invariant mean if it satisfies the following condition ([6]): +(4) ϕ(f ∗ ψ) = ϕ(ψ) ∀ψ ∈ L∞(G) ∀f ∈ P(G). +Let us denote by T (G) the set of all topologically invariant means on L∞(G). Note +that a topologically invariant mean is an invariant mean. In fact, if ϕ is topologically +invariant, then +ϕ(ψs) = ϕ(f ∗ ψs) = ϕ(fs ∗ ψ) = ϕ(ψ) +for each ψ ∈ L∞(R) and s ∈ R. Conversely, For discrete groups G, it is easy to show +that I(G) = T (G) holds true. For every locally compact abelian group G, T (G) ̸= ∅ +is valid, and if G is noncompact, we have T (G) ⊊ I(G). For compact abelian groups +G, T (G) is the singleton consisting of the normalized Haar measure of G. We refer the +reader to [5], [12] for more detailed exposition on these results. +The main objective of this paper is a summability method concerning (topologically) +invariant means. Let l∞ be the set of all bounded functions on the nonnegative integers +Z+ := {n ∈ Z : n ≥ 0}. Lorentz (1948) defined a summability method on l∞ called +almost convergence using Banach limits ([10]). Recall that an element ϕ of l∗ +∞, the dual +space of l∞, is called a Banach limit if the following conditions are satisfied: +(1) ϕ ≥ 0; +(2) ϕ(1) = 1, where 1 is the constant function taking the value 1 everywhere. +(3) ϕ(ψn) = ϕ(ψ) for every n ∈ Z+ and ψ ∈ l∞, +namely, ϕ is a right translation invariant mean on l∞. Let B be the set of all Banach +limits. Then, almost convergence for sequences is defined as follows. +Definition 1.1 (Lorentz, 1948). ψ ∈ l∞ is said to be almost convergent to a (complex) +number α if ϕ(ψ) = α for every ϕ ∈ B. +It is difficult to know from this abstract definition whether a given sequence of +numbers is almost convergent or not. However, Lorentz proved an analytic condition +for almost convergence as follows (his delivation was rather complicated and Sucheston +[16] gave a more simple proof later). +Theorem 1.1 (Lorentz, 1948). ψ ∈ l∞ is almost convergent to α if and only if +lim +k→∞ +1 +k +k−1 +� +i=0 +ψ(n + i) = α +uniformly in n ∈ Z+. +2 + +This assertion can be proved by the Hahn-Banach theorem and is a basic way to +check whether a given sequence actually almost converges. +Note that Banach limits can be regarded as a special kind of invariant means on +L∞(Z), where Z is the additive group of integers. Hence, it is natural to consider +that one can define the notion of almost convergence on an arbitrary locally compact +abelian group or its subsemigroups (note that abelian groups G are amenable, that is, +there exist invariant means on L∞(G)). Specifically, one can define formally almost +convergence of functions in L∞(G) by the following : +Definition 1.2. Let G be a locally compact abelian group. We say that ψ ∈ L∞(G) +is almost convergent to a complex number α if and only if +ϕ(ψ) = α +holds for every ϕ ∈ T (G). In this case, we write as ψ +ac +−→ α. +The reason for adopting topological invariant means instead of (seemingly more +natural) invariant means is that an analogous result of Theorem 1.1 is also valid for +groups not necessarily discrete (see [1], [2]). +The objective of this paper is to provide a new necessary and sufficient condition for +a given bounded function on a locally compact abelian group to be almost convergent. +This is obtained through the thoery of harmonic analysis on locally compact abelian +groups, especially, the theory of spectral synthesis. Furthermore, applying this result, +we obtain complex tauberian theorems for almost convergence on Z and R, which are +related to the famous results of Katznelson-Tzafriri and Wiener-Ikehara. +The paper is organized as follows. In Section 2, we exhibit an analytic condition for +almost convergence. This is a special case of the more general result of Chow [1] when +the underling group is abelian. However, considering the importance of the result, we +include this section for the sake of completeness and consistency of the paper. +In Section 3, following Forelli [4], we develop spectral theory of Cbu(G) and Cbu(G)∗, +which contains a description of subspaces of Cbu(G)∗ defined via spectrum as the an- +nihilator of a certain invariant subspace of Cbu(G). This is somewhat a formal gen- +eralization of his relult to general locally compact abelian groups. In Section 4, as +an application of a result of the previous section, we provide the annihilator of the +subspace of L∞(G)∗ spanned by topologically invariant means on L∞(G). In Section +5, using a result of Section 4, we obtain a new necessary and sufficient condition for al- +most convergence. Section 6 deals with almost convergence on positive subsemigroups +of the special groups Z and R. In Section 7, we deal with complex Tauberian theorems +for almost convergence on Z and R. +2. An analytic condition for almost convergence +Let G be a locally compact abelian group. Since G is an amenable group, there +exists a summing net for G, namely, a net {Kδ}δ∈∆ of nonnull, compact subsets of G +satisfying the following properties (see [5], [12]): +3 + +(1) Kδ ⊆ Kδ′ if δ ≤ δ′ +(2) G = � +δ∈∆ Kδ +(3) limδ +m(Kδ△Kδ,s) +m(Kδ) += 0 uniformly in s on a compact subset of G. +We fix one summing net for G and define a sublinear functional on L∞(G) as +p(ψ) := lim sup +δ +sup +x∈G +1 +m(Kδ) +� +Kδ +ψx(t)dt, +where ψ ∈ L∞(G). We also introduce the functional p(ψ) := −p(−ψ). We note thta p +can be expressed as +p(ψ) := lim inf +δ +inf +x∈G +1 +m(Kδ) +� +Kδ +ψx(t)dt. +Then, we have the following result: +Theorem 2.1. A mean ϕ on Cbu(G) is an ivariant mean if and only if +ϕ(ψ) ≤ p(ψ) +holds for every ψ ∈ Cbu(G). +Proof . First, we show necessity. Let ϕ be an invariant mean on Cbu(G). Note that, +since ψ is in Cbu(G), the mapping G ∋ t �→ ψt ∈ Cbu(G) is continuous and thus, it is +Bochner integrable. Then, we have +ϕ(ψ) = +1 +m(Kδ) +� +Kδ +ϕ(ψt)dt = ϕ +� +1 +m(Kδ) +� +Kδ +ψt(x)dt +� +≤ sup +x +1 +m(Kδ) +� +Kδ +ψx(t)dt +holds true for every Kδ (see [17]). Here we use the elementary fact that for any mean +ϕ on Cbu(R) and ψ in Cbu(R), it holds that +ϕ(ψ) ≤ sup +x∈R +ψ(x). +In fact, let α := supx∈R ψ(x) and then α − ψ ≥ 0, thus by the positivity of ϕ, we have +ϕ(α − ψ) ≥ 0 ⇔ ϕ(α) ≥ ϕ(ψ) +⇔ α ≥ ϕ(ψ). +By taking the limit superior over Kδ +′s, we obtain +ϕ(ψ) ≤ lim sup +δ +sup +x +1 +m(Kδ) +� +Kδ +ψx(t)dt. +4 + +Now, we show sufficiency. Let ϕ satisfy the condition in the theorem. Then, for each +ψ in Cbu(G), we have +ϕ(ψ − ψs) ≤ lim sup +δ +sup +x +1 +m(Kδ) +� +Kδ +(ψx(t) − ψx+s(t))dt += lim sup +δ +sup +x +1 +m(Kδ) +�� +Kδ +ψx(t)dt − +� +Kδ +ψx+s(t)dt +� += lim sup +δ +sup +x +1 +m(Kδ) +�� +Kδ +ψx(t)dt − +� +Kδ,−s +ψx(t)dt +� +≤ lim sup +δ +sup +x +1 +m(Kδ) +� +Kδ△Kδ,−s +|ψx(t)|dt +≤ lim sup +δ +m(Kδ △ Kδ,−s) +m(Kδ) +∥ψ∥∞ = 0. +For the reverse inequality, we consider the relation +ϕ(−ψ) ≤ p(−ψ) ⇔ ϕ(ψ) ≥ −p(−ψ) =: p(ψ). +Then, we can show the inequality +ϕ(ψ − ψs) ≥ 0 +in the same argument as above. Thus, we obtain ϕ(ψ − ψs) = 0, which shows the +translation invariance of ϕ and we complete the proof. +We need the following lemma to obtain a similar condition for topologically invraiant +means. An equivalent assertion was shown in [1] in a more general form. +Lemma 2.1. For any ψ ∈ L∞(G) and f ∈ P(G), +p(ψ − f ∗ ψ) = p(ψ − f ∗ ψ) = 0 +holds true. +Proof . By direct computation, we have +p(ψ − f ∗ ψ) = lim sup +δ +sup +x∈G +1 +m(Kδ) +� +Kδ +{ψx(t) − (f ∗ ψ)x(t)}dt += lim sup +δ +sup +x∈G +1 +m(Kα) +� +Kδ +� +G +{ψx(t) − ψx(t − u)}f(u)dudt += lim sup +δ +sup +x∈G +� +G +f(u)du +1 +m(Kδ) +� +Kδ +{ψx(t) − ψx(t − u)}dt. +The integral +1 +m(Kδ) +� +Kδ +{ψx(t) − ψx(t − u)}dt +5 + +can be evaluated in two ways: +1 +m(Kδ) +� +Kδ +{ψx(t) − ψx(t − u)}dt ≤ +����� +1 +m(Kδ) +� +Kδ△Kδ,−u +ψx(t)dt +����� +≤ m(Kδ △ Kδ,−u) +m(Kδ) +∥ψ∥∞. +(1) +Also, we obtain +1 +m(Kδ) +� +Kδ +{ψx(t) − ψx(t − u)}dt ≤ +1 +m(Kδ)2∥ψ∥∞m(Kδ) = 2∥ψ∥∞. +(2) +Let ε > 0 be given. Take a compact subset Cε such that +� +G\Cε |f(t)|dt < ε. Then, +choose γ such that m(Kδ△Kδ,−u) +m(Kδ) +≤ ε for every δ ≥ γ and u ∈ Cε. Then, by (1) and (2), +we have +lim sup +δ +sup +x∈G +� +G +f(u)du +1 +m(Kδ) +� +Kδ +{ψx(t) − ψx(t − u)}dt += lim sup +δ +sup +x∈G +� � +Cε +f(u)du +1 +m(Kδ) +� +Kδ +{ψx(t) − ψx(t − u)}dt ++ +� +G\Cε +f(u)du +1 +m(Kδ) +� +Kδ +{ψx(t) − ψx(t − u)}dt +� +≤ lim sup +δ +sup +x∈G +� +Cε +|f(u)|m(Kδ △ Kδ,−u) +m(Kδ) +∥ψ∥∞du + 2∥ψ∥∞ +� +G\Cε +|f(u)|du +≤ ε∥ψ∥∞ +� +Cε +|f(u)|du + 2ε∥ψ∥∞ ≤ 3ε∥ψ∥∞. +Since ε > 0 can be arbitrary, we obtain p(ψ−f ∗ψ) ≤ 0. The equation p(ψ−f ∗ψ) ≥ 0 +can be proved similarly. +Since p(ψ) ≤ p(ψ) holds for each ψ ∈ L∞(G), we have +p(ψ − f ∗ ψ) = p(ψ − f ∗ ψ) = 0. We complete the proof. +Combining Theorem 2.1 and Lemma 2.1, we obtain the following result. +Theorem 2.2. A mean ϕ on L∞(G) is a topologically ivariant mean if and only if +ϕ(ψ) ≤ p(ψ) +holds for every ψ ∈ L∞(G). +Proof . First, we show necessity. Assume that ϕ is in T (G). Then, for any ψ ∈ L∞(G) +and f ∈ P(G), we have +ϕ(f ∗ ψ) = ϕ(ψ). +Note that, as stated before, ϕ is an invariant mean on Cbu(G). Since f ∗ ψ ∈ Cbu(G), +by Theorem 2.1, we have +ϕ(f ∗ ψ) ≤ p(f ∗ ψ). +6 + +Since, by Lemma 2.1, p(f ∗ ψ) = p(ψ) holds true, we obtain +ϕ(ψ) = ϕ(f ∗ ψ) ≤ p(f ∗ ψ) = p(ψ). +Now, we show sufficiency. Assume that ϕ(ψ) ≤ p(ψ) holds for each ψ ∈ L∞(G). +Then, by Lemma 2.1, for any f ∈ P(G), we have +0 = p(ψ − f ∗ ψ) ≤ ϕ(ψ − f ∗ ψ) ≤ p(ψ − f ∗ ψ) = 0. +Hence, we obtain ϕ(ψ − f ∗ ψ) = 0 and ϕ is topologically invariant. +Theorem 2.3. Let ψ ∈ L∞ +R (G). Then, ψ is almost convergent to α if and only if +lim +δ +1 +m(Kδ) +� +m(Kδ) +ψ(x + s)dx = α +uniformly in s ∈ G. +Proof . Note that, for any ϕ ∈ T (G) and ψ ∈ L∞(G), we have +p(ψ) ≤ ϕ(ψ) ≤ p(ψ). +Conversely, for any real number α with p(ψ) ≤ α ≤ p(ψ), there exists some ϕ ∈ T (G) +such that ϕ(ψ) = α. In fact, we define ϕ0 on Rψ = {cψ : c ∈ R} by ϕ0(cψ) = cα and +then can extend it to whole L∞(G) such that ϕ(ψ) ≤ p(ψ) holds for every ψ ∈ L∞(G) +by the Hahn-Banach theorem. This extended ϕ is an topogically invariant mean by +Theorem 2.2. +Hence, we have shown that ϕ(ψ) = α for every ϕ ∈ T (G) if and only if p(ψ) = +p(ψ) = α. Now, we show that this is equivalent to the condition given in the theorem. +First, necessity is clear. Recall that the functionals p and p are expressed as folllows: +p(φ) = lim sup +δ +sup +s∈G +1 +m(Kδ) +� +Kδ +ψ(x + s)dx, +p(ψ) = lim inf +δ +inf +s∈G +1 +m(Kδ) +� +Kδ +ψ(x + s)dx. +Let ε > 0 be any positive number. Then, there exist real numbers δ0 and δ1 such that +sup +s∈G +1 +m(Kδ) +� +Kδ +ψ(x + s)dx < α + ε +whenever δ ≥ δ0 and +inf +s∈G +1 +m(Kδ) +� +Kδ +ψ(x + s)dx > α − ε +whenever δ ≥ δ1. Hence, let δ2 = max(δ0, δ1) and we have +−ε < +1 +m(Kδ) +� +Kδ +ψ(x + s)dx − α < ε +for every s ∈ G whenever δ ≥ δ2. +This means uniform convergence of the net in +question. This completes the proof. +7 + +Now, we extend the above result to complex valued functions. +Theorem 2.4. Let ψ ∈ L∞(G). Then, ψ is alomst convergent to α if and only if +lim +δ +1 +m(Kδ) +� +m(Kδ) +ψ(x + s)dx = α +uniformly in s ∈ G. +Proof . Let us denote ψ(x) = u(x)+iv(x), where u, v in L∞ +R (G) are real and imaginary +parts of ψ respectively and let α = β + iτ be a complex number. Then, +ϕ(ψ) = α = β + iτ +for every ϕ ∈ T (G) if and only if +ϕ(u) = β, +ϕ(v) = τ +for every ϕ ∈ T (G). By Theorem 2.3, this is equivalent to +lim +δ +1 +m(Kδ) +� +m(Kδ) +u(x + s)dx = β, +lim +δ +1 +m(Kδ) +� +m(Kδ) +v(x + s)dx = γ +uniformly in s ∈ G, which is equivalent to the assertion of the theorem. +We give examples of summing nets. It can be easily varified that +Kn = [−n, n] (n ∈ N), +Kθ = [−θ, θ] (θ > 0) +are summing nets for G = Z and R, respectively. Then, we have the following formu- +lation of almost convergence on Z and R: +ψ ∈ L∞(Z) is almost convergent to α if and only if +lim +n→∞ +1 +2n + 1 +k+n +� +i=k−n +ψ(i) = α +uniformly in k ∈ Z. Similarly, ψ ∈ L∞(R) is almost convergent to α if and only if +lim +θ→∞ +1 +2θ +� x+θ +x−θ +ψ(t)dt = α +uniformly in x ∈ R. +We note that the former is a two-sided version of Sucheston’s result (Theorem 1.1) +and the latter was given by Raimi [13] (see also [14]). Another kind of analytic expres- +sion for almost convergence on R was given in [19]. +8 + +3. Spectral synthesis of bounded measurable functions +Let G be a locallly compact abelian group and Γ be its dual group. An element λ +in Γ is denoted as χλ(x) when it is viewed as a character on G. For a function f in +L1(G), its Fourier transform ˆf is defined by +ˆf(λ) = +� +G +f(x)χλ(−x)dm(x), +λ ∈ Γ. +The purpose of this section is to develop the spectrum theory of Cbu(G) and Cbu(G)∗. +For the following contents, we refer the readers to [15] as a basic literature. +For f ∈ L1(G), let Z(f) be the zero set of the Fourier transform of f. Let I be +a closed ideal of L1(G). We denote by Z(I) the intersection of the zero sets of the +Fourier transforms of elements in I, namely, +Z(I) = +� +f∈I +Z(f). +By definition, each Z(I) is a closed set of Γ. Note that, in general, Z(I) does not +uniquely determine I. In other words, there exist distinct closed ideals I and I′ such +that Z(I) = Z(I′) holds true. However, there is a closed set C of Γ which is the zero +set of a unique closed ideal of L1(G). Such a closed set is called a spectral synthesis +set. The following result is a fundamental result of spectral synthesis thoery. +Theorem 3.1. Let I be a closed ideal of L1(G) and f ∈ L1(G). If the internal of Z(f) +contains Z(I), then f ∈ I holds true. +For a closed set C of Γ, let I1(C) be the set of all f ∈ L1(G) such that C ⊆ Z(f) +and let I0(C) be the closure of the set of all f ∈ L1(G) such that C ⊆ IntΓZ(f), the +interior of Z(f) in Γ. Then, by definition, I1(C) is the largest closed ideal of L1(G) +with Z(I) = C and, by Theorem 3.1, I0(C) is the smallest closed ideal of L1(G) with +Z(I) = C. Note that C is a spectral synthesis set if and only if I0(C) = I1(C). +Now, following classical theory, we define spectrum of subspaces and elments of +L∞(G). Let Φ be a weak* closed invariant subspace of L∞(G), namely, Φ is a subspace +closed with respect to weak* topology of L∞(G) and ψ ∈ Φ implies ψs ∈ Φ for every +s ∈ G. Related to Φ, we define the closed ideal J(Φ) of L1(G) by the set of all functions +f in L1(G) such that f ∗ ψ = 0 for every ψ ∈ Φ: +J(Φ) = {f ∈ L1(G) : f ∗ ψ = 0 ∀ψ ∈ Φ}. +Then, we define the spectrum sp(Φ) of Φ by Z(J(Φ)). +In the same way, we define spectrum of each member of L∞(G). For each ψ in L∞(G), +let J(ψ) be the set of all functions f in L1(G) such that f ∗ψ = 0. Then, we define the +spectrum sp(ψ) of ψ by Z(J(ψ)). It is easy to confirm that sp(ψ) = sp(Φ(ψ)), where +Φ(ψ) is the weak* closed invariant subspace of L∞(G) generated by ψ. +We give another description of spectrum of Φ. Let ⟨X, X∗⟩ be any dual pair of locally +convex spaces. For subspaces E of X and E∗ of X∗, let us define their annihilators as +9 + +follows: +E⊥ = {ϕ ∈ X∗ : ϕ(x) = 0 ∀x ∈ E}, +E∗⊥ = {x ∈ X : ϕ(x) = 0 ∀ϕ ∈ E∗}. +Note that +(E⊥)⊥ = E, +(E∗⊥)⊥ = E∗ +is an immediate consequence of the Hahn-Banach theorem. +Theorem 3.2. Let Φ be a weak* closed invariant subspace of L∞(G). Then, sp(Φ) is +the set of all chracters contained in Φ : +sp(Φ) = {λ ∈ Γ : χλ ∈ Φ}. +Proof . First, suppose χλ ∈ Φ. Let f be in J(Φ). Then, in particular, we have +f ∗ χλ(x) = +� +G +f(t)χλ(x − t)dm(t) += +� +G +f(t)χλ(x)χλ(−t)dm(t) += χλ(x) ˆf(λ) = 0 +for every x ∈ G. Thus, ˆf(λ) = 0 and λ ∈ Z(J(Φ)) = sp(Φ). +On the other hand, let us assume that χλ ̸∈ Φ. Hence, by the Hahn-Banach theorem, +there exists some f ∈ L1(G) such that ⟨f ∗, ψ⟩ = 0 for every ψ ∈ Φ and ⟨f ∗, χλ⟩ = 1, +where f ∗(x) = f(−x). That is, +� +G +f ∗(t)ψ(t)dt = +� +G +f(−t)ψ(t)dm(t) = f ∗ ψ(0) = 0 +for each ψ ∈ Φ and +� +G +f ∗(t)χλ(t)dt = +� +G +f(−t)χλ(t)dm(t) = ˆf(λ) = 1 +holds true. Since Φ is translation invariant, the first equation also holds for any trans- +late ψx of ψ and we obtain f ∗ ψ = 0. Hence, f is in J(Φ) and ˆf(λ) = 1, it holds that +λ ̸∈ sp(Φ). This completes the proof. +Obviously, the smallest weak* closed subspace of L∞(G) with sp(Φ) = C is the +subspace Φ1(C) which is generated by the characters {χλ}λ∈C. Note that if I = Φ⊥, +or, equivalently, I⊥ = Φ, sp(Φ) = Z(I) holds true. This implies that Φ1(C) = I1(C)⊥. +Further, let Φ0(C) be the annihilaor of I0(C), that is, Φ0(C) = I0(C)⊥, then Φ0(C) is +the largest weak* closed invariant subspace with sp(Φ) = C. We have Φ0(C) = Φ1(C) +if and only if sp(Φ) is a spectral synthesis set. This observation implies the following +assertion: +10 + +Theorem 3.3. Let Φ be a weak* closed invariant subspace of L∞(G). If sp(Φ) is a +spectral synthesis set, then Φ is synthesized by its spectrum. Namely, each member ψ +of Φ is a weak* limit of a net of trigonometric polynomials +ψα(x) = +nα +� +k=1 +cα +keitα +k x +where cα +k ∈ C and tα +k ∈ sp(Φ) for every α and 1 ≤ k ≤ nα. +Following [4], we can also define spectrum for elements of Cbu(G)∗. Let ϕ ∈ Cbu(G)∗ +and f ∈ L1(G). We define their convolution f ∗ϕ, which is also an element of Cbu(G)∗, +by +f ∗ ϕ(ψ) = ϕ(f ∗ ψ), +ψ ∈ L∞(G). +For each ϕ ∈ Cbu(G)∗, let J(ϕ) be the closed ideal of L1(G) consisting of those elements +f for which f ∗ ϕ = 0. Then, we define the spectrum sp(ϕ) of ϕ by Z(J(ϕ)). +The following lemma is basic for arguments what follows. +Lemma 3.1. Let ψ be in L∞(G), ϕ be in Cu(R)∗ and f be in L1(R). +Then, the +following results holds true. +(i) sp(f ∗ ψ) ⊆ sp(ψ) ∩ supp ˆf, +(ii) sp(f ∗ ϕ) ⊆ sp(ϕ) ∩ supp ˆf, +where supp ˆf denotes the support of ˆf. +Proof . (i) First, note that the inclusion sp(f ∗ ψ) ⊆ sp(ψ) is obvious by definition of +spectrum. Fix arbitrary λ ̸∈ supp ˆf. Let g be a funciton in L1(G) such that ˆg = 0 on +a neighborhood of supp ˆf and ˆg(λ) = 1. Then, note that (f ∗ g)�(λ) = ˆf(λ)ˆg(λ) = 0 +for every λ ∈ Γ, which means that f ∗ g = 0 by the uniqueness thorem of the Fourier +transform. Hence, we obtain g ∗ (f ∗ ψ) = (g ∗ f) ∗ ψ = 0 and thus, g is in J(f ∗ ψ). +Since λ ̸∈ Z(g), we conclude that λ /∈ sp(f ∗ ψ). +(ii) This assertion can be proved in the same way as (i) and we omit the proof. +Lemma 3.2. Let ψ be in L∞(G) and ϕ be in Cbu(G)∗. Then, we have the following +results. +(i) If sp(ψ) = ∅, then ψ = 0, +(ii) If sp(ϕ) = ∅, then ϕ = 0. +Proof . (i) Suppose that sp(ψ) = ∅ holds. By definition of spectrum and Wiener’s +Tauberian theorem, which asserts that if a closed ideal I in L1(G) satisfies Z(I) = ∅, +then I = L1(G), we conclude that f ∗ψ = 0 for every f in L1(G). Take an approximate +of identity {fα} of L1(G), observe that +lim +α ∥fα ∗ ψ − ψ∥∞ = 0. +Then, we have +∥ψ∥∞ ≤ ∥ψ − fα ∗ ψ∥∞ + ∥fα ∗ ψ∥∞ += ∥ψ − fα ∗ ψ∥∞, +11 + +which tends to 0 as α → ∞. Thus, it follows that ψ = 0, completing the proof. +(ii) This can be proved similarly as (i). +We define the subspaces of Cbu(G) and Cbu(G)∗ as follows: +EA = {ψ ∈ Cbu(G) : sp(ψ) ⊆ A}; +E∗ +A = {ϕ ∈ Cbu(G)∗ : sp(ϕ) ⊆ A}, +where A is a subset of Γ. +Observe that if A is a closed subset of Γ, then EA = +Φ0(A)∩Cbu(G) holds true and thus, EA is a closed subspace of Cbu(R). For spaces E∗ +A, +we have the following result. +Lemma 3.3. Let A be a closed subset of Γ. Then, E∗ +A is a weak*-closed subspace of +Cbu(G)∗. In particular, it is norm closed. +Proof . Let {ϕα} be a net of elements of E∗ +A such that w∗- limα ϕα = ϕ. We show +that ϕ ∈ E∗ +A, that is, sp(ϕ) ⊆ A. Take arbitrary λ ̸∈ A. Then, there exists a function +f in L1(G) such that ˆf = 0 on a some neighborhood of A and ˆf(λ) = 1. Note that +f ∗ ϕα = 0 for every α. Then, for any ψ in L∞(G), we have +(f ∗ ϕ)(ψ) = ϕ(f ∗ ψ) = lim +α ϕα(f ∗ ψ) = lim +α f ∗ ϕα(ψ) = 0. +Hence, we obtain f ∗ ϕ = 0 and thus, f ∈ J(ϕ). Since λ ̸∈ Z(f), we have λ ̸∈ sp(ϕ), +completing the proof. +The following result will be used in Section 6, which states the continuity of spectrum +with respect to the weak* topology. +Lemma 3.4. Let A be a closed subset of Γ. Suppose that {ψα} be a net of L∞(G) with +sp(ψα) ⊆ A for every α. If w∗- limα ψα = ψ, namely, {ψα} converge to ψ in the weak* +sense, then we have sp(ψ) ⊆ A. +Proof . In fact, for each f ∈ L1(G) with A ⊆ IntΓZ(f), in other words, for f ∈ I0(A), +we have f ∗ ψα = 0 for every α. Thus, we have +f ∗ ψ = lim +α f ∗ ψα = 0, +which means that f ∈ J(ψ). Since ∩f∈I0(A)Z( ˆf) = A, we obtain sp(ψ) = ∩f∈J(ψ)Z( ˆf) ⊆ +A. This complets the proof. +In what follows, we give a description of the spaces E∗ +A in terms of spaces EA. Note +that we consider the dual pair ⟨Cbu(G), Cbu(G)∗⟩. +Theorem 3.4. Let A be a closed subset of Γ. Then, we have +E∗ +A +⊥ = clCbu(G)EAc, +where the symbol clCbu(G)E denotes the closure of a subspace E in Cbu(G). +12 + +Proof . For each ψ ∈ Cbu(G) and ϕ ∈ Cbu(G)∗, Let us define ψ′(x) ∈ Cbu(G) by +ψ′(x) = ϕ(ψx). For any f ∈ L1(G), by the fact mentioned in the proof of Theorem 2.1, +we have +ϕ(f ∗ ψ) = +� +G +ϕ(ψ−t)f(t)dt. +Replacing ψ by ψx where x ∈ G, we obtain +f ∗ ϕ(ψx) = ϕ(f ∗ ψx) = +� +G +ϕ(ψx−t)f(t)dt = +� +G +ψ′(x − t)f(t)dt = f ∗ ψ′(x). +(3) +From this equation, we can deduce the following result: +sp(ψ′) ⊆ sp(ψ) ∩ sp(ϕ). +(4) +In fact, assume that f ∈ J(ψ), that is, f ∗ ψ = 0. Then, for any x ∈ G, it holds that +f ∗ ψx = (f ∗ ψ)x = 0. Thus, by the equation (3), if f ∈ J(ψ), we have f ∗ ψ′ = 0, that +is, f ∈ J(ψ′). Hence, we obtain sp(ψ′) ⊆ sp(ψ) since J(ψ) ⊆ J(ψ′). +Next, assume that f ∈ J(ϕ), that is, f ∗ ϕ = 0. Again, by equation (3), we obtain +f ∗ ψ′ = 0, that is, f ∈ J(ψ′). This shows that sp(ψ′) ⊆ sp(ϕ). We have obtained the +desired relation (4). +Now suppose that ϕ ∈ E∗ +A(G) and ψ ∈ EAc(G). Then, by (4), we have sp(ψ′) ⊆ +sp(ψ) ∩ sp(ϕ) ⊆ A ∩ Ac = ∅. Thus, sp(ψ′) = ∅ holds true. Then, by Lemma 3.2 (1), +we have ψ′ = 0 and ϕ(ψ) = 0, that is, ψ ∈ E∗ +A +⊥. Now we obtain +EAc ⊆ E∗ +A +⊥, +which means that +clCbu(G)EAc ⊆ E∗ +A +⊥. +We now show the reverse inclusion. Let us assume that ϕ = 0 for every ψ ∈ EAc. +We show that sp(ϕ) ⊆ A. Let λ ̸∈ A and take f ∈ L1(G) such that ˆf = 0 on some +neighborhood U of A with λ ̸∈ U and ˆf(λ) = 1. Then, f ∗ ϕ(ψ) = ϕ(f ∗ ψ) = 0 for +every ψ ∈ Cbu(G) since sp(f ∗ ψ) ⊆ supp ˆf ⊆ Uc ⊆ Ac. Thus, f is a function in J(ϕ) +with ˆf(λ) = 1 and we conclude that λ ̸∈ sp(ϕ). Now, we have shown that +EAc⊥ ⊆ E∗ +A. +Considering the annihilators of the both sides of the above equation and then taking +the closure, we obtain the inclusion relation +E∗ +A +⊥ ⊆ clCbu(G)EAc. +This completes the proof. +4. An application to topologically invariant means +We apply Theorem 3.4 to the case C = {0}, that is, the singleton consisting of only +the unit of G. The following result is essentially due to Muhly [11], who proved the +special case of G = R. +Theorem 4.1. For ϕ ∈ Cbu(G)∗, ϕ is an invariant mean if and only if sp(ϕ) = {0}. +13 + +Proof . First, assume that ϕ is invariant. Then, for any f ∈ L1(G) for which Z(f) +contains {0}, that is, +� +G f(x)dm(x) = 0, we have +f ∗ ϕ(ψ) = ϕ(f ∗ ψ) = +� +G +ϕ(ψ−t)f(t)dt += +� +G +ϕ(ψ)f(t)dt = ϕ(ψ) +� +G +f(t)dt = 0. +Therefore, we conclude that sp(ϕ) ⊆ {0}. Note that if sp(ϕ) = ∅, by Lemma 3.2, ϕ = 0 +holds true. Thus, we have the desired conclusion that sp(ϕ) = {0}. +Conversely, assume that sp(ϕ) = {0}. Then, for any f ∈ I0({0}) and ψ ∈ Cbu(G), +we have +f ∗ ϕ(ψ) = +� +R +ϕ(ψ−t)f(t)dt = 0. +For any x ∈ G, set ψ′(x) = ϕ(ψx) and we have +f ∗ ϕ(ψx) = +� +R +ϕ(ψx−t)f(t)dt = +� +R +ψ′(x − t)f(t)dt = 0. +This means that sp(ψ′) = {0}. Since {0} is a spectral synthesis set, we have Φ(ψ′) = +Φ1({0}) = R. Hence, ψ′(x) is a constant function, which means that ϕ is translation +invariant. This completes the proof. +Let T(G) be the closed subspace of L∞(G)∗ spaned by T (G). We determine the +annihilator of T(G) via Theorem 3.4. For any subset A of Γ, let us define the subspace +E′ +A of L∞(G) by +E′ +A = {ψ ∈ L∞(G) : sp(ψ) ⊆ A}. +Theorem 4.2. For a locally compact abelian group G, the following assertion holds +true. +T(G)⊥ = clL∞(G)E′ +G\{0}. +Proof . Let ϕ ∈ T(G). Fix ψ ∈ E′ +G\{0}. For any f ∈ P(G), we have f ∗ ψ ∈ Cbu(G) +and sp(f ∗ ψ) ⊆ G \ {0} by Lemma 3.1. Hence, for each ϕ ∈ T(G), we have +ϕ(ψ) = ϕ(f ∗ ψ) = 0 +by Theorems 3.4 and 4.1. Now, we obtain that +T(G)⊥ ⊇ clL∞(G)E′ +G\{0}. +We show the reverse inclusion. Suppose ϕ ∈ L∞(G) vanishes on E′ +G\{0}. We show that +ϕ is in T(G), that is, ϕ(ψ −f ∗ ψ) = 0 for every f ∈ P(G) and ψ ∈ L∞(G). First, note +that for a function g in L1(G) such that ˆg = 1 in some neighborhood U of 0, we have +ϕ(ψ − g ∗ ψ) = 0 for every ψ in L∞(G). In fact, to show this, it is sufficient to show +that ψ − g ∗ ψ is in E′ +G\{0}, in other words, sp(ψ − g ∗ ψ) ⊆ G \ {0}. Take a function h +in L1(G) such that ˆh = 0 outside U and ˆh(0) = 1. Observe that +h ∗ (ψ − g ∗ ψ) = (h − g ∗ h) ∗ ψ. +14 + +Here, we have (h−g ∗h)�(λ) = ˆh(λ)(1− ˆg(λ)) = 0 for every λ ∈ Γ. Hence, h−g ∗h = 0 +and thus h ∈ J(ψ − g ∗ ψ). Since ˆh(0) = 1, we conclude that 0 ̸∈ sp(ψ − g ∗ ψ). We +obtain the desired assertion. +For any f ∈ P(G) and ε > 0, there exists a function h in L1(G) such that ˆh = 1 on +some neighborhood of 0 and ∥f − h∥1 < ε(See [15], Theorem 2.6.5). Hence, we have +|ϕ(ψ − f ∗ ψ)| = |ϕ(ψ − h ∗ ψ) + ϕ(h ∗ ψ − f ∗ ψ)| +≤ |ϕ(ψ − h ∗ ψ)| + |ϕ(h ∗ ψ − f ∗ ψ)| += |ϕ(h ∗ ψ − f ∗ ψ)| += |ϕ((h − f) ∗ ψ))| +≤ ∥ϕ∥∥h − f∥1∥ψ∥∞ +≤ ∥ϕ∥∥ψ∥∞ε. +Since ε > 0 can be arbitrary, we obtain ϕ(ψ − f ∗ ψ) = 0, which means that ψ is in +T(G). This completes the proof. +5. A functional analytic condition for almost convergence +In this section, applying Theorem 4.2, we obtain the second necessary and sufficient +condition on almost convergence. Let us define the subspaces of L∞(G) as follows: +E(G) := {ψ ∈ L∞(G) : ψ is almost convergent} +E0(G) := {ψ ∈ L∞(G) : ψ is almost convergent to 0} +It is clear that the following decomposition holds true: +E(G) = R ⊕ E0(G), +ψ �→ α + (ψ − α), +where α is the number to which ψ almost converges. Theorem 4.2 can be reformulated +in terms of almost convergence as follows: +Theorem 5.1. Let G be a locally compact abelian group. ψ ∈ L∞(G) is almost con- +vergent to 0 if and only if +ψ ∈ clL∞(G)E′ +G\{0}. +In other words, ψ is almost convergent to 0 if and only if for any ε > 0, there exists +ψ1 ∈ L∞(G) whose spectrum does not intersect some neighborhood of 0 such that ∥ψ − +ψ1∥∞ < ε. +In what follows, we provide some applications of the theorem. For the proof of the +following result, see [15]. +Lemma 5.1. Let µ ∈ M(Γ). Let us define ˆµ∗(x) ∈ Cbu(G) by +ˆµ∗(x) = +� +G +χλ(x)dµ(λ), +x ∈ G. +Then, sp(ˆµ∗) = supp µ holds true, where supp µ is the support of the measure µ. +15 + +The following result was due to Eberlein [3]. He showed that every weakly almost +periodic function on R almost converges and the functions ˆµ∗ defined above is weakly +almost periodic. Here, we will give a more direct proof using Theorem 5.1. +Theorem 5.2. For each µ ∈ M(G), ˆµ∗ is almost convergent to µ({0}). +Proof . Let µ0 = µ − µ({0}). Then, for any ε > 0, there exits a neighborhood U of 0 +such that |µ0|(U) < ε. Let us define µUc(A) = µ(A ∩ Uc). Then, sp(ˆµ∗ +Uc) ∩ U = ∅ by +Lemma 5.1 and we have +|ˆµ∗ +0(x) − ˆµ∗ +Uc(x)| ≤ +� +U +|χλ(x)|d|µ|(x) = |µ0|(U) < ε. +Since supp ˆµUc(x) ⊆ Uc, by Theorem 5.1, we have ˆµ∗ +0 +ac +−→ 0, which in turn implies that +ˆµ∗ ac +−→ µ({0}), completing the proof. +Theorem 5.3. Let {λn}∞ +n=1 be a sequence of Γ. Let ψ ∈ L∞(G) be defined by +ψ(x) = +∞ +� +n=1 +cnχλn(x) (uniformly bounded and pointwise convergence). +Then, we have sp(ψ) ⊆ clΓ{λn}∞ +n=1. +Proof . Let Φ be the weak* closed invariant subspace of L∞(G) generated by {χλn}∞ +n=1. +Then, we have obviously sp(Φ) = clΓ{λn}∞ +n=1. Since ψ ∈ Φ by the above expression of +ψ, Φ(ψ) ⊆ Φ holds true. Thus, we obtain +sp(ψ) = sp(Φ(ψ)) ⊆ sp(Φ) = clΓ{λn}∞ +n=1. +This completes the proof. +Theorem 5.4. Let {λn}∞ +n=0 (λ0 = 0) be a sequence of Γ which does not accumulate to +0. If ψ ∈ L∞(G) is expressed by +ψ(x) = +∞ +� +n=0 +cnχλn(x) (uniformly bounded and pointwise convergence). +Then, ψ is almost convergent to c0. +Proof . By Theorem 5.4, sp(ψ0) ⊆ clΓ{λn}∞ +n=1, where ψ0(x) = ψ(x) − c0. By the +assumption that {χλn}∞ +n=1 does not accumulate to 0, there exists some neighborhood +U of 0 such that U ∩ {λn}∞ +n=1 = ∅. Hence, we have sp(ψ0) ⊆ Uc and by Theorem 5.1, +we have ψ0 +ac +−→ 0, which shows the desired assertion. +We give an example in the case G = R. One of the important examples of functions +on R which is expressed by the sum of exponential functions is Dirichlet series. +16 + +Corollary 5.1. Let {an}∞ +n=1 be a sequence of complex numbers and ψ(s) is the Dirichlet +series whose coefficients are {an}∞ +n=1. +ψ(s) = a1 +1s + a2 +2s + · · · an +ns + · · · += a1 +1σ e−it log 1 + a2 +2σ e−it log 2 + · · · an +nσ e−it log n + · · · (s = σ + it). +If ψ(s) converges uniformly boundedly and pointwisely on the line Re(s) = σ, then, +ψσ(t) = ψ(σ + it) is almost convergent to a1. +6. One-sided almost convergence +In the following two sections, we consider the special groups of G = Z and R. For +these groups, it is convenient to define almost convergence for functions defined on +positive numbers Z+ and R+ := {x ∈ R : x ≥ 0}. In what follows, the symbols G +stands for Z or R and G+ stands for Z+ or R+. +Let us define three closed subspaces of L∞(G) as follows: +L∞ +0 (G) := {φ ∈ L∞(G) : lim +|x|→∞ φ(x) = 0}, +L∞ +0,+(G) := {φ ∈ L∞(G) : lim +x→∞ φ(x) = 0}, +L∞ +0,−(G) := {φ ∈ L∞(G) : lim +x→−∞ φ(x) = 0}. +For simplicity, we will also use the symbols L∞, L∞ +0 , L∞ +0,+ and L∞ +0,− as abbreviations of +L∞(G), L∞ +0 (G), L∞ +0,+(G) and L∞ +0,−(G), respectively. Now observe that the isomorphism +L∞/L∞ +0 ∼= L∞/(L∞ +0,+ ∩ L∞ +0,−) +∼= (L∞/L∞ +0,+) ⊕ (L∞/L∞ +0,−) +holds. Here, the image of an equivalence class [ψ] in L∞/L∞ +0 is given by [ψ+] + [ψ−], +where ψ+ := ψ · IG+ and ψ− := ψ · I−G+, where IE denotes the characteristic function +of a subset E of G. +Therefore, for any ϕ in L∞(G)∗ with ϕ = 0 on L∞ +0 , we have the decomposition of ϕ +given as follows: +(L∞/L∞ +0 )∗ ∼= (L∞/L∞ +0,+)∗ ⊕ (L∞/L∞ +0,−)∗, +ϕ = ϕ+ + ϕ−, +where ϕ+(ψ) := ϕ(ψ+) and ϕ−(ψ) := ϕ(ψ−) for each ψ ∈ L∞. We note that the spaces +(L∞/L∞ +0,±)∗ can be identified with the subspace of L∞(G)∗ consisting of those elements +vanishing on L∞ +0,±, respectively. +By Theorem 2.2, for each ϕ ∈ T , we have ϕ = 0 on L∞ +0 and hence, ϕ is decomposed +into the sum of ϕ+ ∈ (L∞/L∞ +0,+)∗ and ϕ− ∈ (L∞/L∞ +0,−)∗. We use the symbols +T+ := T ∩ (L∞/L∞ +0,+)∗, +T− := T ∩ (L∞/L∞ +0,−)∗. +Of course, T = co(T+ ∪ T−) holds true. +Now we define one-sided almost convergence for elements in L∞(G). +17 + +Definition 6.1. We say that ψ is one-sidedly almost convergent to the number α if +ϕ(ψ) = α for every ϕ ∈ T+. +In this case, we write ψ +oac +→ α. We also define one-sided almost convergence for the +functions in L∞(G+), the space of essentially bounded functions defined on the positive +part G+ of G, by identifying ψ as the function on G defined by +˜ψ(x) = +� +ψ(x), +x ≥ 0, +0, +x < 0. +Lemma 6.1. Let ψ be in L∞(G) which vanishes on the negative part −G+ of G. Then, +ψ +ac +−→ 0 if and only if ψ +oac +−−→ 0 holds true. +Proof . Sufficiency is obvious. Suppose that ψ ∈ L∞(G) is one-sidedly almost conver- +gent to 0, that is, ϕ(ψ) = 0 for every ϕ ∈ T+. Since ψ is in L∞ +0,−(G), we have ϕ(ψ) = 0 +for every T−. Hence, it follows that ψ +ac +−→ 0 by the fact that T = co(T+ ∪ T−). +We can provide a similar condition to Theorem 2.3 for a given ψ ∈ L∞(G) to be +one-sidedly almost convergent. To this end, we need a following elementary lemma +concerning extreme values of means in (L∞/L∞ +0,+)∗. +Lemma 6.2. Let ϕ be a mean on L∞(G) which vanishes on L∞ +0,+(G). Then, we have +ϕ(ψ) ≤ sup +x∈R+ +ψ(x) +for every ψ ∈ L∞(G). +Theorem 6.1. A mean ϕ on L∞(G)∗ is in T+ if and only if +(i) G = Z +ϕ(ψ) ≤ p+(ψ) := lim sup +k→∞ +sup +n∈Z+ +1 +k +k−1 +� +i=0 +ψ(n + i) +holds for every ψ ∈ L∞(Z). +(ii) G = R +ϕ(ψ) ≤ p+(ψ) := lim sup +θ→∞ +sup +x∈R+ +1 +θ +� x+θ +x +ψ(t)dt +holds for every ψ ∈ L∞(R). +Proof . Using Lemma 6.2, necessity can be proved in a similar way to the proof of +Theorem 2.1. For sufficiency, observe that we have p+(ψ) ≤ p(ψ) for every ψ ∈ L∞(G) +and thus, by Theorem 2.2, it follows that a mean ϕ satisfying the condition in the +theorem is in T . Furthermore, it is clear that p+(ψ) = p+(ψ)(:= −p+(−ψ)) = 0 holds +for every ψ ∈ L∞ +0,+(G) and thus, such a ϕ is in (L∞/L∞ +0,+)∗. As a result, we obtain that +ϕ is in T+ = T ∩ (L∞/L∞ +0,+)∗. This completes the proof. +18 + +Now, we can show the following result just as Theorem 2.3 was derived from Theorem +2.2. +Theorem 6.2. Let ψ be in L∞(G+). Then, ψ is one-sidedly almost convergent to α if +and only if +(i) G = Z +lim +k→∞ +1 +k +k−1 +� +i=0 +ψ(i + n) = α +uniformly in n ∈ Z+. +(ii) G = R +lim +θ→∞ +1 +θ +� x+θ +x +ψ(t)dt = α +uniformly in x ∈ R+. +We remark that one-sided almost convergence for l∞ = L∞(Z+) is equivalent to +Lorentz’s almost convergence and (i) of the above theorem is the very result Lorentz +proved (Theorem 1.1). +7. Complex Tauberian Thoerem for almost convergence +In this section, we study so-called complex Tauberian theorems for almost conver- +gence on the groups Z and R. One of the main theorems of this section (Theorem 7.5) +can be viewed as an anlogue of the celebrated Wiener-Ikehara theorem. First, following +[8], we introduce the following definition. +Definition 7.1. We say that a function f(z) defined on the unit disc D := {z ∈ C : +|z| < 1} has H1 boundary behavior at the point z0 = eit0 (t0 ∈ [0, 2π]) if there exist a +number δ > 0 and a function F(eit) in L1(t0 − δ, t0 + δ) such that f(reit) converges to +F(eit) in L1(t0 − δ, t0 + δ) as r → 1−. +Theorem 7.1. Let ψ be in l∞ and denote an = ψ(n) for n ≥ 0. Let ˆψ(z) be the +function on the unit disc D defined by +ˆψ(z) = +∞ +� +n=0 +anzn. +If ˆψ has H1 boundary behaviour at z = 1, then ψ is almost convergent to 0. +Proof . For each numbers r with 0 < r < 1, define the functions ψr(n) in L1(Z) and +ˆψr(θ) in L1(T) by +ψr(n) := +� +anrn +(n ≥ 0), +0 +(n < 0). +ˆψr(θ) := ˆψ(reiθ) = +∞ +� +n=0 +anrneinθ (θ ∈ T), +19 + +respectively. Let δ0 > 0 be a number such that +lim +r→1− +� δ0 +−δ0 +| ˆψr(θ) − ˆψ(θ)|dθ +2π = 0 +holds true for some ˆψ ∈ L1(−δ0, δ0). Fix ε > 0 and choose a number 0 < δ < δ0 such +that +� δ +−δ +| ˆψr(θ)|dθ +2π < ε +for every 0 < r < 1. Let Cδ be a function in L1(Z) such that ˆCδ = 1 on (− δ +2, δ +2), ˆCδ = 0 +outside [−δ, δ] and 0 ≤ ˆCδ ≤ 1 on T. For each 0 < r < 1, put +ˆψr,δ(θ) = ˆψr(θ)(1 − ˆCδ(θ)), +and +ψr,δ(n) = ( ˆψr,δ)�(n), +n ∈ Z. +Then, we have +sp(ψr,δ) ∩ +� +−δ +2, δ +2 +� += ∅ (0 < r < 1). +In fact, by Lemma 5.1, sp(ψr,δ) = supp ˆψr,δ ⊆ T \ (− δ +2, δ +2). By definition of ψr,δ, we +have +ψr,δ(n) = +� +T +ˆψr,δ(θ)e−inθ dθ +2π += +� +T +ˆψr(θ)e−inθ dθ +2π − +� +T +ˆψr(θ) ˆCδ(θ)e−inθ dθ +2π += ψr(n) − (ψr ∗ Cδ)(n). +Letting r → 1−, we obtain +ψδ(n) := lim +r→1−{ψr(n) − (ψr ∗ Cδ)(n)} = ψ(n) − (ψ ∗ Cδ)(n) (n ∈ Z) +and +sp(ψδ) ∩ +� +−δ +2, δ +2 +� += ∅ +by Lemma 3.4. For each r ∈ (0, 1) and n ∈ Z, we have +|ψr ∗ Cδ(n)| = +���� +� +T +ˆψr ˆCδ(θ)e−inθ dθ +2π +���� +≤ +� +T +| ˆψr(θ) ˆCδ(θ)|dθ +2π +≤ +� δ +−δ +| ˆψr(θ)|dθ +2π < ε. +Taking limit as r → 1−, we obtain |ψ ∗ Cδ(n)| ≤ ε. Then, it follows that +|ψδ(n) − ψ(n)| = |ψ ∗ Cδ(n)| < ε (n ∈ Z), +20 + +which means ∥ψδ − ψ∥∞ ≤ ε. Therefore, by Theorem 5.1, we conclude that ψ +ac +−→ 0. +We complete the proof. +Theorem 7.2. Let ψ be in l∞ and denote an = ψ(n) for n ≥ 0. Let ˆψ(z) be the +function on the unit disc D defined by +ˆψ(z) = +∞ +� +n=0 +anzn. +If ˆψ(z) − +α +1−z has H1 boundary behaviour at z = 1, then ψ is one-sidedly almost +convergent to α, that is, +lim +k→∞ +1 +k +k−1 +� +i=0 +ψ(i + n) = α +uniformly in n ≥ 0. +Proof . Let ψ1(n) = ψ(n) − α · IZ+(n). Then, we have ˆψ1(z) = ˆψ(z) − +α +1−z. Hence, +by Theorem 7.1, we obtain ψ1 +ac +−→ 0. Since, ψ1(n) = 0 for n < 0, we have ψ1 +oac +−−→ 0 by +Lemma 6.1. Then, by Theorem 6.2 we obtain +1 +k +k−1 +� +i=0 +ψ(i + n) = α +uniformly in n ≥ 0. We complete the proof. +Corollary 7.1. Let ψ be in l∞ and denote an = ψ(n) for n ≥ 0. Let ˆψ(z) be the +function on the unit disc D defined by +ˆψ(z) = +∞ +� +n=0 +anzn. +If ˆψ has at most a pole of order 1 at z = 1, then ψ +oac +−−→ −α, where α is the residue of +ˆψ(z) at z = 1. +Proof . By the assumption, on a neighborhood U of 1, we can write as +ˆf(z) = +α +z − 1 + b0 + b1(z − 1) + b2(z − 1)2 + · · · . +Hence, we obtain +ˆf(z) − −α +1 − z = b0 + b1(z − 1) + b1(z − 1)2 + · · · . +Since the right-hand side of the above equation is continuous on U, we have ˆf(z)− −α +1−z ∈ +H1 +{1}(D). We obtain the result by Theorem 7.2. +Now we mention an interesting result of Katznelson and Tzafriri related to this +theorem, which can be regarded as a dual of the above theorem. (see [7], [8]). +21 + +Theorem 7.3. Let {an}n≥0 be a bounded sequence of complex numbers. Suppose that +the analytic function +f(z) = +∞ +� +n=0 +anzn +has H1 boundary behavior on the unit circle except for z = 1. Then, the equation +lim +n→∞(an+1 − an) = 0. +holds true. +Analogous results concerning G = R can be obtained. Since proofs are similar to the +case G = Z above, in what follows, we exhibit only results without their proofs. First, +we define H1 boundary behavior for the functions defined on the right-half plane. +Definition 7.2. We say that a function f(z) defined on the right-half plane C+ := +{z ∈ C : Re(z) > 0} has H1 boundary behavior at the point z0 = iy0 (y0 ∈ R) if there +exist a number δ > 0 and a function F(y) in L1(y0 − δ, y0 + δ) such that f(x + iy) +converges to F(y) in L1(y0 − δ, y0 + δ) as x → 0+. +Theorem 7.4. Let ψ ∈ L∞(R+). Let Lψ(s) be the Laplace transform of ψ, the analytic +function on the right half plane C+ defined by +Lψ(s) = +� +R+ +ψ(t)e−stdt, +s ∈ C+. +If Lψ(s) has H1 boundary behaviour at s = 0, then ψ is almost convergent to 0. +Theorem 7.5. Let ψ ∈ L∞(R+) and let Lψ(s) be its Laplace transform. If Lψ(s) − α +s +has H1 boundary behaviour at s = 0, then ψ is one-sidedly almost convergent to α, +that is, +lim +θ→∞ +1 +θ +� x+θ +x +ψ(t)dt = α +uniformly in x ≥ 0. +Corollary 7.2. Let ψ ∈ L∞(R+) and let Lψ(s) be its Laplace transform. If Lψ(s) has +at most a pole of order 1 at s = 0, then ψ +oac +−−→ α, where α is the residue of Lψ(s) at +s = 0. +Finally, we remark on the relation among Tauberian theorems concerning the be- +havior of analytic functions f(z) = �∞ +n=0 anzn or that of Lψ(s) near its boundaries. +First, we mention the following classical and famous result (See [8]). +Theorem 7.6 (Hardy-Littlewood). Let {an}n≥0 be a sequence of real numbers with +an ≥ C (n ≥ 0) for some constant C. Suppose that f(x) = �∞ +n=0 anxn converges for +|x| < 1 (x ∈ R) and +lim +x→1(1 − x)f(x) = α. +22 + +Then, it holds that +lim +k→∞ +1 +k +k−1 +� +i=0 +ai = α. +Let H1(D) be the Hardy space on the unit disc and H1 +{1}(D) be the class of functions +whose elements are the analytic functions on D having H1 boundary behavour at the +point z = 1. Then, we can summarize as follows: Here let {an}n≥0 be a bounded +sequence of complex numbers and f(z) = �∞ +n=0 anzn. + + + + + +f(x) − +α +1−x = o( +1 +1−x) (x → 1−) +=⇒ limk→∞ +1 +k +�k−1 +i=0 ai = α, +f(z) − +α +1−z ∈ H1 +{1}(D) +=⇒ limk→∞ +1 +k +�k−1 +i=0 an+i = α uniformly in n ∈ Z+, +f(z) − +a +1−z ∈ H1(D) +=⇒ limk→∞ ak = α. +The first one follows from Theorem 7.6. The second is due to Theorem 7.2. The third is +by the Riemann-Lebesgue lemma. Now, It is easy to show that for a bounded sequence +{an}, limn→∞ an = 0 if and only if an +oac +−−→ 0 and limn→∞ an+1 − an = 0. Hence, we see +that the third assertion above is decomposed into the two assertions of Theorems 7.2 +and 7.3. +The corresponding results for the case of R is in order: First, the right half plane +version of Hardy-Littlewood’s theorem reads as follows (see [8]): +Theorem 7.7. Let ψ be a locally integrable function on the half line R+ and ψ(x) ≥ +C (x ≥ 0) for some contstant C. If Lψ(x) exists for all x > 0 and +lim +x→0+ xLψ(x) = lim +x→0+ x +� ∞ +0 +e−xtψ(t)dt = α, +then we have +lim +θ→∞ +1 +θ +� θ +0 +ψ(t)dt = α. +Let H1 +{0}(C+) and H1 +R(C+) be the set of analytic functions on the right half plane +C+ having H1 boundary behavior at s = 0 and the whole line R, respectively. Then, +the following result holds true. + + + + + +Lφ(x) − α +x = o( 1 +x) (x → 0+) +=⇒ limθ→∞ +1 +θ +� θ +0 φ(t)dt = α, +Lψ(s) − α +s ∈ H1 +{0}(C+) +=⇒ limθ→∞ +1 +θ +� x+θ +x +ψ(t)dt = α uniformly in x ∈ R+, +Lψ(s) − α +s ∈ H1 +R(C+) +=⇒ w∗- limx→∞ φx = α. +We remark that the third one follows from the Wiener-Ikehara theorem. Briefly, we can +say that Theorem 7.4 is an intermediate result between Hardy-Littlewood’s theorem +and the Wiener-Ikehara theorem. +23 + +References +[1] C. Chow, On topologically invariant means on a locally compact group, Trans. Amer. Math. Sot. +151 (1970), 443-456. +[2] C. Chow, Weakly almost periodic functions and almost convergent functions on a group, Trans. +Amer. Math. Sot. 206 (1975), 175-200. +[3] W. F. Eberlein, Abstract ergodic theorems and weakly almost periodic functions, Trans. Amer. +Math. Soc. 67 (1949), 217-240. +[4] F. Forelli, Analytic and quasi-invariant measures, Acta Math. 118 (1967), 33-59. +[5] F. D. Greenleaf, Invariant means on topological groups and their applications, Van Nostrand, +Princeton, N. J. (1969). +[6] A. Hulanicki, Means and Folner condition on locally compact groups, Studia Math. 27 (1966), +87-104. +[7] Y. Katznelson, L. Tzafriri, On power bounded operators, J. Funct. Anal. 68 (1986) 313-328. +[8] J. Korevaar, Tauberian theory, Springer, Berlin, 2004. +[9] R. Kunisada, Invariant linear functionals on L∞(R+), J. Math. Anal. Appl. 481 (2020), 123452. +[10] G. G. Lorentz, A contribution to the theory of divergent series, Acta Math. 80 (1948), 167-190. +[11] P. Muhly, Function algebras and flows, Acta Sci. Math. (Szeged) 35 (1975), 55-66. +[12] A. T. Paterson, Amenability, American Mathematical Society (1988). +[13] R. A. Raimi, Mean values and Banach limits, Proc. Amer. Math. Soc. 8 (1957), 1029-1036. +[14] R. A. Raimi, On Banach’s generalized limits, Duke Math. J. 26 (1959), 17-28. +[15] W. Rudin, Fourier analysis on groups, Interscience, New York (1962). +[16] L. Sucheston, Banach limits, Amer. Math. Monthly 74 (1967), 308-311. +[17] K. Yosida, Functional analysis, Springer, Berlin (1965). +Faculty of Liberal Arts, Tsuru University, Tsuru-shi, Yamanashi-ken 402-8555, Japan +Email address: tk-waseda@ruri.waseda.jp +24 + diff --git a/ydAzT4oBgHgl3EQftP1k/content/tmp_files/load_file.txt b/ydAzT4oBgHgl3EQftP1k/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..15ba2c752ed99041c6aa161cdb99921e2ac2331d --- /dev/null +++ b/ydAzT4oBgHgl3EQftP1k/content/tmp_files/load_file.txt @@ -0,0 +1,696 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf,len=695 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='01672v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='FA] 4 Jan 2023 ON ALMOST CONVERGENCE ON LOCALLY COMPACT ABELIAN GROUPS RYOICHI KUNISADA Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' We study a summability method called almost convergence for bounded measurable functions defined on a locally compact abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' We define almost convergence using topologically invariant means and exhibit two different kinds of necessary and sufficient conditions, one is analytic and the other is functional analytic, for a given function to be almost convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' As an application, we show complex Tauberian theorems for almost convergence on the integers and the real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' In particular, the latter one can be viewed as an analogue of the Wiener-Ikehara theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Introduction For a locally compact abelian group G, let L1(G) be the group algebra of G and L∞(G) be the set of all essentially bounded measurable functions on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let Cbu(G) be the set of all bounded, uniformly continuous functions on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' The uniformity U of G is the set of all subsetes of G2 given by {(x, y) ∈ G2 : x − y ∈ U}, where U is a neighborhood of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Note that both spaces L∞(G) and Cbu(G) are Ba- nach spaces with respect to the supremum norm ∥ · ∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Here we consider complex- valued functions in general and we denote by L∞ R (G) the space of real-valued essentially bounded measurable functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' A general element of L1(G) is denotedy by the symbol f(we also use g, h if necessary) and that of L∞(G) is denoted by ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' For each s ∈ R, we use the symbols fs(x) := f(x+s) and ψs(x) := ψ(x + s), the translates of f and ψ by s, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let L∞(G)∗ and Cbu(G)∗ be the dual spaces of L∞(G) and Cbu(G), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' An element ϕ of L∞(G)∗ (Cbu(G)∗) is said to be a mean on L∞(G) (Cbu(G)) if it satisfies (1) ϕ ≥ 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=', ϕ(ψ) ≥ 0 for every positive ψ ∈ L∞(G) (Cbu(G)));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' (2) ϕ(1) = 1, where 1 is the constant function taking the value 1 everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Further, ϕ is called an invariant mean on L∞(G) (Cbu(G)) if it is a mean such that (3) ϕ(ψs) = ϕ(ψ) for every ψ ∈ L∞(G) (Cbu(G)) and s ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let us denote by I(G) (I0(G)) the set of all invariant means on L∞(G) (Cbu(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Primary 40H05;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Secondary 22B05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Almost convergence, topologically invariant means, tauberian theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' 1 Now, we introduce another class of means on L∞(G) which is more important for our purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' For f ∈ L1(G) and ψ ∈ L∞(G), their convolution f ∗ ψ is defined by f ∗ ψ(x) = � G f(t)ψ(x − t)dm(t), x ∈ G, where m is the Haar measure of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let P(G) be the set of positive elements f in L1(G) such that � G f(x)dm(x) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' A mean ϕ on L∞(G) is said to be a topologically invariant mean if it satisfies the following condition ([6]): (4) ϕ(f ∗ ψ) = ϕ(ψ) ∀ψ ∈ L∞(G) ∀f ∈ P(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let us denote by T (G) the set of all topologically invariant means on L∞(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Note that a topologically invariant mean is an invariant mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' In fact, if ϕ is topologically invariant, then ϕ(ψs) = ϕ(f ∗ ψs) = ϕ(fs ∗ ψ) = ϕ(ψ) for each ψ ∈ L∞(R) and s ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Conversely, For discrete groups G, it is easy to show that I(G) = T (G) holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' For every locally compact abelian group G, T (G) ̸= ∅ is valid, and if G is noncompact, we have T (G) ⊊ I(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' For compact abelian groups G, T (G) is the singleton consisting of the normalized Haar measure of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' We refer the reader to [5], [12] for more detailed exposition on these results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' The main objective of this paper is a summability method concerning (topologically) invariant means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let l∞ be the set of all bounded functions on the nonnegative integers Z+ := {n ∈ Z : n ≥ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Lorentz (1948) defined a summability method on l∞ called almost convergence using Banach limits ([10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Recall that an element ϕ of l∗ ∞, the dual space of l∞, is called a Banach limit if the following conditions are satisfied: (1) ϕ ≥ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' (2) ϕ(1) = 1, where 1 is the constant function taking the value 1 everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' (3) ϕ(ψn) = ϕ(ψ) for every n ∈ Z+ and ψ ∈ l∞, namely, ϕ is a right translation invariant mean on l∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let B be the set of all Banach limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Then, almost convergence for sequences is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='1 (Lorentz, 1948).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' ψ ∈ l∞ is said to be almost convergent to a (complex) number α if ϕ(ψ) = α for every ϕ ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' It is difficult to know from this abstract definition whether a given sequence of numbers is almost convergent or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' However, Lorentz proved an analytic condition for almost convergence as follows (his delivation was rather complicated and Sucheston [16] gave a more simple proof later).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='1 (Lorentz, 1948).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' ψ ∈ l∞ is almost convergent to α if and only if lim k→∞ 1 k k−1 � i=0 ψ(n + i) = α uniformly in n ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' 2 This assertion can be proved by the Hahn-Banach theorem and is a basic way to check whether a given sequence actually almost converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Note that Banach limits can be regarded as a special kind of invariant means on L∞(Z), where Z is the additive group of integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Hence, it is natural to consider that one can define the notion of almost convergence on an arbitrary locally compact abelian group or its subsemigroups (note that abelian groups G are amenable, that is, there exist invariant means on L∞(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Specifically, one can define formally almost convergence of functions in L∞(G) by the following : Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let G be a locally compact abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' We say that ψ ∈ L∞(G) is almost convergent to a complex number α if and only if ϕ(ψ) = α holds for every ϕ ∈ T (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' In this case, we write as ψ ac −→ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' The reason for adopting topological invariant means instead of (seemingly more natural) invariant means is that an analogous result of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='1 is also valid for groups not necessarily discrete (see [1], [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' The objective of this paper is to provide a new necessary and sufficient condition for a given bounded function on a locally compact abelian group to be almost convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' This is obtained through the thoery of harmonic analysis on locally compact abelian groups, especially, the theory of spectral synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Furthermore, applying this result, we obtain complex tauberian theorems for almost convergence on Z and R, which are related to the famous results of Katznelson-Tzafriri and Wiener-Ikehara.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' In Section 2, we exhibit an analytic condition for almost convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' This is a special case of the more general result of Chow [1] when the underling group is abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' However, considering the importance of the result, we include this section for the sake of completeness and consistency of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' In Section 3, following Forelli [4], we develop spectral theory of Cbu(G) and Cbu(G)∗, which contains a description of subspaces of Cbu(G)∗ defined via spectrum as the an- nihilator of a certain invariant subspace of Cbu(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' This is somewhat a formal gen- eralization of his relult to general locally compact abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' In Section 4, as an application of a result of the previous section, we provide the annihilator of the subspace of L∞(G)∗ spanned by topologically invariant means on L∞(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' In Section 5, using a result of Section 4, we obtain a new necessary and sufficient condition for al- most convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Section 6 deals with almost convergence on positive subsemigroups of the special groups Z and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' In Section 7, we deal with complex Tauberian theorems for almost convergence on Z and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' An analytic condition for almost convergence Let G be a locally compact abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Since G is an amenable group, there exists a summing net for G, namely, a net {Kδ}δ∈∆ of nonnull, compact subsets of G satisfying the following properties (see [5], [12]): 3 (1) Kδ ⊆ Kδ′ if δ ≤ δ′ (2) G = � δ∈∆ Kδ (3) limδ m(Kδ△Kδ,s) m(Kδ) = 0 uniformly in s on a compact subset of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' We fix one summing net for G and define a sublinear functional on L∞(G) as p(ψ) := lim sup δ sup x∈G 1 m(Kδ) � Kδ ψx(t)dt, where ψ ∈ L∞(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' We also introduce the functional p(ψ) := −p(−ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' We note thta p can be expressed as p(ψ) := lim inf δ inf x∈G 1 m(Kδ) � Kδ ψx(t)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Then, we have the following result: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' A mean ϕ on Cbu(G) is an ivariant mean if and only if ϕ(ψ) ≤ p(ψ) holds for every ψ ∈ Cbu(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Proof .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' First, we show necessity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let ϕ be an invariant mean on Cbu(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Note that, since ψ is in Cbu(G), the mapping G ∋ t �→ ψt ∈ Cbu(G) is continuous and thus, it is Bochner integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Then, we have ϕ(ψ) = 1 m(Kδ) � Kδ ϕ(ψt)dt = ϕ � 1 m(Kδ) � Kδ ψt(x)dt � ≤ sup x 1 m(Kδ) � Kδ ψx(t)dt holds true for every Kδ (see [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Here we use the elementary fact that for any mean ϕ on Cbu(R) and ψ in Cbu(R), it holds that ϕ(ψ) ≤ sup x∈R ψ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' In fact, let α := supx∈R ψ(x) and then α − ψ ≥ 0, thus by the positivity of ϕ, we have ϕ(α − ψ) ≥ 0 ⇔ ϕ(α) ≥ ϕ(ψ) ⇔ α ≥ ϕ(ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' By taking the limit superior over Kδ ′s, we obtain ϕ(ψ) ≤ lim sup δ sup x 1 m(Kδ) � Kδ ψx(t)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' 4 Now, we show sufficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let ϕ satisfy the condition in the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Then, for each ψ in Cbu(G), we have ϕ(ψ − ψs) ≤ lim sup δ sup x 1 m(Kδ) � Kδ (ψx(t) − ψx+s(t))dt = lim sup δ sup x 1 m(Kδ) �� Kδ ψx(t)dt − � Kδ ψx+s(t)dt � = lim sup δ sup x 1 m(Kδ) �� Kδ ψx(t)dt − � Kδ,−s ψx(t)dt � ≤ lim sup δ sup x 1 m(Kδ) � Kδ△Kδ,−s |ψx(t)|dt ≤ lim sup δ m(Kδ △ Kδ,−s) m(Kδ) ∥ψ∥∞ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' For the reverse inequality, we consider the relation ϕ(−ψ) ≤ p(−ψ) ⇔ ϕ(ψ) ≥ −p(−ψ) =: p(ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Then, we can show the inequality ϕ(ψ − ψs) ≥ 0 in the same argument as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Thus, we obtain ϕ(ψ − ψs) = 0, which shows the translation invariance of ϕ and we complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' We need the following lemma to obtain a similar condition for topologically invraiant means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' An equivalent assertion was shown in [1] in a more general form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' For any ψ ∈ L∞(G) and f ∈ P(G), p(ψ − f ∗ ψ) = p(ψ − f ∗ ψ) = 0 holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Proof .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' By direct computation, we have p(ψ − f ∗ ψ) = lim sup δ sup x∈G 1 m(Kδ) � Kδ {ψx(t) − (f ∗ ψ)x(t)}dt = lim sup δ sup x∈G 1 m(Kα) � Kδ � G {ψx(t) − ψx(t − u)}f(u)dudt = lim sup δ sup x∈G � G f(u)du 1 m(Kδ) � Kδ {ψx(t) − ψx(t − u)}dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' The integral 1 m(Kδ) � Kδ {ψx(t) − ψx(t − u)}dt 5 can be evaluated in two ways: 1 m(Kδ) � Kδ {ψx(t) − ψx(t − u)}dt ≤ ����� 1 m(Kδ) � Kδ△Kδ,−u ψx(t)dt ����� ≤ m(Kδ △ Kδ,−u) m(Kδ) ∥ψ∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' (1) Also, we obtain 1 m(Kδ) � Kδ {ψx(t) − ψx(t − u)}dt ≤ 1 m(Kδ)2∥ψ∥∞m(Kδ) = 2∥ψ∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' (2) Let ε > 0 be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Take a compact subset Cε such that � G\\Cε |f(t)|dt < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Then, choose γ such that m(Kδ△Kδ,−u) m(Kδ) ≤ ε for every δ ≥ γ and u ∈ Cε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Then, by (1) and (2), we have lim sup δ sup x∈G � G f(u)du 1 m(Kδ) � Kδ {ψx(t) − ψx(t − u)}dt = lim sup δ sup x∈G � � Cε f(u)du 1 m(Kδ) � Kδ {ψx(t) − ψx(t − u)}dt + � G\\Cε f(u)du 1 m(Kδ) � Kδ {ψx(t) − ψx(t − u)}dt � ≤ lim sup δ sup x∈G � Cε |f(u)|m(Kδ △ Kδ,−u) m(Kδ) ∥ψ∥∞du + 2∥ψ∥∞ � G\\Cε |f(u)|du ≤ ε∥ψ∥∞ � Cε |f(u)|du + 2ε∥ψ∥∞ ≤ 3ε∥ψ∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Since ε > 0 can be arbitrary, we obtain p(ψ−f ∗ψ) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' The equation p(ψ−f ∗ψ) ≥ 0 can be proved similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Since p(ψ) ≤ p(ψ) holds for each ψ ∈ L∞(G), we have p(ψ − f ∗ ψ) = p(ψ − f ∗ ψ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' We complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Combining Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='1 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='1, we obtain the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' A mean ϕ on L∞(G) is a topologically ivariant mean if and only if ϕ(ψ) ≤ p(ψ) holds for every ψ ∈ L∞(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Proof .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' First, we show necessity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Assume that ϕ is in T (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Then, for any ψ ∈ L∞(G) and f ∈ P(G), we have ϕ(f ∗ ψ) = ϕ(ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Note that, as stated before, ϕ is an invariant mean on Cbu(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Since f ∗ ψ ∈ Cbu(G), by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='1, we have ϕ(f ∗ ψ) ≤ p(f ∗ ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' 6 Since, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='1, p(f ∗ ψ) = p(ψ) holds true, we obtain ϕ(ψ) = ϕ(f ∗ ψ) ≤ p(f ∗ ψ) = p(ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Now, we show sufficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Assume that ϕ(ψ) ≤ p(ψ) holds for each ψ ∈ L∞(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Then, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='1, for any f ∈ P(G), we have 0 = p(ψ − f ∗ ψ) ≤ ϕ(ψ − f ∗ ψ) ≤ p(ψ − f ∗ ψ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Hence, we obtain ϕ(ψ − f ∗ ψ) = 0 and ϕ is topologically invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let ψ ∈ L∞ R (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Then, ψ is almost convergent to α if and only if lim δ 1 m(Kδ) � m(Kδ) ψ(x + s)dx = α uniformly in s ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Proof .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Note that, for any ϕ ∈ T (G) and ψ ∈ L∞(G), we have p(ψ) ≤ ϕ(ψ) ≤ p(ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Conversely, for any real number α with p(ψ) ≤ α ≤ p(ψ), there exists some ϕ ∈ T (G) such that ϕ(ψ) = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' In fact, we define ϕ0 on Rψ = {cψ : c ∈ R} by ϕ0(cψ) = cα and then can extend it to whole L∞(G) such that ϕ(ψ) ≤ p(ψ) holds for every ψ ∈ L∞(G) by the Hahn-Banach theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' This extended ϕ is an topogically invariant mean by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Hence, we have shown that ϕ(ψ) = α for every ϕ ∈ T (G) if and only if p(ψ) = p(ψ) = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Now, we show that this is equivalent to the condition given in the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' First, necessity is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Recall that the functionals p and p are expressed as folllows: p(φ) = lim sup δ sup s∈G 1 m(Kδ) � Kδ ψ(x + s)dx, p(ψ) = lim inf δ inf s∈G 1 m(Kδ) � Kδ ψ(x + s)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let ε > 0 be any positive number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Then, there exist real numbers δ0 and δ1 such that sup s∈G 1 m(Kδ) � Kδ ψ(x + s)dx < α + ε whenever δ ≥ δ0 and inf s∈G 1 m(Kδ) � Kδ ψ(x + s)dx > α − ε whenever δ ≥ δ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Hence, let δ2 = max(δ0, δ1) and we have −ε < 1 m(Kδ) � Kδ ψ(x + s)dx − α < ε for every s ∈ G whenever δ ≥ δ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' This means uniform convergence of the net in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' 7 Now, we extend the above result to complex valued functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let ψ ∈ L∞(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Then, ψ is alomst convergent to α if and only if lim δ 1 m(Kδ) � m(Kδ) ψ(x + s)dx = α uniformly in s ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Proof .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let us denote ψ(x) = u(x)+iv(x), where u, v in L∞ R (G) are real and imaginary parts of ψ respectively and let α = β + iτ be a complex number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Then, ϕ(ψ) = α = β + iτ for every ϕ ∈ T (G) if and only if ϕ(u) = β, ϕ(v) = τ for every ϕ ∈ T (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='3, this is equivalent to lim δ 1 m(Kδ) � m(Kδ) u(x + s)dx = β, lim δ 1 m(Kδ) � m(Kδ) v(x + s)dx = γ uniformly in s ∈ G, which is equivalent to the assertion of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' We give examples of summing nets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' It can be easily varified that Kn = [−n, n] (n ∈ N), Kθ = [−θ, θ] (θ > 0) are summing nets for G = Z and R, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Then, we have the following formu- lation of almost convergence on Z and R: ψ ∈ L∞(Z) is almost convergent to α if and only if lim n→∞ 1 2n + 1 k+n � i=k−n ψ(i) = α uniformly in k ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Similarly, ψ ∈ L∞(R) is almost convergent to α if and only if lim θ→∞ 1 2θ � x+θ x−θ ψ(t)dt = α uniformly in x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' We note that the former is a two-sided version of Sucheston’s result (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='1) and the latter was given by Raimi [13] (see also [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Another kind of analytic expres- sion for almost convergence on R was given in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Spectral synthesis of bounded measurable functions Let G be a locallly compact abelian group and Γ be its dual group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' An element λ in Γ is denoted as χλ(x) when it is viewed as a character on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' For a function f in L1(G), its Fourier transform ˆf is defined by ˆf(λ) = � G f(x)χλ(−x)dm(x), λ ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' The purpose of this section is to develop the spectrum theory of Cbu(G) and Cbu(G)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' For the following contents, we refer the readers to [15] as a basic literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' For f ∈ L1(G), let Z(f) be the zero set of the Fourier transform of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let I be a closed ideal of L1(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' We denote by Z(I) the intersection of the zero sets of the Fourier transforms of elements in I, namely, Z(I) = � f∈I Z(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' By definition, each Z(I) is a closed set of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Note that, in general, Z(I) does not uniquely determine I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' In other words, there exist distinct closed ideals I and I′ such that Z(I) = Z(I′) holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' However, there is a closed set C of Γ which is the zero set of a unique closed ideal of L1(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Such a closed set is called a spectral synthesis set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' The following result is a fundamental result of spectral synthesis thoery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let I be a closed ideal of L1(G) and f ∈ L1(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' If the internal of Z(f) contains Z(I), then f ∈ I holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' For a closed set C of Γ, let I1(C) be the set of all f ∈ L1(G) such that C ⊆ Z(f) and let I0(C) be the closure of the set of all f ∈ L1(G) such that C ⊆ IntΓZ(f), the interior of Z(f) in Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Then, by definition, I1(C) is the largest closed ideal of L1(G) with Z(I) = C and, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='1, I0(C) is the smallest closed ideal of L1(G) with Z(I) = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Note that C is a spectral synthesis set if and only if I0(C) = I1(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Now, following classical theory, we define spectrum of subspaces and elments of L∞(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let Φ be a weak* closed invariant subspace of L∞(G), namely, Φ is a subspace closed with respect to weak* topology of L∞(G) and ψ ∈ Φ implies ψs ∈ Φ for every s ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Related to Φ, we define the closed ideal J(Φ) of L1(G) by the set of all functions f in L1(G) such that f ∗ ψ = 0 for every ψ ∈ Φ: J(Φ) = {f ∈ L1(G) : f ∗ ψ = 0 ∀ψ ∈ Φ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Then, we define the spectrum sp(Φ) of Φ by Z(J(Φ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' In the same way, we define spectrum of each member of L∞(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' For each ψ in L∞(G), let J(ψ) be the set of all functions f in L1(G) such that f ∗ψ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Then, we define the spectrum sp(ψ) of ψ by Z(J(ψ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' It is easy to confirm that sp(ψ) = sp(Φ(ψ)), where Φ(ψ) is the weak* closed invariant subspace of L∞(G) generated by ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' We give another description of spectrum of Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let ⟨X, X∗⟩ be any dual pair of locally convex spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' For subspaces E of X and E∗ of X∗, let us define their annihilators as 9 follows: E⊥ = {ϕ ∈ X∗ : ϕ(x) = 0 ∀x ∈ E}, E∗⊥ = {x ∈ X : ϕ(x) = 0 ∀ϕ ∈ E∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Note that (E⊥)⊥ = E, (E∗⊥)⊥ = E∗ is an immediate consequence of the Hahn-Banach theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let Φ be a weak* closed invariant subspace of L∞(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Then, sp(Φ) is the set of all chracters contained in Φ : sp(Φ) = {λ ∈ Γ : χλ ∈ Φ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Proof .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' First, suppose χλ ∈ Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let f be in J(Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Then, in particular, we have f ∗ χλ(x) = � G f(t)χλ(x − t)dm(t) = � G f(t)χλ(x)χλ(−t)dm(t) = χλ(x) ˆf(λ) = 0 for every x ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Thus, ˆf(λ) = 0 and λ ∈ Z(J(Φ)) = sp(Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' On the other hand, let us assume that χλ ̸∈ Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Hence, by the Hahn-Banach theorem, there exists some f ∈ L1(G) such that ⟨f ∗, ψ⟩ = 0 for every ψ ∈ Φ and ⟨f ∗, χλ⟩ = 1, where f ∗(x) = f(−x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' That is, � G f ∗(t)ψ(t)dt = � G f(−t)ψ(t)dm(t) = f ∗ ψ(0) = 0 for each ψ ∈ Φ and � G f ∗(t)χλ(t)dt = � G f(−t)χλ(t)dm(t) = ˆf(λ) = 1 holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Since Φ is translation invariant, the first equation also holds for any trans- late ψx of ψ and we obtain f ∗ ψ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Hence, f is in J(Φ) and ˆf(λ) = 1, it holds that λ ̸∈ sp(Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Obviously, the smallest weak* closed subspace of L∞(G) with sp(Φ) = C is the subspace Φ1(C) which is generated by the characters {χλ}λ∈C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Note that if I = Φ⊥, or, equivalently, I⊥ = Φ, sp(Φ) = Z(I) holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' This implies that Φ1(C) = I1(C)⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Further, let Φ0(C) be the annihilaor of I0(C), that is, Φ0(C) = I0(C)⊥, then Φ0(C) is the largest weak* closed invariant subspace with sp(Φ) = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' We have Φ0(C) = Φ1(C) if and only if sp(Φ) is a spectral synthesis set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' This observation implies the following assertion: 10 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let Φ be a weak* closed invariant subspace of L∞(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' If sp(Φ) is a spectral synthesis set, then Φ is synthesized by its spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Namely, each member ψ of Φ is a weak* limit of a net of trigonometric polynomials ψα(x) = nα � k=1 cα keitα k x where cα k ∈ C and tα k ∈ sp(Φ) for every α and 1 ≤ k ≤ nα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Following [4], we can also define spectrum for elements of Cbu(G)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let ϕ ∈ Cbu(G)∗ and f ∈ L1(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' We define their convolution f ∗ϕ, which is also an element of Cbu(G)∗, by f ∗ ϕ(ψ) = ϕ(f ∗ ψ), ψ ∈ L∞(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' For each ϕ ∈ Cbu(G)∗, let J(ϕ) be the closed ideal of L1(G) consisting of those elements f for which f ∗ ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Then, we define the spectrum sp(ϕ) of ϕ by Z(J(ϕ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' The following lemma is basic for arguments what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let ψ be in L∞(G), ϕ be in Cu(R)∗ and f be in L1(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Then, the following results holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' (i) sp(f ∗ ψ) ⊆ sp(ψ) ∩ supp ˆf, (ii) sp(f ∗ ϕ) ⊆ sp(ϕ) ∩ supp ˆf, where supp ˆf denotes the support of ˆf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Proof .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' (i) First, note that the inclusion sp(f ∗ ψ) ⊆ sp(ψ) is obvious by definition of spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Fix arbitrary λ ̸∈ supp ˆf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let g be a funciton in L1(G) such that ˆg = 0 on a neighborhood of supp ˆf and ˆg(λ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Then, note that (f ∗ g)�(λ) = ˆf(λ)ˆg(λ) = 0 for every λ ∈ Γ, which means that f ∗ g = 0 by the uniqueness thorem of the Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Hence, we obtain g ∗ (f ∗ ψ) = (g ∗ f) ∗ ψ = 0 and thus, g is in J(f ∗ ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Since λ ̸∈ Z(g), we conclude that λ /∈ sp(f ∗ ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' (ii) This assertion can be proved in the same way as (i) and we omit the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let ψ be in L∞(G) and ϕ be in Cbu(G)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Then, we have the following results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' (i) If sp(ψ) = ∅, then ψ = 0, (ii) If sp(ϕ) = ∅, then ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Proof .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' (i) Suppose that sp(ψ) = ∅ holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' By definition of spectrum and Wiener’s Tauberian theorem, which asserts that if a closed ideal I in L1(G) satisfies Z(I) = ∅, then I = L1(G), we conclude that f ∗ψ = 0 for every f in L1(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Take an approximate of identity {fα} of L1(G), observe that lim α ∥fα ∗ ψ − ψ∥∞ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Then, we have ∥ψ∥∞ ≤ ∥ψ − fα ∗ ψ∥∞ + ∥fα ∗ ψ∥∞ = ∥ψ − fα ∗ ψ∥∞, 11 which tends to 0 as α → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Thus, it follows that ψ = 0, completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' (ii) This can be proved similarly as (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' We define the subspaces of Cbu(G) and Cbu(G)∗ as follows: EA = {ψ ∈ Cbu(G) : sp(ψ) ⊆ A};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' E∗ A = {ϕ ∈ Cbu(G)∗ : sp(ϕ) ⊆ A}, where A is a subset of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Observe that if A is a closed subset of Γ, then EA = Φ0(A)∩Cbu(G) holds true and thus, EA is a closed subspace of Cbu(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' For spaces E∗ A, we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let A be a closed subset of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Then, E∗ A is a weak*-closed subspace of Cbu(G)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' In particular, it is norm closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Proof .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let {ϕα} be a net of elements of E∗ A such that w∗- limα ϕα = ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' We show that ϕ ∈ E∗ A, that is, sp(ϕ) ⊆ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Take arbitrary λ ̸∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Then, there exists a function f in L1(G) such that ˆf = 0 on a some neighborhood of A and ˆf(λ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Note that f ∗ ϕα = 0 for every α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Then, for any ψ in L∞(G), we have (f ∗ ϕ)(ψ) = ϕ(f ∗ ψ) = lim α ϕα(f ∗ ψ) = lim α f ∗ ϕα(ψ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Hence, we obtain f ∗ ϕ = 0 and thus, f ∈ J(ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Since λ ̸∈ Z(f), we have λ ̸∈ sp(ϕ), completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' The following result will be used in Section 6, which states the continuity of spectrum with respect to the weak* topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let A be a closed subset of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Suppose that {ψα} be a net of L∞(G) with sp(ψα) ⊆ A for every α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' If w∗- limα ψα = ψ, namely, {ψα} converge to ψ in the weak* sense, then we have sp(ψ) ⊆ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Proof .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' In fact, for each f ∈ L1(G) with A ⊆ IntΓZ(f), in other words, for f ∈ I0(A), we have f ∗ ψα = 0 for every α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Thus, we have f ∗ ψ = lim α f ∗ ψα = 0, which means that f ∈ J(ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Since ∩f∈I0(A)Z( ˆf) = A, we obtain sp(ψ) = ∩f∈J(ψ)Z( ˆf) ⊆ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' This complets the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' In what follows, we give a description of the spaces E∗ A in terms of spaces EA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Note that we consider the dual pair ⟨Cbu(G), Cbu(G)∗⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let A be a closed subset of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Then, we have E∗ A ⊥ = clCbu(G)EAc, where the symbol clCbu(G)E denotes the closure of a subspace E in Cbu(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' 12 Proof .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' For each ψ ∈ Cbu(G) and ϕ ∈ Cbu(G)∗, Let us define ψ′(x) ∈ Cbu(G) by ψ′(x) = ϕ(ψx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' For any f ∈ L1(G), by the fact mentioned in the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='1, we have ϕ(f ∗ ψ) = � G ϕ(ψ−t)f(t)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Replacing ψ by ψx where x ∈ G, we obtain f ∗ ϕ(ψx) = ϕ(f ∗ ψx) = � G ϕ(ψx−t)f(t)dt = � G ψ′(x − t)f(t)dt = f ∗ ψ′(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' (3) From this equation, we can deduce the following result: sp(ψ′) ⊆ sp(ψ) ∩ sp(ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' (4) In fact, assume that f ∈ J(ψ), that is, f ∗ ψ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Then, for any x ∈ G, it holds that f ∗ ψx = (f ∗ ψ)x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Thus, by the equation (3), if f ∈ J(ψ), we have f ∗ ψ′ = 0, that is, f ∈ J(ψ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Hence, we obtain sp(ψ′) ⊆ sp(ψ) since J(ψ) ⊆ J(ψ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Next, assume that f ∈ J(ϕ), that is, f ∗ ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Again, by equation (3), we obtain f ∗ ψ′ = 0, that is, f ∈ J(ψ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' This shows that sp(ψ′) ⊆ sp(ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' We have obtained the desired relation (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Now suppose that ϕ ∈ E∗ A(G) and ψ ∈ EAc(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Then, by (4), we have sp(ψ′) ⊆ sp(ψ) ∩ sp(ϕ) ⊆ A ∩ Ac = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Thus, sp(ψ′) = ∅ holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Then, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='2 (1), we have ψ′ = 0 and ϕ(ψ) = 0, that is, ψ ∈ E∗ A ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Now we obtain EAc ⊆ E∗ A ⊥, which means that clCbu(G)EAc ⊆ E∗ A ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' We now show the reverse inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let us assume that ϕ = 0 for every ψ ∈ EAc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' We show that sp(ϕ) ⊆ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let λ ̸∈ A and take f ∈ L1(G) such that ˆf = 0 on some neighborhood U of A with λ ̸∈ U and ˆf(λ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Then, f ∗ ϕ(ψ) = ϕ(f ∗ ψ) = 0 for every ψ ∈ Cbu(G) since sp(f ∗ ψ) ⊆ supp ˆf ⊆ Uc ⊆ Ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Thus, f is a function in J(ϕ) with ˆf(λ) = 1 and we conclude that λ ̸∈ sp(ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Now, we have shown that EAc⊥ ⊆ E∗ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Considering the annihilators of the both sides of the above equation and then taking the closure, we obtain the inclusion relation E∗ A ⊥ ⊆ clCbu(G)EAc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' An application to topologically invariant means We apply Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='4 to the case C = {0}, that is, the singleton consisting of only the unit of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' The following result is essentially due to Muhly [11], who proved the special case of G = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' For ϕ ∈ Cbu(G)∗, ϕ is an invariant mean if and only if sp(ϕ) = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' 13 Proof .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' First, assume that ϕ is invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Then, for any f ∈ L1(G) for which Z(f) contains {0}, that is, � G f(x)dm(x) = 0, we have f ∗ ϕ(ψ) = ϕ(f ∗ ψ) = � G ϕ(ψ−t)f(t)dt = � G ϕ(ψ)f(t)dt = ϕ(ψ) � G f(t)dt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Therefore, we conclude that sp(ϕ) ⊆ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Note that if sp(ϕ) = ∅, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='2, ϕ = 0 holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Thus, we have the desired conclusion that sp(ϕ) = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Conversely, assume that sp(ϕ) = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Then, for any f ∈ I0({0}) and ψ ∈ Cbu(G), we have f ∗ ϕ(ψ) = � R ϕ(ψ−t)f(t)dt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' For any x ∈ G, set ψ′(x) = ϕ(ψx) and we have f ∗ ϕ(ψx) = � R ϕ(ψx−t)f(t)dt = � R ψ′(x − t)f(t)dt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' This means that sp(ψ′) = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Since {0} is a spectral synthesis set, we have Φ(ψ′) = Φ1({0}) = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Hence, ψ′(x) is a constant function, which means that ϕ is translation invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let T(G) be the closed subspace of L∞(G)∗ spaned by T (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' We determine the annihilator of T(G) via Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' For any subset A of Γ, let us define the subspace E′ A of L∞(G) by E′ A = {ψ ∈ L∞(G) : sp(ψ) ⊆ A}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' For a locally compact abelian group G, the following assertion holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' T(G)⊥ = clL∞(G)E′ G\\{0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Proof .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let ϕ ∈ T(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Fix ψ ∈ E′ G\\{0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' For any f ∈ P(G), we have f ∗ ψ ∈ Cbu(G) and sp(f ∗ ψ) ⊆ G \\ {0} by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Hence, for each ϕ ∈ T(G), we have ϕ(ψ) = ϕ(f ∗ ψ) = 0 by Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='4 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Now, we obtain that T(G)⊥ ⊇ clL∞(G)E′ G\\{0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' We show the reverse inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Suppose ϕ ∈ L∞(G) vanishes on E′ G\\{0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' We show that ϕ is in T(G), that is, ϕ(ψ −f ∗ ψ) = 0 for every f ∈ P(G) and ψ ∈ L∞(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' First, note that for a function g in L1(G) such that ˆg = 1 in some neighborhood U of 0, we have ϕ(ψ − g ∗ ψ) = 0 for every ψ in L∞(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' In fact, to show this, it is sufficient to show that ψ − g ∗ ψ is in E′ G\\{0}, in other words, sp(ψ − g ∗ ψ) ⊆ G \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Take a function h in L1(G) such that ˆh = 0 outside U and ˆh(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Observe that h ∗ (ψ − g ∗ ψ) = (h − g ∗ h) ∗ ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' 14 Here, we have (h−g ∗h)�(λ) = ˆh(λ)(1− ˆg(λ)) = 0 for every λ ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Hence, h−g ∗h = 0 and thus h ∈ J(ψ − g ∗ ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Since ˆh(0) = 1, we conclude that 0 ̸∈ sp(ψ − g ∗ ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' We obtain the desired assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' For any f ∈ P(G) and ε > 0, there exists a function h in L1(G) such that ˆh = 1 on some neighborhood of 0 and ∥f − h∥1 < ε(See [15], Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Hence, we have |ϕ(ψ − f ∗ ψ)| = |ϕ(ψ − h ∗ ψ) + ϕ(h ∗ ψ − f ∗ ψ)| ≤ |ϕ(ψ − h ∗ ψ)| + |ϕ(h ∗ ψ − f ∗ ψ)| = |ϕ(h ∗ ψ − f ∗ ψ)| = |ϕ((h − f) ∗ ψ))| ≤ ∥ϕ∥∥h − f∥1∥ψ∥∞ ≤ ∥ϕ∥∥ψ∥∞ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Since ε > 0 can be arbitrary, we obtain ϕ(ψ − f ∗ ψ) = 0, which means that ψ is in T(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' A functional analytic condition for almost convergence In this section, applying Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='2, we obtain the second necessary and sufficient condition on almost convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let us define the subspaces of L∞(G) as follows: E(G) := {ψ ∈ L∞(G) : ψ is almost convergent} E0(G) := {ψ ∈ L∞(G) : ψ is almost convergent to 0} It is clear that the following decomposition holds true: E(G) = R ⊕ E0(G), ψ �→ α + (ψ − α), where α is the number to which ψ almost converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='2 can be reformulated in terms of almost convergence as follows: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let G be a locally compact abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' ψ ∈ L∞(G) is almost con- vergent to 0 if and only if ψ ∈ clL∞(G)E′ G\\{0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' In other words, ψ is almost convergent to 0 if and only if for any ε > 0, there exists ψ1 ∈ L∞(G) whose spectrum does not intersect some neighborhood of 0 such that ∥ψ − ψ1∥∞ < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' In what follows, we provide some applications of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' For the proof of the following result, see [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let µ ∈ M(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let us define ˆµ∗(x) ∈ Cbu(G) by ˆµ∗(x) = � G χλ(x)dµ(λ), x ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Then, sp(ˆµ∗) = supp µ holds true, where supp µ is the support of the measure µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' 15 The following result was due to Eberlein [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' He showed that every weakly almost periodic function on R almost converges and the functions ˆµ∗ defined above is weakly almost periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Here, we will give a more direct proof using Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' For each µ ∈ M(G), ˆµ∗ is almost convergent to µ({0}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Proof .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let µ0 = µ − µ({0}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Then, for any ε > 0, there exits a neighborhood U of 0 such that |µ0|(U) < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let us define µUc(A) = µ(A ∩ Uc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Then, sp(ˆµ∗ Uc) ∩ U = ∅ by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='1 and we have |ˆµ∗ 0(x) − ˆµ∗ Uc(x)| ≤ � U |χλ(x)|d|µ|(x) = |µ0|(U) < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Since supp ˆµUc(x) ⊆ Uc, by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='1, we have ˆµ∗ 0 ac −→ 0, which in turn implies that ˆµ∗ ac −→ µ({0}), completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let {λn}∞ n=1 be a sequence of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let ψ ∈ L∞(G) be defined by ψ(x) = ∞ � n=1 cnχλn(x) (uniformly bounded and pointwise convergence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Then, we have sp(ψ) ⊆ clΓ{λn}∞ n=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Proof .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let Φ be the weak* closed invariant subspace of L∞(G) generated by {χλn}∞ n=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Then, we have obviously sp(Φ) = clΓ{λn}∞ n=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Since ψ ∈ Φ by the above expression of ψ, Φ(ψ) ⊆ Φ holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Thus, we obtain sp(ψ) = sp(Φ(ψ)) ⊆ sp(Φ) = clΓ{λn}∞ n=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let {λn}∞ n=0 (λ0 = 0) be a sequence of Γ which does not accumulate to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' If ψ ∈ L∞(G) is expressed by ψ(x) = ∞ � n=0 cnχλn(x) (uniformly bounded and pointwise convergence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Then, ψ is almost convergent to c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Proof .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' By Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='4, sp(ψ0) ⊆ clΓ{λn}∞ n=1, where ψ0(x) = ψ(x) − c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' By the assumption that {χλn}∞ n=1 does not accumulate to 0, there exists some neighborhood U of 0 such that U ∩ {λn}∞ n=1 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Hence, we have sp(ψ0) ⊆ Uc and by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='1, we have ψ0 ac −→ 0, which shows the desired assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' We give an example in the case G = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' One of the important examples of functions on R which is expressed by the sum of exponential functions is Dirichlet series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' 16 Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let {an}∞ n=1 be a sequence of complex numbers and ψ(s) is the Dirichlet series whose coefficients are {an}∞ n=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' ψ(s) = a1 1s + a2 2s + · · · an ns + · · · = a1 1σ e−it log 1 + a2 2σ e−it log 2 + · · · an nσ e−it log n + · · · (s = σ + it).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' If ψ(s) converges uniformly boundedly and pointwisely on the line Re(s) = σ, then, ψσ(t) = ψ(σ + it) is almost convergent to a1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' One-sided almost convergence In the following two sections, we consider the special groups of G = Z and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' For these groups, it is convenient to define almost convergence for functions defined on positive numbers Z+ and R+ := {x ∈ R : x ≥ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' In what follows, the symbols G stands for Z or R and G+ stands for Z+ or R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let us define three closed subspaces of L∞(G) as follows: L∞ 0 (G) := {φ ∈ L∞(G) : lim |x|→∞ φ(x) = 0}, L∞ 0,+(G) := {φ ∈ L∞(G) : lim x→∞ φ(x) = 0}, L∞ 0,−(G) := {φ ∈ L∞(G) : lim x→−∞ φ(x) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' For simplicity, we will also use the symbols L∞, L∞ 0 , L∞ 0,+ and L∞ 0,− as abbreviations of L∞(G), L∞ 0 (G), L∞ 0,+(G) and L∞ 0,−(G), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Now observe that the isomorphism L∞/L∞ 0 ∼= L∞/(L∞ 0,+ ∩ L∞ 0,−) ∼= (L∞/L∞ 0,+) ⊕ (L∞/L∞ 0,−) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Here, the image of an equivalence class [ψ] in L∞/L∞ 0 is given by [ψ+] + [ψ−], where ψ+ := ψ · IG+ and ψ− := ψ · I−G+, where IE denotes the characteristic function of a subset E of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Therefore, for any ϕ in L∞(G)∗ with ϕ = 0 on L∞ 0 , we have the decomposition of ϕ given as follows: (L∞/L∞ 0 )∗ ∼= (L∞/L∞ 0,+)∗ ⊕ (L∞/L∞ 0,−)∗, ϕ = ϕ+ + ϕ−, where ϕ+(ψ) := ϕ(ψ+) and ϕ−(ψ) := ϕ(ψ−) for each ψ ∈ L∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' We note that the spaces (L∞/L∞ 0,±)∗ can be identified with the subspace of L∞(G)∗ consisting of those elements vanishing on L∞ 0,±, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='2, for each ϕ ∈ T , we have ϕ = 0 on L∞ 0 and hence, ϕ is decomposed into the sum of ϕ+ ∈ (L∞/L∞ 0,+)∗ and ϕ− ∈ (L∞/L∞ 0,−)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' We use the symbols T+ := T ∩ (L∞/L∞ 0,+)∗, T− := T ∩ (L∞/L∞ 0,−)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Of course, T = co(T+ ∪ T−) holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Now we define one-sided almost convergence for elements in L∞(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' 17 Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' We say that ψ is one-sidedly almost convergent to the number α if ϕ(ψ) = α for every ϕ ∈ T+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' In this case, we write ψ oac → α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' We also define one-sided almost convergence for the functions in L∞(G+), the space of essentially bounded functions defined on the positive part G+ of G, by identifying ψ as the function on G defined by ˜ψ(x) = � ψ(x), x ≥ 0, 0, x < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let ψ be in L∞(G) which vanishes on the negative part −G+ of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Then, ψ ac −→ 0 if and only if ψ oac −−→ 0 holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Proof .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Sufficiency is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Suppose that ψ ∈ L∞(G) is one-sidedly almost conver- gent to 0, that is, ϕ(ψ) = 0 for every ϕ ∈ T+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Since ψ is in L∞ 0,−(G), we have ϕ(ψ) = 0 for every T−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Hence, it follows that ψ ac −→ 0 by the fact that T = co(T+ ∪ T−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' We can provide a similar condition to Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='3 for a given ψ ∈ L∞(G) to be one-sidedly almost convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' To this end, we need a following elementary lemma concerning extreme values of means in (L∞/L∞ 0,+)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let ϕ be a mean on L∞(G) which vanishes on L∞ 0,+(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Then, we have ϕ(ψ) ≤ sup x∈R+ ψ(x) for every ψ ∈ L∞(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' A mean ϕ on L∞(G)∗ is in T+ if and only if (i) G = Z ϕ(ψ) ≤ p+(ψ) := lim sup k→∞ sup n∈Z+ 1 k k−1 � i=0 ψ(n + i) holds for every ψ ∈ L∞(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' (ii) G = R ϕ(ψ) ≤ p+(ψ) := lim sup θ→∞ sup x∈R+ 1 θ � x+θ x ψ(t)dt holds for every ψ ∈ L∞(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Proof .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Using Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='2, necessity can be proved in a similar way to the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' For sufficiency, observe that we have p+(ψ) ≤ p(ψ) for every ψ ∈ L∞(G) and thus, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='2, it follows that a mean ϕ satisfying the condition in the theorem is in T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Furthermore, it is clear that p+(ψ) = p+(ψ)(:= −p+(−ψ)) = 0 holds for every ψ ∈ L∞ 0,+(G) and thus, such a ϕ is in (L∞/L∞ 0,+)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' As a result, we obtain that ϕ is in T+ = T ∩ (L∞/L∞ 0,+)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' 18 Now, we can show the following result just as Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='3 was derived from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let ψ be in L∞(G+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Then, ψ is one-sidedly almost convergent to α if and only if (i) G = Z lim k→∞ 1 k k−1 � i=0 ψ(i + n) = α uniformly in n ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' (ii) G = R lim θ→∞ 1 θ � x+θ x ψ(t)dt = α uniformly in x ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' We remark that one-sided almost convergence for l∞ = L∞(Z+) is equivalent to Lorentz’s almost convergence and (i) of the above theorem is the very result Lorentz proved (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Complex Tauberian Thoerem for almost convergence In this section, we study so-called complex Tauberian theorems for almost conver- gence on the groups Z and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' One of the main theorems of this section (Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='5) can be viewed as an anlogue of the celebrated Wiener-Ikehara theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' First, following [8], we introduce the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' We say that a function f(z) defined on the unit disc D := {z ∈ C : |z| < 1} has H1 boundary behavior at the point z0 = eit0 (t0 ∈ [0, 2π]) if there exist a number δ > 0 and a function F(eit) in L1(t0 − δ, t0 + δ) such that f(reit) converges to F(eit) in L1(t0 − δ, t0 + δ) as r → 1−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let ψ be in l∞ and denote an = ψ(n) for n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let ˆψ(z) be the function on the unit disc D defined by ˆψ(z) = ∞ � n=0 anzn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' If ˆψ has H1 boundary behaviour at z = 1, then ψ is almost convergent to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Proof .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' For each numbers r with 0 < r < 1, define the functions ψr(n) in L1(Z) and ˆψr(θ) in L1(T) by ψr(n) := � anrn (n ≥ 0), 0 (n < 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' ˆψr(θ) := ˆψ(reiθ) = ∞ � n=0 anrneinθ (θ ∈ T), 19 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let δ0 > 0 be a number such that lim r→1− � δ0 −δ0 | ˆψr(θ) − ˆψ(θ)|dθ 2π = 0 holds true for some ˆψ ∈ L1(−δ0, δ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Fix ε > 0 and choose a number 0 < δ < δ0 such that � δ −δ | ˆψr(θ)|dθ 2π < ε for every 0 < r < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let Cδ be a function in L1(Z) such that ˆCδ = 1 on (− δ 2, δ 2), ˆCδ = 0 outside [−δ, δ] and 0 ≤ ˆCδ ≤ 1 on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' For each 0 < r < 1, put ˆψr,δ(θ) = ˆψr(θ)(1 − ˆCδ(θ)), and ψr,δ(n) = ( ˆψr,δ)�(n), n ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Then, we have sp(ψr,δ) ∩ � −δ 2, δ 2 � = ∅ (0 < r < 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' In fact, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='1, sp(ψr,δ) = supp ˆψr,δ ⊆ T \\ (− δ 2, δ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' By definition of ψr,δ, we have ψr,δ(n) = � T ˆψr,δ(θ)e−inθ dθ 2π = � T ˆψr(θ)e−inθ dθ 2π − � T ˆψr(θ) ˆCδ(θ)e−inθ dθ 2π = ψr(n) − (ψr ∗ Cδ)(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Letting r → 1−, we obtain ψδ(n) := lim r→1−{ψr(n) − (ψr ∗ Cδ)(n)} = ψ(n) − (ψ ∗ Cδ)(n) (n ∈ Z) and sp(ψδ) ∩ � −δ 2, δ 2 � = ∅ by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' For each r ∈ (0, 1) and n ∈ Z, we have |ψr ∗ Cδ(n)| = ���� � T ˆψr ˆCδ(θ)e−inθ dθ 2π ���� ≤ � T | ˆψr(θ) ˆCδ(θ)|dθ 2π ≤ � δ −δ | ˆψr(θ)|dθ 2π < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Taking limit as r → 1−, we obtain |ψ ∗ Cδ(n)| ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Then, it follows that |ψδ(n) − ψ(n)| = |ψ ∗ Cδ(n)| < ε (n ∈ Z), 20 which means ∥ψδ − ψ∥∞ ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Therefore, by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='1, we conclude that ψ ac −→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' We complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let ψ be in l∞ and denote an = ψ(n) for n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let ˆψ(z) be the function on the unit disc D defined by ˆψ(z) = ∞ � n=0 anzn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' If ˆψ(z) − α 1−z has H1 boundary behaviour at z = 1, then ψ is one-sidedly almost convergent to α, that is, lim k→∞ 1 k k−1 � i=0 ψ(i + n) = α uniformly in n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Proof .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let ψ1(n) = ψ(n) − α · IZ+(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Then, we have ˆψ1(z) = ˆψ(z) − α 1−z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Hence, by Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='1, we obtain ψ1 ac −→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Since, ψ1(n) = 0 for n < 0, we have ψ1 oac −−→ 0 by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Then, by Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='2 we obtain 1 k k−1 � i=0 ψ(i + n) = α uniformly in n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' We complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let ψ be in l∞ and denote an = ψ(n) for n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let ˆψ(z) be the function on the unit disc D defined by ˆψ(z) = ∞ � n=0 anzn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' If ˆψ has at most a pole of order 1 at z = 1, then ψ oac −−→ −α, where α is the residue of ˆψ(z) at z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Proof .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' By the assumption, on a neighborhood U of 1, we can write as ˆf(z) = α z − 1 + b0 + b1(z − 1) + b2(z − 1)2 + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Hence, we obtain ˆf(z) − −α 1 − z = b0 + b1(z − 1) + b1(z − 1)2 + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Since the right-hand side of the above equation is continuous on U, we have ˆf(z)− −α 1−z ∈ H1 {1}(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' We obtain the result by Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Now we mention an interesting result of Katznelson and Tzafriri related to this theorem, which can be regarded as a dual of the above theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' (see [7], [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' 21 Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let {an}n≥0 be a bounded sequence of complex numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Suppose that the analytic function f(z) = ∞ � n=0 anzn has H1 boundary behavior on the unit circle except for z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Then, the equation lim n→∞(an+1 − an) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Analogous results concerning G = R can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Since proofs are similar to the case G = Z above, in what follows, we exhibit only results without their proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' First, we define H1 boundary behavior for the functions defined on the right-half plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' We say that a function f(z) defined on the right-half plane C+ := {z ∈ C : Re(z) > 0} has H1 boundary behavior at the point z0 = iy0 (y0 ∈ R) if there exist a number δ > 0 and a function F(y) in L1(y0 − δ, y0 + δ) such that f(x + iy) converges to F(y) in L1(y0 − δ, y0 + δ) as x → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let ψ ∈ L∞(R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let Lψ(s) be the Laplace transform of ψ, the analytic function on the right half plane C+ defined by Lψ(s) = � R+ ψ(t)e−stdt, s ∈ C+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' If Lψ(s) has H1 boundary behaviour at s = 0, then ψ is almost convergent to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let ψ ∈ L∞(R+) and let Lψ(s) be its Laplace transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' If Lψ(s) − α s has H1 boundary behaviour at s = 0, then ψ is one-sidedly almost convergent to α, that is, lim θ→∞ 1 θ � x+θ x ψ(t)dt = α uniformly in x ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let ψ ∈ L∞(R+) and let Lψ(s) be its Laplace transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' If Lψ(s) has at most a pole of order 1 at s = 0, then ψ oac −−→ α, where α is the residue of Lψ(s) at s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Finally, we remark on the relation among Tauberian theorems concerning the be- havior of analytic functions f(z) = �∞ n=0 anzn or that of Lψ(s) near its boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' First, we mention the following classical and famous result (See [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='6 (Hardy-Littlewood).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let {an}n≥0 be a sequence of real numbers with an ≥ C (n ≥ 0) for some constant C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Suppose that f(x) = �∞ n=0 anxn converges for |x| < 1 (x ∈ R) and lim x→1(1 − x)f(x) = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' 22 Then, it holds that lim k→∞ 1 k k−1 � i=0 ai = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let H1(D) be the Hardy space on the unit disc and H1 {1}(D) be the class of functions whose elements are the analytic functions on D having H1 boundary behavour at the point z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Then, we can summarize as follows: Here let {an}n≥0 be a bounded sequence of complex numbers and f(z) = �∞ n=0 anzn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 f(x) − α 1−x = o( 1 1−x) (x → 1−) =⇒ limk→∞ 1 k �k−1 i=0 ai = α, f(z) − α 1−z ∈ H1 {1}(D) =⇒ limk→∞ 1 k �k−1 i=0 an+i = α uniformly in n ∈ Z+, f(z) − a 1−z ∈ H1(D) =⇒ limk→∞ ak = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' The first one follows from Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' The second is due to Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' The third is by the Riemann-Lebesgue lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Now, It is easy to show that for a bounded sequence {an}, limn→∞ an = 0 if and only if an oac −−→ 0 and limn→∞ an+1 − an = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Hence, we see that the third assertion above is decomposed into the two assertions of Theorems 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='2 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' The corresponding results for the case of R is in order: First, the right half plane version of Hardy-Littlewood’s theorem reads as follows (see [8]): Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let ψ be a locally integrable function on the half line R+ and ψ(x) ≥ C (x ≥ 0) for some contstant C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' If Lψ(x) exists for all x > 0 and lim x→0+ xLψ(x) = lim x→0+ x � ∞ 0 e−xtψ(t)dt = α, then we have lim θ→∞ 1 θ � θ 0 ψ(t)dt = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Let H1 {0}(C+) and H1 R(C+) be the set of analytic functions on the right half plane C+ having H1 boundary behavior at s = 0 and the whole line R, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Then, the following result holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 Lφ(x) − α x = o( 1 x) (x → 0+) =⇒ limθ→∞ 1 θ � θ 0 φ(t)dt = α, Lψ(s) − α s ∈ H1 {0}(C+) =⇒ limθ→∞ 1 θ � x+θ x ψ(t)dt = α uniformly in x ∈ R+, Lψ(s) − α s ∈ H1 R(C+) =⇒ w∗- limx→∞ φx = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' We remark that the third one follows from the Wiener-Ikehara theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Briefly, we can say that Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='4 is an intermediate result between Hardy-Littlewood’s theorem and the Wiener-Ikehara theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' 23 References [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Chow, On topologically invariant means on a locally compact group, Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Amer.' metadata={'source': 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limits, Duke Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' 26 (1959), 17-28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' [15] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Rudin, Fourier analysis on groups, Interscience, New York (1962).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' [16] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Sucheston, Banach limits, Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Monthly 74 (1967), 308-311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' [17] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Yosida, Functional analysis, Springer, Berlin (1965).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content=' Faculty of Liberal Arts, Tsuru University, Tsuru-shi, Yamanashi-ken 402-8555, Japan Email address: tk-waseda@ruri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='waseda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} +page_content='jp 24' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQftP1k/content/2301.01672v1.pdf'} diff --git a/zdAzT4oBgHgl3EQf8P6x/content/tmp_files/load_file.txt b/zdAzT4oBgHgl3EQf8P6x/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0e45948bcc147980cf30e46a55717149eb0fa3bf --- /dev/null +++ b/zdAzT4oBgHgl3EQf8P6x/content/tmp_files/load_file.txt @@ -0,0 +1,440 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf,len=439 +page_content='What’s in a Text-to-Image Prompt?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' The Potential of Stable Diffusion in Visual Arts Education Nassim Dehouche 1,*, Kullathida Dehouche 2 1 Business Administration Division, Mahidol University International College, Salaya, Thailand;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' nassim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='deh@mahidol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='edu 2 Poh-Chang Academy of Arts, Rajamangala University of Technology Ra�anakosin, Bangkok, Thailand;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' kullathida.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='mee@rmutr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='th Correspondence: nassim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='deh@mahidol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='edu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Mahidol University International College, 999 Phu�amonthon 4 Road, Salaya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' 73170, Thailand Abstract: Text-to-Image artificial intelligence (AI) recently saw a major breakthrough with the release of Dall-E and its open-source counterpart, Stable Diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' These programs allow anyone to create original visual art pieces by simply providing descriptions in natural language (prompts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Using a sample of 72,980 Stable Diffusion prompts, we propose a formalization of this new medium of art creation and assess its potential for teaching the history of art, aesthetics, and technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Our findings indicate that text-to-Image AI has the potential to revolutionize the way art is taught, offering new, cost-effective possibilities for experimentation and expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' However, it also raises important questions about the ownership of artistic works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' As more and more art is created using these programs, it will be crucial to establish new legal and economic models to protect the rights of artists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Keywords: artificial intelligence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' art;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' education;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' computational creativity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' intellectual property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=" “It is, in the first place, 'by a word conceived in intellect' that the artist, whether human or divine, works.” Ananda K." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Coomaraswamy [1] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Introduction The traditional view of art, espoused by Coomaraswamy [1], is that of (human) art as imitation (of divine creation), with the word as a starting point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' This view, notably challenged by contemporary expressionist and formalist perspectives [2], was given a new technical expression with recent advances in artificial intelligence (AI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Indeed, AI has made impressive strides in the realm of creativity, with computers now able to generate relevant and original text [3] and images [4, 5], in response to simple natural language prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Some of these outputs have even been indistinguishable from human creations, leading to their recognition in traditional art contests [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' AI-generated art remains a controversial topic, with notable debates over whether it can truly be considered art in the first place [7], but despite the increasing academic interest in generative AI models, li�le a�ention has been given to their potential use in visual arts education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' In our view, these models contain a compressed version of centuries of human artistic creations, which presents an undeniable interest for art education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Thus, in this paper, we explore the possibilities of incorporating them in visual art education, particularly for the teaching of art history, aesthetics, and technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Following this introductory section, the remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Section 2 situates recent developments in the field of Text-to-Image in the broader 2 of 11 history of AI-generated art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Section 3 focuses specifically on Stable Diffusion, an advanced, open-source Text-to-Image system, and illustrates its basic capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Section 4 describes the methods and data of our analysis of 72,980 Stable Diffusion interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Based on this analysis, Section 5 proposes a formalization and procedural framework for Stable Diffusion prompts that can serve as a basis for their formal usage in educational software or curricula, and discusses some of its potential uses for the teaching of subjects such as the history of art, aesthetics, and technique, as well as its implications for the protection of the intellectual property of artists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Lastly, Section 6 concludes this paper by outlining the work that remains to be done, in our view, to facilitate the integration of Text-to-Image AI in art education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' A Brief History of AI-generated Art The first a�empts at using Artificial Intelligence to create coherent, original content from human prompts can be traced back to the 1950s, when researchers at the MIT Artificial Intelligence Laboratory created a program called ELIZA [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' ELIZA was able to generate simple responses to text input, using pa�ern matching and natural language processing techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' While not strictly art, ELIZA was an early example of Text-to-Text: software that could generate original text output that was intended to be interpreted by humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' One of the first examples of AI-generated art proper was a program called AARON, developed by artist Harold Cohen in the 1970s [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' AARON was a computer program that was capable of generating complex drawings and paintings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' AARON used a set of rules and constraints to create its art, and was able to learn from its own outputs to improve over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' As AI technology advanced in the 1980s and 1990s, more complex and sophisticated AI-generated art began to emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' For instance, Karl Sims generated unique 3D images and animations based on evolutionary algorithms [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' In recent years, the advent of deep learning has led to even more realistic outputs, and consequently, AI-generated art gained increasing a�ention from both the art world and the general public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' In 2015, a team at Google used deep learning techniques to train a neural network on a dataset of over 10,000 paintings, with the goal of generating original works of art from input images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' The resulting program, known as DeepDream [11], was able to create surreal, visually striking images from input images (Image-to-Image).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Another notable example is the work of a Paris-based art collective named "Obvious," which resulted in a software-generated portrait that sold for over $432,000 at a Christie\'s auction, in 2018 [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Year 2020 saw a major qualitative leap in Text-to-Text capabilities, with the release of the third generation Generative Pretrained Transformer (GPT-3), by private research firm OpenAI [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' GPT-3 constitutes an important advance in terms of the generality of Text-to-Text models, and is able to generate text that is highly coherent, in response to virtually any prompt in natural language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' This was made possible by the sheer size of the model, which consisted of 175 billion parameters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' an order of magnitude more than the second largest similar model to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' This vast number of parameters allowed GPT-3 to comprehend language tasks it was not particularly trained for, and ushered in the era of Large Language Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' These models have the ability to generate high-quality, human-like text, which can be used in a variety of applications, including machine translation, text summarization, and creative writing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' The success of GPT-3 led to the development of CLIP [13], another breakthrough model by OpenAI, which was designed to link text to images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' CLIP (Contrastive Language–Image Pretraining) is a general-purpose image-text model trained on 400 million text-image pairs from the internet, allowing it to perform image classification with any user-provided label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' It can also generate text that accurately describes any input image (Image-to-Text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Based on these advances, OpenAI released DALL-E [4], which is able to generate convincing 3 of 11 images from text descriptions (Text-to-Image).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' While DALL-E remains a proprietary, closed-source software, the code of CLIP was released open-source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' This allowed artificial intelligence firm Stability AI to develop and train Stable Diffusion [5], an open-source Text-to-Image model, with comparable performance to DALL-E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Stable Diffusion was released under a permissive license allowing commercial and non-commercial usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Although they represent an important technical breakthrough, CLIP, and the Text-to-Image systems based on it, also raise important ethical and societal concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Because of its training on mass, indiscriminate internet data, CLIP has a propensity to reproduce biased and unfair stereotypes present in culture and society [14], and its possible unfair usage of protected works has alerted legal experts [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' These systems also have the potential to be used for nefarious purposes, such as creating fake news or spreading misinformation [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Stable Diffusion Stable Diffusion is a text-to-image model, released in 2022, that uses a deep learning technique called latent diffusion [5] to generate images based on text descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=" Unlike some previous Text-to-Image models, Stable Diffusion's code and model weights are publicly available and can be run on most consumer hardware." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' To generate images, Stable Diffusion uses CLIP [12] to project a text prompt into a joint text-image embedding space, and select a rough, noisy image that is semantically close to the input prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' This image is then subject to a denoising method based on the latent diffusion model to produce the final image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' In addition to a text prompt, the Text-to-Image generation script within Stable Diffusion allows users to input various parameters such as sampling type, output image dimensions, and seed value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' This la�er integer parameter is typically set randomly, but a constant seed value allows for reproducibility, and the conservation of some aspects of the generated images across prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' For instance, in Figure 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' and Figure 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=', we use Stable Diffusion version 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='1 to generate two images, with the respective prompts “detailed photograph of an older woman/man wearing a leather jacket, waist shot, forest background, in the style of Brandon Stanton, Humans of New York”, and set the seed in both images at a value 1034.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' We can see that this seed value conserves some of the facial features of the subject, across prompts and genders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Additionally, a constant seed value can be useful to maintain a subject’s appearance in different poses and se�ings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' For instance, Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' shows images resulting from the prompt “digital illustration of an older woman/man wearing a leather jacket, Victorian aesthetics, waist shot, forest background, in the style of Magali Villeneuve” and a constant seed value of 1242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' It should be noted that, even for a constant seed value, text prompts typically generate some random artifacts and imperfection, such as the variations in the neckwear of the character between Figure 2(a) and 2(b), as well as Figure 2(c) and 2(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' These imperfections can require additional post-processing of the generated images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' 4 of 11 (a) (b) (c) (d) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Images generated in Stable Diffusion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=', with the prompts “detailed photograph of an older woman/man wearing a leather jacket, waist shot, forest background, in the style of Brandon Stanton, Humans of New York”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Additional inpainting was applied to generate Figures (c) and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Some front-end implementations of Stable Diffusion, such as DreamStudio1, offer additional functions for post-processing tasks, such as inpainting and outpainting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Inpainting involves altering a specific part of an image by filling in a masked area with new content based on a user-provided prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Outpainting, on the other hand, involves generating new content to extend an image beyond its original dimensions based on a user-provided prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Both of these functions use the Stable Diffusion model to generate the new content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' For instance, with Figure 1(b) as a starting point, we can add accessories to the subject, as illustrated in Figure 1(c), or change the background of the image, as in Figure 1(d), with the respective inpainting prompts “man wearing a yellow hat” and “man in a colorful street corner”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' 1 h�ps://beta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='dreamstudio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='ai/ FU5 of 11 (a) (b) (c) (d) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Images generated in Stable Diffusion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=', with the prompts “digital illustration of an older woman/man wearing a leather jacket, Victorian aesthetics, waist shot, forest background, in the style of Magali Villeneuve”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Moreover, Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' and Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' illustrate Stable Diffusion’s ability to reproduce the style of contemporary, practicing artists (photographer Brandon Stanton and illustrator Magali Villeneuve, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' This controversial aspect of generative AI [17] is analyzed more thoroughly in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Data and Methods Stable Diffusion’s output images are highly sensitive to the wording of text prompts, so we set out to examine the format and semantic content of this form of input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' To this end, we gathered a dataset of 72,980 Stable Diffusion prompts from Lexica2, a search engine that features curated Stable Diffusion outputs submi�ed by users along with the prompts that generated them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' We conducted our analysis in three steps: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Tokenization: Each prompt is broken down into “tokens”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' atomic linguistic terms, which can be words, phrases, symbols, or other meaningful elements of the prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' This step is performed using the BERT Tokenizer [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Topic extraction: The goal of this step is to automatically identify the main topics or themes present in the 72,980 prompts, with the prior knowledge that they represent detailed description of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' This is performed using the GPT-3 [3] API3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' 3 h�ps://openai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='com/api/ 2 h�ps://lexica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='art/ 6 of 11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Classification: Tokens, from each prompt, are classified into one or several of the linguistic topics identified in step 1, using the GPT-3 API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Additionally, the ability of Stable Diffusion to accurately reproduce the style of specific artists, whose work was used for its training, has been a controversial issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=" To specifically examine the usage of this feature in prompts, we identified tokens that represent the name of an artist, brand, or collective using BERT's named-entity recognition function [18] and calculated the frequency of each of these entities in the 72,980 prompts under consideration." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Results and Discussion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Formalizing Stable Diffusion Prompts Topic extraction allows us to identify the primary elements (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' semantic categories of tokens) described in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' These are the most frequent categories of keywords in the 72,980 considered prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Primary elements in 72,980 Stable Diffusion prompts Topic Description Subject The characters and objects in the image, such as “a cyborg”, “two dogs”, “a car”, “a wizard”, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Medium The type of visual object that is the image, such as “digital illustration”, “photograph”, “3D render”, “concept art”, “poster”, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Technique The tools and software used to create the image, such as “Blender”, “pincushion lens”, “Unreal engine”, “Octane”, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Genre Aesthetic features that describe the artistic genre of the image, such as “anime”, “surreal”, “baroque”, “photorealistic”, sci-fi, black and white, epic fantasy, film noir, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Mood Features that describe the atmosphere and emotions of the image, such as “beautiful”, “eerie”, “bleak”, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Tone Features that describe the chromatic composition of the image, such as “pastel”, “synthwave colors”, “ethereal colors”, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Lighting The use of light and shadows in the image “dark”, "cinematic lighting", "realistic shaded lighting", "studio lighting", radiant light, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Resolution Features that describe the level of detail of the image, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' "highly-detailed", "photorealistic", "100 mm\'\', “8K”, “16K”, “HQ”, “sharp focus”, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Artistic References Artists or works of art to use as inspiration, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' “Greg Rutkowski”, “Studio Ghibli”, “Artgerm”, “Zaha Hadid”, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Reception/Popularity Awards, recognition, or trending status on art-focused platforms,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' "trending on artstation", “masterpiece”, "award-winning”, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Less frequent topics, that are extensions or additional details of the previous main topics are listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' 7 of 11 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Secondary elements in 72,980 Stable Diffusion prompts Topic Examples Physical a�ributes of the subject race, age, clothing, accessories, “cute”, “glamorous”, “chonky”, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Emotional or psychological traits of the subject “happy”, “anxious”, “triumphant”, “pensive”, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Environment/Se�ing time, weather, “medieval”, “post-apocalyptic”, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Symmetry/Repetition “symmetry”, “symmetrical”, “pa�ern”, “motif”, “fractal”, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Depth of field “blurred background”, “deep focus”, “aperture”, “F/4”, “F/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='8, "sharp focus", "bokeh", etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Angle “ultra wide angle”, “zenith view”, “cinematic view”, “close up”, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Message/Meaning “propaganda”, “religious’, “advertisement”, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Proposed Procedural Classification The identified prompt elements align remarkably well with traditional photography concepts, and can be procedurally classified as in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Proposed creative process for Text-to-Image prompts based on the semantic elements in 72,980 Stable Diffusion prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Mise-en-scène: Mise-en-scène is a term commonly used in the study of photography, film, and theater to refer to the arrangement of objects, se�ings, and actors within a shot or scene [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' This category thus includes the visual and compositional elements that will appear in the frame to create the intended cultural object, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' “a defiant older woman/man wearing a leather jacket, in a post-apocalyptic city, bleak lighting”, illustrated in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Mise-en-scene Dispositif Cultural object (M) (2moH) (chuM) Subject Technique Medium Physical attributes Resolution Genre Emotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Or psych.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' traits Angle Artistic References Environment/Setting Depth of field Message/meaning Symmetry/Repetition Mood Reception/popularity Lighting Tone8 of 11 (a) (b) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Images generated in Stable Diffusion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=', with the prompts “digital painting of a defiant older woman/man wearing a leather jacket, in a post-apocalyptic city, bleak lighting, trending on artstation, Greg Rutkowski” Dispositif: In photography and film, the concept of dispositif pertains to the configuration of the material technology [19] used to capture an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Within our more general classification, this category can also possibly include software tools and post-processing techniques for digital images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' If mise-en-scène is what is displayed in the image, the dispositif would be how it is created, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' “close up, black and white, wide aperture, 8K, sharp edges”, illustrated in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' (a) (b) Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Images generated in Stable Diffusion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=', with the prompts “portrait photograph of a happy, pensive older woman/man wearing a leather jacket, forest background, close up, black and white, wide aperture, 8K, sharp edges, Robert Doisneau”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Cultural object: These elements describe the “object” of the artist’s creation, understood in its double meaning of “artifact” and “purpose”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' the la�er understanding includes descriptions of the medium and genre of the image, as well as its positioning in the history of art through artistic references (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' “a photograph by Annie Leibovi�” or “a renaissance painting by Michelangelo”);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' the former descriptions of the message/meaning and reception/popularity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' “religious, award-winning”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' These two combinations of prompts are illustrated in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' 9 of 11 (a) (b) Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' “portrait of an older woman wearing a leather jacket, religious, award-winning”, as (a) “a photograph by Annie Leibovi�” and (b) “a renaissance painting by Michelangelo”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' It is important to note that the elements in our proposed procedural classification are not independent or exclusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=" For example, using an artist's name as an artistic reference can influence the mood and tone of the resulting image." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' It can be interesting to explore unusual or conflicting combinations of these elements for creative purposes, but it is worth remembering that the initial image associated with a text by CLIP is a noisy pixel soup, and the prompts are meant to guide its denoising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Therefore, the more coherent the prompt, the be�er the outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Mastering Text-to-Image involves understanding the interplay of these elements, which includes a degree of randomness, in order to generate the most coherent art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' The Need for New Economic Models for Visual Arts The word cloud in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' shows the frequency of named entities used as artistic references in the 72,980 prompts under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' We found that these named entities predominantly refer to contemporary, practicing artists who frequently post their work on digital art platform ArtStation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' For example, Polish painter Greg Rutkowski, including slight misspellings of his name and mentions alongside other artists, appears in 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='06% of the prompts, while mentions of ArtStation as an element of Reception/Popularity appear in 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='35% of the prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Sachin adeevJohn Ruar Yoji Jamle weth Fenghua ZhongShinkai Studio John Harris rehz Cushar Jar RutkowskiAlphonse ushart ArtemZhongRuan Jespel ua Elvgre EEising Wadim Kashin Cushart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='Krenz riam Alejand BellBeeple S Singer Sargen uis Rovo ate Android AL Aramaki Tran Fenghua WLOP Rossdraws tudio Ghibl Mucha leremyLipkin Demura phonse enz Genshin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Impact Arexander Andrel Anto Jansson Aublet hlnoy Greg Jeremy Lipking MakotoShinkai :AoshimaTerry Gurney Agua Rutkowski ish hin S Gensh Loish Mullins StanleyLauStalenhag James Gil Elvgren ames Hanu ea ristanEaton Gcto Rutknowski Rodger Alex Ross James Gurney Jansson J1m rAoshima John.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='Berkey ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='esperEjsing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='Henson ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='Dave ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='JimHenson ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='Mohrbache ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='Marina Abramovic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='Greg ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='Magall ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='eneuve ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='Artgerm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='Greg Rutknowski ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='CharlieBowater ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='Klimt Nixel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='om ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='Bagshaw ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='Melyin John ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='Sachin Teng ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='Guweiz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='Josar ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='RoSS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='Craig Mullins Seb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='Albert Aublet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='Pak ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='MC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='Syd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='James ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='Jean ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='Simon ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='Stalenhag ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='Melvin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='Dave ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='Giger ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='RusS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='Mill ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='Mike Mignola ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='JohnHarney ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='Donato Giancola ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='StanleyArtgerm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='Impact ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='Jami ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='Royo ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='Bierstadt Bill ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='yincent Di ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='Antoni ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='Moebius ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='Michael ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='Huang ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='Guang1iar ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='Giileard ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='Chiho ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='ames ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='Mark Brooks10 of 11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content='Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Word cloud of artists, brands, or collective names used for inspiration in 72,980 Stable Diffusion prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' The popularity of these keywords can be a�ributed to the fact that platforms like ArtStation encourage artists to include detailed labels describing their work in order to make it more accessible to persons with disabilities, which makes these creations particularly useful for training Text-to-Image AI models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Thus, ArtStation artists are somehow penalized for their virtue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' The legal question of whether this training constitutes plagiarism is still open [15] and may take years, if not decades, to be resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' In addition to possible unfair usage of the intellectual property of these artists, the widespread use of Stable Diffusion also leads to the original creations of these artists being overshadowed in search engine results by AI-generated works that bear their names in the prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' While incomplete, as it does not account for works that are used implicitly in the creation of an image, a simple technical solution to these issues could be to devise compensation models for artists based on the frequency of their names appearing as a Style Reference in commercial Text-to-Image applications, similar to music streaming economic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Conclusions This paper aims to provide a structured approach to understanding this new medium of art creation and connect it to established art education concepts, despite the fact that the output of a Stable Diffusion prompt is random to some extent and highly sensitive to its wording.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Stable Diffusion’s “understanding” of art is no doubt superficial and essentially situated at the level of gimmicks (which, as noted by [21], remain "capitalism\'s most successful aesthetic category").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' However, with proper guidance and curation from educators, it can represent a valuable, didactic tool for the transmission of technical concepts, as well as more experiential concepts of artistic genres, movements, and aesthetics that characterize a cultural object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Additionally, variations on the elements of mise-en-scène and dispositif, for a constant seed integer, can constitute a fast and cheap method of experimentation and prototyping, before using costly studio time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQf8P6x/content/2301.01902v1.pdf'} +page_content=' Notwithstanding their potential, 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